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1 The GNU Emacs Calculator
**************************

"Calc" is an advanced desk calculator and mathematical tool written by
Dave Gillespie that runs as part of the GNU Emacs environment.

   This manual, also written (mostly) by Dave Gillespie, is divided into
three major parts: "Getting Started," the "Calc Tutorial," and the
"Calc Reference."  The Tutorial introduces all the major aspects of
Calculator use in an easy, hands-on way.  The remainder of the manual is
a complete reference to the features of the Calculator.

   For help in the Emacs Info system (which you are using to read this
file), type `?'.  (You can also type `h' to run through a longer Info
tutorial.)

   This file documents Calc, the GNU Emacs calculator.

   Copyright (C) 1990, 1991, 2001, 2002, 2003, 2004, 2005, 2006, 2007,
2008, 2009 Free Software Foundation, Inc.

     Permission is granted to copy, distribute and/or modify this
     document under the terms of the GNU Free Documentation License,
     Version 1.3 or any later version published by the Free Software
     Foundation; with the Invariant Sections being just "GNU GENERAL
     PUBLIC LICENSE", with the Front-Cover texts being "A GNU Manual,"
     and with the Back-Cover Texts as in (a) below.  A copy of the
     license is included in the section entitled "GNU Free
     Documentation License."

     (a) The FSF's Back-Cover Text is: "You have the freedom to copy and
     modify this GNU manual.  Buying copies from the FSF supports it in
     developing GNU and promoting software freedom."

* Menu:

* Getting Started::       General description and overview.

* Interactive Tutorial::
* Tutorial::              A step-by-step introduction for beginners.

* Introduction::          Introduction to the Calc reference manual.
* Data Types::            Types of objects manipulated by Calc.
* Stack and Trail::       Manipulating the stack and trail buffers.
* Mode Settings::         Adjusting display format and other modes.
* Arithmetic::            Basic arithmetic functions.
* Scientific Functions::  Transcendentals and other scientific functions.
* Matrix Functions::      Operations on vectors and matrices.
* Algebra::               Manipulating expressions algebraically.
* Units::                 Operations on numbers with units.
* Store and Recall::      Storing and recalling variables.
* Graphics::              Commands for making graphs of data.
* Kill and Yank::         Moving data into and out of Calc.
* Keypad Mode::           Operating Calc from a keypad.
* Embedded Mode::         Working with formulas embedded in a file.
* Programming::           Calc as a programmable calculator.

* Copying::               How you can copy and share Calc.
* GNU Free Documentation License:: The license for this documentation.
* Customizing Calc::      Customizing Calc.
* Reporting Bugs::        How to report bugs and make suggestions.

* Summary::               Summary of Calc commands and functions.

* Key Index::             The standard Calc key sequences.
* Command Index::         The interactive Calc commands.
* Function Index::        Functions (in algebraic formulas).
* Concept Index::         General concepts.
* Variable Index::        Variables used by Calc (both user and internal).
* Lisp Function Index::   Internal Lisp math functions.

File: calc,  Node: Getting Started,  Next: Interactive Tutorial,  Prev: Top,  Up: Top

2 Getting Started
*****************

This chapter provides a general overview of Calc, the GNU Emacs
Calculator:  What it is, how to start it and how to exit from it, and
what are the various ways that it can be used.

* Menu:

* What is Calc::
* About This Manual::
* Notations Used in This Manual::
* Demonstration of Calc::
* Using Calc::
* History and Acknowledgements::

File: calc,  Node: What is Calc,  Next: About This Manual,  Prev: Getting Started,  Up: Getting Started

2.1 What is Calc?
=================

"Calc" is an advanced calculator and mathematical tool that runs as
part of the GNU Emacs environment.  Very roughly based on the HP-28/48
series of calculators, its many features include:

   * Choice of algebraic or RPN (stack-based) entry of calculations.

   * Arbitrary precision integers and floating-point numbers.

   * Arithmetic on rational numbers, complex numbers (rectangular and
     polar), error forms with standard deviations, open and closed
     intervals, vectors and matrices, dates and times, infinities,
     sets, quantities with units, and algebraic formulas.

   * Mathematical operations such as logarithms and trigonometric
     functions.

   * Programmer's features (bitwise operations, non-decimal numbers).

   * Financial functions such as future value and internal rate of
     return.

   * Number theoretical features such as prime factorization and
     arithmetic modulo M for any M.

   * Algebraic manipulation features, including symbolic calculus.

   * Moving data to and from regular editing buffers.

   * Embedded mode for manipulating Calc formulas and data directly
     inside any editing buffer.

   * Graphics using GNUPLOT, a versatile (and free) plotting program.

   * Easy programming using keyboard macros, algebraic formulas,
     algebraic rewrite rules, or extended Emacs Lisp.

   Calc tries to include a little something for everyone; as a result
it is large and might be intimidating to the first-time user.  If you
plan to use Calc only as a traditional desk calculator, all you really
need to read is the "Getting Started" chapter of this manual and
possibly the first few sections of the tutorial.  As you become more
comfortable with the program you can learn its additional features.
Calc does not have the scope and depth of a fully-functional symbolic
math package, but Calc has the advantages of convenience, portability,
and freedom.

File: calc,  Node: About This Manual,  Next: Notations Used in This Manual,  Prev: What is Calc,  Up: Getting Started

2.2 About This Manual
=====================

This document serves as a complete description of the GNU Emacs
Calculator.  It works both as an introduction for novices and as a
reference for experienced users.  While it helps to have some
experience with GNU Emacs in order to get the most out of Calc, this
manual ought to be readable even if you don't know or use Emacs
regularly.

   This manual is divided into three major parts: the "Getting Started"
chapter you are reading now, the Calc tutorial, and the Calc reference
manual.

   If you are in a hurry to use Calc, there is a brief "demonstration"
below which illustrates the major features of Calc in just a couple of
pages.  If you don't have time to go through the full tutorial, this
will show you everything you need to know to begin.  *Note
Demonstration of Calc::.

   The tutorial chapter walks you through the various parts of Calc
with lots of hands-on examples and explanations.  If you are new to
Calc and you have some time, try going through at least the beginning
of the tutorial.  The tutorial includes about 70 exercises with
answers.  These exercises give you some guided practice with Calc, as
well as pointing out some interesting and unusual ways to use its
features.

   The reference section discusses Calc in complete depth.  You can read
the reference from start to finish if you want to learn every aspect of
Calc.  Or, you can look in the table of contents or the Concept Index
to find the parts of the manual that discuss the things you need to
know.

   Every Calc keyboard command is listed in the Calc Summary, and also
in the Key Index.  Algebraic functions, `M-x' commands, and variables
also have their own indices.

   You can access this manual on-line at any time within Calc by
pressing the `h i' key sequence.  Outside of the Calc window, you can
press `C-x * i' to read the manual on-line.  From within Calc the
command `h t' will jump directly to the Tutorial; from outside of Calc
the command `C-x * t' will jump to the Tutorial and start Calc if
necessary.  Pressing `h s' or `C-x * s' will take you directly to the
Calc Summary.  Within Calc, you can also go to the part of the manual
describing any Calc key, function, or variable using `h k', `h f', or
`h v', respectively.  *Note Help Commands::.

   The Calc manual can be printed, but because the manual is so large,
you should only make a printed copy if you really need it.  To print the
manual, you will need the TeX typesetting program (this is a free
program by Donald Knuth at Stanford University) as well as the
`texindex' program and `texinfo.tex' file, both of which can be
obtained from the FSF as part of the `texinfo' package.  To print the
Calc manual in one huge tome, you will need the source code to this
manual, `calc.texi', available as part of the Emacs source.  Once you
have this file, type `texi2dvi calc.texi'.  Alternatively, change to
the `man' subdirectory of the Emacs source distribution, and type `make
calc.dvi'. (Don't worry if you get some "overfull box" warnings while
TeX runs.)  The result will be a device-independent output file called
`calc.dvi', which you must print in whatever way is right for your
system.  On many systems, the command is

     lpr -d calc.dvi

or

     dvips calc.dvi

File: calc,  Node: Notations Used in This Manual,  Next: Demonstration of Calc,  Prev: About This Manual,  Up: Getting Started

2.3 Notations Used in This Manual
=================================

This section describes the various notations that are used throughout
the Calc manual.

   In keystroke sequences, uppercase letters mean you must hold down
the shift key while typing the letter.  Keys pressed with Control held
down are shown as `C-x'.  Keys pressed with Meta held down are shown as
`M-x'.  Other notations are <RET> for the Return key, <SPC> for the
space bar, <TAB> for the Tab key, <DEL> for the Delete key, and <LFD>
for the Line-Feed key.  The <DEL> key is called Backspace on some
keyboards, it is whatever key you would use to correct a simple typing
error when regularly using Emacs.

   (If you don't have the <LFD> or <TAB> keys on your keyboard, the
`C-j' and `C-i' keys are equivalent to them, respectively.  If you
don't have a Meta key, look for Alt or Extend Char.  You can also press
<ESC> or `C-[' first to get the same effect, so that `M-x', `<ESC> x',
and `C-[ x' are all equivalent.)

   Sometimes the <RET> key is not shown when it is "obvious" that you
must press <RET> to proceed.  For example, the <RET> is usually omitted
in key sequences like `M-x calc-keypad <RET>'.

   Commands are generally shown like this:  `p' (`calc-precision') or
`C-x * k' (`calc-keypad').  This means that the command is normally
used by pressing the `p' key or `C-x * k' key sequence, but it also has
the full-name equivalent shown, e.g., `M-x calc-precision'.

   Commands that correspond to functions in algebraic notation are
written:  `C' (`calc-cos') [`cos'].  This means the `C' key is
equivalent to `M-x calc-cos', and that the corresponding function in an
algebraic-style formula would be `cos(X)'.

   A few commands don't have key equivalents:  `calc-sincos' [`sincos'].

File: calc,  Node: Demonstration of Calc,  Next: Using Calc,  Prev: Notations Used in This Manual,  Up: Getting Started

2.4 A Demonstration of Calc
===========================

This section will show some typical small problems being solved with
Calc.  The focus is more on demonstration than explanation, but
everything you see here will be covered more thoroughly in the Tutorial.

   To begin, start Emacs if necessary (usually the command `emacs' does
this), and type `C-x * c' to start the Calculator.  (You can also use
`M-x calc' if this doesn't work.  *Note Starting Calc::, for various
ways of starting the Calculator.)

   Be sure to type all the sample input exactly, especially noting the
difference between lower-case and upper-case letters.  Remember, <RET>,
<TAB>, <DEL>, and <SPC> are the Return, Tab, Delete, and Space keys.

   *RPN calculation.*  In RPN, you type the input number(s) first, then
the command to operate on the numbers.

Type `2 <RET> 3 + Q' to compute the square root of 2+3, which is
2.2360679775.

Type `P 2 ^' to compute the value of `pi' squared, 9.86960440109.

Type <TAB> to exchange the order of these two results.

Type `- I H S' to subtract these results and compute the Inverse
Hyperbolic sine of the difference, 2.72996136574.

Type <DEL> to erase this result.

   *Algebraic calculation.*  You can also enter calculations using
conventional "algebraic" notation.  To enter an algebraic formula, use
the apostrophe key.

Type `' sqrt(2+3) <RET>' to compute the square root of 2+3.

Type `' pi^2 <RET>' to enter `pi' squared.  To evaluate this symbolic
formula as a number, type `='.

Type `' arcsinh($ - $$) <RET>' to subtract the second-most-recent
result from the most-recent and compute the Inverse Hyperbolic sine.

   *Keypad mode.*  If you are using the X window system, press
`C-x * k' to get Keypad mode.  (If you don't use X, skip to the next
section.)

Click on the <2>, <ENTER>, <3>, <+>, and <SQRT> "buttons" using your
left mouse button.

Click on <PI>, <2>, and y^x.

Click on <INV>, then <ENTER> to swap the two results.

Click on <->, <INV>, <HYP>, and <SIN>.

Click on <<-> to erase the result, then click <OFF> to turn the Keypad
Calculator off.

   *Grabbing data.*  Type `C-x * x' if necessary to exit Calc.  Now
select the following numbers as an Emacs region:  "Mark" the front of
the list by typing `C-<SPC>' or `C-@' there, then move to the other end
of the list.  (Either get this list from the on-line copy of this
manual, accessed by `C-x * i', or just type these numbers into a
scratch file.)  Now type `C-x * g' to "grab" these numbers into Calc.

     1.23  1.97
     1.6   2
     1.19  1.08

The result `[1.23, 1.97, 1.6, 2, 1.19, 1.08]' is a Calc "vector."  Type
`V R +' to compute the sum of these numbers.

Type `U' to Undo this command, then type `V R *' to compute the product
of the numbers.

You can also grab data as a rectangular matrix.  Place the cursor on
the upper-leftmost `1' and set the mark, then move to just after the
lower-right `8' and press `C-x * r'.

Type `v t' to transpose this 3x2 matrix into a 2x3 matrix.  Type `v u'
to unpack the rows into two separate vectors.  Now type
`V R + <TAB> V R +' to compute the sums of the two original columns.
(There is also a special grab-and-sum-columns command, `C-x * :'.)

   *Units conversion.*  Units are entered algebraically.  Type
`' 43 mi/hr <RET>' to enter the quantity 43 miles-per-hour.  Type
`u c km/hr <RET>'.  Type `u c m/s <RET>'.

   *Date arithmetic.*  Type `t N' to get the current date and time.
Type `90 +' to find the date 90 days from now.  Type `' <25 dec 87>
<RET>' to enter a date, then `- 7 /' to see how many weeks have passed
since then.

   *Algebra.*  Algebraic entries can also include formulas or equations
involving variables.  Type `' [x + y = a, x y = 1] <RET>' to enter a
pair of equations involving three variables.  (Note the leading
apostrophe in this example; also, note that the space in `x y' is
required.)  Type `a S x,y <RET>' to solve these equations for the
variables `x' and `y'.

Type `d B' to view the solutions in more readable notation.  Type `d C'
to view them in C language notation, `d T' to view them in the notation
for the TeX typesetting system, and `d L' to view them in the notation
for the LaTeX typesetting system.  Type `d N' to return to normal
notation.

Type `7.5', then `s l a <RET>' to let `a = 7.5' in these formulas.
(That's the letter `l', not the numeral `1'.)

   *Help functions.*  You can read about any command in the on-line
manual.  Remember to type the letter `l', then `C-x * c', to return
here after each of these commands: `h k t N' to read about the `t N'
command, `h f sqrt <RET>' to read about the `sqrt' function, and `h s'
to read the Calc summary.

   Press <DEL> repeatedly to remove any leftover results from the stack.
To exit from Calc, press `q' or `C-x * c' again.

File: calc,  Node: Using Calc,  Next: History and Acknowledgements,  Prev: Demonstration of Calc,  Up: Getting Started

2.5 Using Calc
==============

Calc has several user interfaces that are specialized for different
kinds of tasks.  As well as Calc's standard interface, there are Quick
mode, Keypad mode, and Embedded mode.

* Menu:

* Starting Calc::
* The Standard Interface::
* Quick Mode Overview::
* Keypad Mode Overview::
* Standalone Operation::
* Embedded Mode Overview::
* Other C-x * Commands::

File: calc,  Node: Starting Calc,  Next: The Standard Interface,  Prev: Using Calc,  Up: Using Calc

2.5.1 Starting Calc
-------------------

On most systems, you can type `C-x *' to start the Calculator.  The key
sequence `C-x *' is bound to the command `calc-dispatch', which can be
rebound if convenient (*note Customizing Calc::).

   When you press `C-x *', Emacs waits for you to press a second key to
complete the command.  In this case, you will follow `C-x *' with a
letter (upper- or lower-case, it doesn't matter for `C-x *') that says
which Calc interface you want to use.

   To get Calc's standard interface, type `C-x * c'.  To get Keypad
mode, type `C-x * k'.  Type `C-x * ?' to get a brief list of the
available options, and type a second `?' to get a complete list.

   To ease typing, `C-x * *' also works to start Calc.  It starts the
same interface (either `C-x * c' or `C-x * k') that you last used,
selecting the `C-x * c' interface by default.

   If `C-x *' doesn't work for you, you can always type explicit
commands like `M-x calc' (for the standard user interface) or
`M-x calc-keypad' (for Keypad mode).  First type `M-x' (that's Meta
with the letter `x'), then, at the prompt, type the full command (like
`calc-keypad') and press Return.

   The same commands (like `C-x * c' or `C-x * *') that start the
Calculator also turn it off if it is already on.

File: calc,  Node: The Standard Interface,  Next: Quick Mode Overview,  Prev: Starting Calc,  Up: Using Calc

2.5.2 The Standard Calc Interface
---------------------------------

Calc's standard interface acts like a traditional RPN calculator,
operated by the normal Emacs keyboard.  When you type `C-x * c' to
start the Calculator, the Emacs screen splits into two windows with the
file you were editing on top and Calc on the bottom.


     ...
     --**-Emacs: myfile             (Fundamental)----All----------------------
     --- Emacs Calculator Mode ---                   |Emacs Calculator Trail
     2:  17.3                                        |    17.3
     1:  -5                                          |    3
         .                                           |    2
                                                     |    4
                                                     |  * 8
                                                     |  ->-5
                                                     |
     --%*-Calc: 12 Deg       (Calculator)----All----- --%*- *Calc Trail*

   In this figure, the mode-line for `myfile' has moved up and the
"Calculator" window has appeared below it.  As you can see, Calc
actually makes two windows side-by-side.  The lefthand one is called
the "stack window" and the righthand one is called the "trail window."
The stack holds the numbers involved in the calculation you are
currently performing.  The trail holds a complete record of all
calculations you have done.  In a desk calculator with a printer, the
trail corresponds to the paper tape that records what you do.

   In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
were first entered into the Calculator, then the 2 and 4 were
multiplied to get 8, then the 3 and 8 were subtracted to get -5.  (The
`>' symbol shows that this was the most recent calculation.)  The net
result is the two numbers 17.3 and -5 sitting on the stack.

   Most Calculator commands deal explicitly with the stack only, but
there is a set of commands that allow you to search back through the
trail and retrieve any previous result.

   Calc commands use the digits, letters, and punctuation keys.
Shifted (i.e., upper-case) letters are different from lowercase
letters.  Some letters are "prefix" keys that begin two-letter
commands.  For example, `e' means "enter exponent" and shifted `E'
means `e^x'.  With the `d' ("display modes") prefix the letter "e"
takes on very different meanings:  `d e' means "engineering notation"
and `d E' means ""eqn" language mode."

   There is nothing stopping you from switching out of the Calc window
and back into your editing window, say by using the Emacs `C-x o'
(`other-window') command.  When the cursor is inside a regular window,
Emacs acts just like normal.  When the cursor is in the Calc stack or
trail windows, keys are interpreted as Calc commands.

   When you quit by pressing `C-x * c' a second time, the Calculator
windows go away but the actual Stack and Trail are not gone, just
hidden.  When you press `C-x * c' once again you will get the same
stack and trail contents you had when you last used the Calculator.

   The Calculator does not remember its state between Emacs sessions.
Thus if you quit Emacs and start it again, `C-x * c' will give you a
fresh stack and trail.  There is a command (`m m') that lets you save
your favorite mode settings between sessions, though.  One of the
things it saves is which user interface (standard or Keypad) you last
used; otherwise, a freshly started Emacs will always treat `C-x * *'
the same as `C-x * c'.

   The `q' key is another equivalent way to turn the Calculator off.

   If you type `C-x * b' first and then `C-x * c', you get a
full-screen version of Calc (`full-calc') in which the stack and trail
windows are still side-by-side but are now as tall as the whole Emacs
screen.  When you press `q' or `C-x * c' again to quit, the file you
were editing before reappears.  The `C-x * b' key switches back and
forth between "big" full-screen mode and the normal partial-screen mode.

   Finally, `C-x * o' (`calc-other-window') is like `C-x * c' except
that the Calc window is not selected.  The buffer you were editing
before remains selected instead.  If you are in a Calc window, then
`C-x * o' will switch you out of it, being careful not to switch you to
the Calc Trail window.  So `C-x * o' is a handy way to switch out of
Calc momentarily to edit your file; you can then type `C-x * c' to
switch back into Calc when you are done.

File: calc,  Node: Quick Mode Overview,  Next: Keypad Mode Overview,  Prev: The Standard Interface,  Up: Using Calc

2.5.3 Quick Mode (Overview)
---------------------------

"Quick mode" is a quick way to use Calc when you don't need the full
complexity of the stack and trail.  To use it, type `C-x * q'
(`quick-calc') in any regular editing buffer.

   Quick mode is very simple:  It prompts you to type any formula in
standard algebraic notation (like `4 - 2/3') and then displays the
result at the bottom of the Emacs screen (3.33333333333 in this case).
You are then back in the same editing buffer you were in before, ready
to continue editing or to type `C-x * q' again to do another quick
calculation.  The result of the calculation will also be in the Emacs
"kill ring" so that a `C-y' command at this point will yank the result
into your editing buffer.

   Calc mode settings affect Quick mode, too, though you will have to
go into regular Calc (with `C-x * c') to change the mode settings.

   *Note Quick Calculator::, for further information.

File: calc,  Node: Keypad Mode Overview,  Next: Standalone Operation,  Prev: Quick Mode Overview,  Up: Using Calc

2.5.4 Keypad Mode (Overview)
----------------------------

"Keypad mode" is a mouse-based interface to the Calculator.  It is
designed for use with terminals that support a mouse.  If you don't
have a mouse, you will have to operate Keypad mode with your arrow keys
(which is probably more trouble than it's worth).

   Type `C-x * k' to turn Keypad mode on or off.  Once again you get
two new windows, this time on the righthand side of the screen instead
of at the bottom.  The upper window is the familiar Calc Stack; the
lower window is a picture of a typical calculator keypad.

     |--- Emacs Calculator Mode ---
     |2:  17.3
     |1:  -5
     |    .
     |--%*-Calc: 12 Deg       (Calcul
     |----+----+--Calc---+----+----1
     |FLR |CEIL|RND |TRNC|CLN2|FLT |
     |----+----+----+----+----+----|
     | LN |EXP |    |ABS |IDIV|MOD |
     |----+----+----+----+----+----|
     |SIN |COS |TAN |SQRT|y^x |1/x |
     |----+----+----+----+----+----|
     |  ENTER  |+/- |EEX |UNDO| <- |
     |-----+---+-+--+--+-+---++----|
     | INV |  7  |  8  |  9  |  /  |
     |-----+-----+-----+-----+-----|
     | HYP |  4  |  5  |  6  |  *  |
     |-----+-----+-----+-----+-----|
     |EXEC |  1  |  2  |  3  |  -  |
     |-----+-----+-----+-----+-----|
     | OFF |  0  |  .  | PI  |  +  |
     |-----+-----+-----+-----+-----+

   Keypad mode is much easier for beginners to learn, because there is
no need to memorize lots of obscure key sequences.  But not all
commands in regular Calc are available on the Keypad.  You can always
switch the cursor into the Calc stack window to use standard Calc
commands if you need.  Serious Calc users, though, often find they
prefer the standard interface over Keypad mode.

   To operate the Calculator, just click on the "buttons" of the keypad
using your left mouse button.  To enter the two numbers shown here you
would click `1 7 . 3 ENTER 5 +/- ENTER'; to add them together you would
then click `+' (to get 12.3 on the stack).

   If you click the right mouse button, the top three rows of the
keypad change to show other sets of commands, such as advanced math
functions, vector operations, and operations on binary numbers.

   Because Keypad mode doesn't use the regular keyboard, Calc leaves
the cursor in your original editing buffer.  You can type in this
buffer in the usual way while also clicking on the Calculator keypad.
One advantage of Keypad mode is that you don't need an explicit command
to switch between editing and calculating.

   If you press `C-x * b' first, you get a full-screen Keypad mode
(`full-calc-keypad') with three windows:  The keypad in the lower left,
the stack in the lower right, and the trail on top.

   *Note Keypad Mode::, for further information.

File: calc,  Node: Standalone Operation,  Next: Embedded Mode Overview,  Prev: Keypad Mode Overview,  Up: Using Calc

2.5.5 Standalone Operation
--------------------------

If you are not in Emacs at the moment but you wish to use Calc, you
must start Emacs first.  If all you want is to run Calc, you can give
the commands:

     emacs -f full-calc

or

     emacs -f full-calc-keypad

which run a full-screen Calculator (as if by `C-x * b C-x * c') or a
full-screen X-based Calculator (as if by `C-x * b C-x * k').  In
standalone operation, quitting the Calculator (by pressing `q' or
clicking on the keypad <EXIT> button) quits Emacs itself.

File: calc,  Node: Embedded Mode Overview,  Next: Other C-x * Commands,  Prev: Standalone Operation,  Up: Using Calc

2.5.6 Embedded Mode (Overview)
------------------------------

"Embedded mode" is a way to use Calc directly from inside an editing
buffer.  Suppose you have a formula written as part of a document like
this:

     The derivative of

                                        ln(ln(x))

     is

and you wish to have Calc compute and format the derivative for you and
store this derivative in the buffer automatically.  To do this with
Embedded mode, first copy the formula down to where you want the result
to be, leaving a blank line before and after the formula:

     The derivative of

                                        ln(ln(x))

     is

                                        ln(ln(x))

   Now, move the cursor onto this new formula and press `C-x * e'.
Calc will read the formula (using the surrounding blank lines to tell
how much text to read), then push this formula (invisibly) onto the Calc
stack.  The cursor will stay on the formula in the editing buffer, but
the line with the formula will now appear as it would on the Calc stack
(in this case, it will be left-aligned) and the buffer's mode line will
change to look like the Calc mode line (with mode indicators like `12
Deg' and so on).  Even though you are still in your editing buffer, the
keyboard now acts like the Calc keyboard, and any new result you get is
copied from the stack back into the buffer.  To take the derivative,
you would type `a d x <RET>'.

     The derivative of

                                        ln(ln(x))

     is

     1 / ln(x) x

   (Note that by default, Calc gives division lower precedence than
multiplication, so that `1 / ln(x) x' is equivalent to `1 / (ln(x) x)'.)

   To make this look nicer, you might want to press `d =' to center the
formula, and even `d B' to use Big display mode.

     The derivative of

                                        ln(ln(x))

     is
     % [calc-mode: justify: center]
     % [calc-mode: language: big]

                                            1
                                         -------
                                         ln(x) x

   Calc has added annotations to the file to help it remember the modes
that were used for this formula.  They are formatted like comments in
the TeX typesetting language, just in case you are using TeX or LaTeX.
(In this example TeX is not being used, so you might want to move these
comments up to the top of the file or otherwise put them out of the
way.)

   As an extra flourish, we can add an equation number using a
righthand label:  Type `d } (1) <RET>'.

     % [calc-mode: justify: center]
     % [calc-mode: language: big]
     % [calc-mode: right-label: " (1)"]

                                            1
                                         -------                      (1)
                                         ln(x) x

   To leave Embedded mode, type `C-x * e' again.  The mode line and
keyboard will revert to the way they were before.

   The related command `C-x * w' operates on a single word, which
generally means a single number, inside text.  It searches for an
expression which "looks" like a number containing the point.  Here's an
example of its use:

     A slope of one-third corresponds to an angle of 1 degrees.

   Place the cursor on the `1', then type `C-x * w' to enable Embedded
mode on that number.  Now type `3 /' (to get one-third), and `I T' (the
Inverse Tangent converts a slope into an angle), then `C-x * w' again
to exit Embedded mode.

     A slope of one-third corresponds to an angle of 18.4349488229 degrees.

   *Note Embedded Mode::, for full details.

File: calc,  Node: Other C-x * Commands,  Prev: Embedded Mode Overview,  Up: Using Calc

2.5.7 Other `C-x *' Commands
----------------------------

Two more Calc-related commands are `C-x * g' and `C-x * r', which
"grab" data from a selected region of a buffer into the Calculator.
The region is defined in the usual Emacs way, by a "mark" placed at one
end of the region, and the Emacs cursor or "point" placed at the other.

   The `C-x * g' command reads the region in the usual left-to-right,
top-to-bottom order.  The result is packaged into a Calc vector of
numbers and placed on the stack.  Calc (in its standard user interface)
is then started.  Type `v u' if you want to unpack this vector into
separate numbers on the stack.  Also, `C-u C-x * g' interprets the
region as a single number or formula.

   The `C-x * r' command reads a rectangle, with the point and mark
defining opposite corners of the rectangle.  The result is a matrix of
numbers on the Calculator stack.

   Complementary to these is `C-x * y', which "yanks" the value at the
top of the Calc stack back into an editing buffer.  If you type
`C-x * y' while in such a buffer, the value is yanked at the current
position.  If you type `C-x * y' while in the Calc buffer, Calc makes
an educated guess as to which editing buffer you want to use.  The Calc
window does not have to be visible in order to use this command, as
long as there is something on the Calc stack.

   Here, for reference, is the complete list of `C-x *' commands.  The
shift, control, and meta keys are ignored for the keystroke following
`C-x *'.

Commands for turning Calc on and off:

`*'
     Turn Calc on or off, employing the same user interface as last
     time.

`=, +, -, /, \, &, #'
     Alternatives for `*'.

`C'
     Turn Calc on or off using its standard bottom-of-the-screen
     interface.  If Calc is already turned on but the cursor is not in
     the Calc window, move the cursor into the window.

`O'
     Same as `C', but don't select the new Calc window.  If Calc is
     already turned on and the cursor is in the Calc window, move it
     out of that window.

`B'
     Control whether `C-x * c' and `C-x * k' use the full screen.

`Q'
     Use Quick mode for a single short calculation.

`K'
     Turn Calc Keypad mode on or off.

`E'
     Turn Calc Embedded mode on or off at the current formula.

`J'
     Turn Calc Embedded mode on or off, select the interesting part.

`W'
     Turn Calc Embedded mode on or off at the current word (number).

`Z'
     Turn Calc on in a user-defined way, as defined by a `Z I' command.

`X'
     Quit Calc; turn off standard, Keypad, or Embedded mode if on.
     (This is like `q' or <OFF> inside of Calc.)

Commands for moving data into and out of the Calculator:

`G'
     Grab the region into the Calculator as a vector.

`R'
     Grab the rectangular region into the Calculator as a matrix.

`:'
     Grab the rectangular region and compute the sums of its columns.

`_'
     Grab the rectangular region and compute the sums of its rows.

`Y'
     Yank a value from the Calculator into the current editing buffer.

Commands for use with Embedded mode:

`A'
     "Activate" the current buffer.  Locate all formulas that contain
     `:=' or `=>' symbols and record their locations so that they can
     be updated automatically as variables are changed.

`D'
     Duplicate the current formula immediately below and select the
     duplicate.

`F'
     Insert a new formula at the current point.

`N'
     Move the cursor to the next active formula in the buffer.

`P'
     Move the cursor to the previous active formula in the buffer.

`U'
     Update (i.e., as if by the `=' key) the formula at the current
     point.

``'
     Edit (as if by `calc-edit') the formula at the current point.

Miscellaneous commands:

`I'
     Run the Emacs Info system to read the Calc manual.  (This is the
     same as `h i' inside of Calc.)

`T'
     Run the Emacs Info system to read the Calc Tutorial.

`S'
     Run the Emacs Info system to read the Calc Summary.

`L'
     Load Calc entirely into memory.  (Normally the various parts are
     loaded only as they are needed.)

`M'
     Read a region of written keystroke names (like `C-n a b c <RET>')
     and record them as the current keyboard macro.

`0'
     (This is the "zero" digit key.)  Reset the Calculator to its
     initial state:  Empty stack, and initial mode settings.

File: calc,  Node: History and Acknowledgements,  Prev: Using Calc,  Up: Getting Started

2.6 History and Acknowledgements
================================

Calc was originally started as a two-week project to occupy a lull in
the author's schedule.  Basically, a friend asked if I remembered the
value of `2^32'.  I didn't offhand, but I said, "that's easy, just call
up an `xcalc'."  `Xcalc' duly reported that the answer to our question
was `4.294967e+09'--with no way to see the full ten digits even though
we knew they were there in the program's memory!  I was so annoyed, I
vowed to write a calculator of my own, once and for all.

   I chose Emacs Lisp, a) because I had always been curious about it
and b) because, being only a text editor extension language after all,
Emacs Lisp would surely reach its limits long before the project got
too far out of hand.

   To make a long story short, Emacs Lisp turned out to be a
distressingly solid implementation of Lisp, and the humble task of
calculating turned out to be more open-ended than one might have
expected.

   Emacs Lisp didn't have built-in floating point math (now it does), so
this had to be simulated in software.  In fact, Emacs integers would
only comfortably fit six decimal digits or so--not enough for a decent
calculator.  So I had to write my own high-precision integer code as
well, and once I had this I figured that arbitrary-size integers were
just as easy as large integers.  Arbitrary floating-point precision was
the logical next step.  Also, since the large integer arithmetic was
there anyway it seemed only fair to give the user direct access to it,
which in turn made it practical to support fractions as well as floats.
All these features inspired me to look around for other data types that
might be worth having.

   Around this time, my friend Rick Koshi showed me his nifty new HP-28
calculator.  It allowed the user to manipulate formulas as well as
numerical quantities, and it could also operate on matrices.  I decided
that these would be good for Calc to have, too.  And once things had
gone this far, I figured I might as well take a look at serious algebra
systems for further ideas.  Since these systems did far more than I
could ever hope to implement, I decided to focus on rewrite rules and
other programming features so that users could implement what they
needed for themselves.

   Rick complained that matrices were hard to read, so I put in code to
format them in a 2D style.  Once these routines were in place, Big mode
was obligatory.  Gee, what other language modes would be useful?

   Scott Hemphill and Allen Knutson, two friends with a strong
mathematical bent, contributed ideas and algorithms for a number of
Calc features including modulo forms, primality testing, and
float-to-fraction conversion.

   Units were added at the eager insistence of Mass Sivilotti.  Later,
Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
expert assistance with the units table.  As far as I can remember, the
idea of using algebraic formulas and variables to represent units dates
back to an ancient article in Byte magazine about muMath, an early
algebra system for microcomputers.

   Many people have contributed to Calc by reporting bugs and suggesting
features, large and small.  A few deserve special mention:  Tim Peters,
who helped develop the ideas that led to the selection commands, rewrite
rules, and many other algebra features; Francois Pinard, who
contributed an early prototype of the Calc Summary appendix as well as
providing valuable suggestions in many other areas of Calc; Carl Witty,
whose eagle eyes discovered many typographical and factual errors in
the Calc manual; Tim Kay, who drove the development of Embedded mode;
Ove Ewerlid, who made many suggestions relating to the algebra commands
and contributed some code for polynomial operations; Randal Schwartz,
who suggested the `calc-eval' function; Juha Sarlin, who first worked
out how to split Calc into quickly-loading parts; Bob Weiner, who
helped immensely with the Lucid Emacs port; and Robert J. Chassell, who
suggested the Calc Tutorial and exercises as well as many other things.

   Among the books used in the development of Calc were Knuth's _Art of
Computer Programming_ (especially volume II, _Seminumerical
Algorithms_); _Numerical Recipes_ by Press, Flannery, Teukolsky, and
Vetterling; Bevington's _Data Reduction and Error Analysis for the
Physical Sciences_; _Concrete Mathematics_ by Graham, Knuth, and
Patashnik; Steele's _Common Lisp, the Language_; the _CRC Standard Math
Tables_ (William H. Beyer, ed.); and Abramowitz and Stegun's venerable
_Handbook of Mathematical Functions_.  Also, of course, Calc could not
have been written without the excellent _GNU Emacs Lisp Reference
Manual_, by Bil Lewis and Dan LaLiberte.

   Final thanks go to Richard Stallman, without whose fine
implementations of the Emacs editor, language, and environment, Calc
would have been finished in two weeks.

File: calc,  Node: Interactive Tutorial,  Next: Tutorial,  Prev: Getting Started,  Up: Top

3 Tutorial
**********

Some brief instructions on using the Emacs Info system for this
tutorial:

   Press the space bar and Delete keys to go forward and backward in a
section by screenfuls (or use the regular Emacs scrolling commands for
this).

   Press `n' or `p' to go to the Next or Previous section.  If the
section has a "menu", press a digit key like `1' or `2' to go to a
sub-section from the menu.  Press `u' to go back up from a sub-section
to the menu it is part of.

   Exercises in the tutorial all have cross-references to the
appropriate page of the "answers" section.  Press `f', then the
exercise number, to see the answer to an exercise.  After you have
followed a cross-reference, you can press the letter `l' to return to
where you were before.

   You can press `?' at any time for a brief summary of Info commands.

   Press the number `1' now to enter the first section of the Tutorial.

* Menu:

* Tutorial::

File: calc,  Node: Tutorial,  Next: Introduction,  Prev: Interactive Tutorial,  Up: Top

4 Tutorial
**********

This chapter explains how to use Calc and its many features, in a
step-by-step, tutorial way.  You are encouraged to run Calc and work
along with the examples as you read (*note Starting Calc::).  If you
are already familiar with advanced calculators, you may wish to skip on
to the rest of this manual.

   This tutorial describes the standard user interface of Calc only.
The Quick mode and Keypad mode interfaces are fairly self-explanatory.
*Note Embedded Mode::, for a description of the Embedded mode interface.

   The easiest way to read this tutorial on-line is to have two windows
on your Emacs screen, one with Calc and one with the Info system.  Press
`C-x * t' to set this up; the on-line tutorial will be opened in the
current window and Calc will be started in another window.  From the
Info window, the command `C-x * c' can be used to switch to the Calc
window and `C-x * o' can be used to switch back to the Info window.
(If you have a printed copy of the manual you can use that instead; in
that case you only need to press `C-x * c' to start Calc.)

   This tutorial is designed to be done in sequence.  But the rest of
this manual does not assume you have gone through the tutorial.  The
tutorial does not cover everything in the Calculator, but it touches on
most general areas.

   You may wish to print out a copy of the Calc Summary and keep notes
on it as you learn Calc.  *Note About This Manual::, to see how to make
a printed summary.  *Note Summary::.

* Menu:

* Basic Tutorial::
* Arithmetic Tutorial::
* Vector/Matrix Tutorial::
* Types Tutorial::
* Algebra Tutorial::
* Programming Tutorial::

* Answers to Exercises::

File: calc,  Node: Basic Tutorial,  Next: Arithmetic Tutorial,  Prev: Tutorial,  Up: Tutorial

4.1 Basic Tutorial
==================

In this section, we learn how RPN and algebraic-style calculations
work, how to undo and redo an operation done by mistake, and how to
control various modes of the Calculator.

* Menu:

* RPN Tutorial::            Basic operations with the stack.
* Algebraic Tutorial::      Algebraic entry; variables.
* Undo Tutorial::           If you make a mistake: Undo and the trail.
* Modes Tutorial::          Common mode-setting commands.

File: calc,  Node: RPN Tutorial,  Next: Algebraic Tutorial,  Prev: Basic Tutorial,  Up: Basic Tutorial

4.1.1 RPN Calculations and the Stack
------------------------------------

Calc normally uses RPN notation.  You may be familiar with the RPN
system from Hewlett-Packard calculators, FORTH, or PostScript.
(Reverse Polish Notation, RPN, is named after the Polish mathematician
Jan Lukasiewicz.)

The central component of an RPN calculator is the "stack".  A
calculator stack is like a stack of dishes.  New dishes (numbers) are
added at the top of the stack, and numbers are normally only removed
from the top of the stack.

   In an operation like `2+3', the 2 and 3 are called the "operands"
and the `+' is the "operator".  In an RPN calculator you always enter
the operands first, then the operator.  Each time you type a number,
Calc adds or "pushes" it onto the top of the Stack.  When you press an
operator key like `+', Calc "pops" the appropriate number of operands
from the stack and pushes back the result.

   Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
`2 <RET> 3 <RET> +'.  (The <RET> key, Return, corresponds to the
<ENTER> key on traditional RPN calculators.)  Try this now if you wish;
type `C-x * c' to switch into the Calc window (you can type `C-x * c'
again or `C-x * o' to switch back to the Tutorial window).  The first
four keystrokes "push" the numbers 2 and 3 onto the stack.  The `+' key
"pops" the top two numbers from the stack, adds them, and pushes the
result (5) back onto the stack.  Here's how the stack will look at
various points throughout the calculation:

         .          1:  2          2:  2          1:  5              .
                        .          1:  3              .
                                       .

       C-x * c          2 <RET>          3 <RET>            +             <DEL>

   The `.' symbol is a marker that represents the top of the stack.
Note that the "top" of the stack is really shown at the bottom of the
Stack window.  This may seem backwards, but it turns out to be less
distracting in regular use.

   The numbers `1:' and `2:' on the left are "stack level numbers".
Old RPN calculators always had four stack levels called `x', `y', `z',
and `t'.  Calc's stack can grow as large as you like, so it uses
numbers instead of letters.  Some stack-manipulation commands accept a
numeric argument that says which stack level to work on.  Normal
commands like `+' always work on the top few levels of the stack.

   The Stack buffer is just an Emacs buffer, and you can move around in
it using the regular Emacs motion commands.  But no matter where the
cursor is, even if you have scrolled the `.' marker out of view, most
Calc commands always move the cursor back down to level 1 before doing
anything.  It is possible to move the `.' marker upwards through the
stack, temporarily "hiding" some numbers from commands like `+'.  This
is called "stack truncation" and we will not cover it in this tutorial;
*note Truncating the Stack::, if you are interested.

   You don't really need the second <RET> in `2 <RET> 3 <RET> +'.
That's because if you type any operator name or other non-numeric key
when you are entering a number, the Calculator automatically enters
that number and then does the requested command.  Thus `2 <RET> 3 +'
will work just as well.

   Examples in this tutorial will often omit <RET> even when the stack
displays shown would only happen if you did press <RET>:

     1:  2          2:  2          1:  5
         .          1:  3              .
                        .

       2 <RET>            3              +

Here, after pressing `3' the stack would really show `1:  2' with
`Calc: 3' in the minibuffer.  In these situations, you can press the
optional <RET> to see the stack as the figure shows.

   (*) *Exercise 1.*  (This tutorial will include exercises at various
points.  Try them if you wish.  Answers to all the exercises are
located at the end of the Tutorial chapter.  Each exercise will include
a cross-reference to its particular answer.  If you are reading with
the Emacs Info system, press `f' and the exercise number to go to the
answer, then the letter `l' to return to where you were.)

Here's the first exercise:  What will the keystrokes `1 <RET> 2 <RET> 3
<RET> 4 + * -' compute?  (`*' is the symbol for multiplication.)
Figure it out by hand, then try it with Calc to see if you're right.
*Note 1: RPN Answer 1. (*)

   (*) *Exercise 2.*  Compute `2*4 + 7*9.5 + 5/4' using the stack.
*Note 2: RPN Answer 2. (*)

   The <DEL> key is called Backspace on some keyboards.  It is whatever
key you would use to correct a simple typing error when regularly using
Emacs.  The <DEL> key pops and throws away the top value on the stack.
(You can still get that value back from the Trail if you should need it
later on.)  There are many places in this tutorial where we assume you
have used <DEL> to erase the results of the previous example at the
beginning of a new example.  In the few places where it is really
important to use <DEL> to clear away old results, the text will remind
you to do so.

   (It won't hurt to let things accumulate on the stack, except that
whenever you give a display-mode-changing command Calc will have to
spend a long time reformatting such a large stack.)

   Since the `-' key is also an operator (it subtracts the top two
stack elements), how does one enter a negative number?  Calc uses the
`_' (underscore) key to act like the minus sign in a number.  So,
typing `-5 <RET>' won't work because the `-' key will try to do a
subtraction, but `_5 <RET>' works just fine.

   You can also press `n', which means "change sign."  It changes the
number at the top of the stack (or the number being entered) from
positive to negative or vice-versa:  `5 n <RET>'.

   If you press <RET> when you're not entering a number, the effect is
to duplicate the top number on the stack.  Consider this calculation:

     1:  3          2:  3          1:  9          2:  9          1:  81
         .          1:  3              .          1:  9              .
                        .                             .

       3 <RET>           <RET>             *             <RET>             *

(Of course, an easier way to do this would be `3 <RET> 4 ^', to raise 3
to the fourth power.)

   The space-bar key (denoted <SPC> here) performs the same function as
<RET>; you could replace all three occurrences of <RET> in the above
example with <SPC> and the effect would be the same.

   Another stack manipulation key is <TAB>.  This exchanges the top two
stack entries.  Suppose you have computed `2 <RET> 3 +' to get 5, and
then you realize what you really wanted to compute was `20 / (2+3)'.

     1:  5          2:  5          2:  20         1:  4
         .          1:  20         1:  5              .
                        .              .

      2 <RET> 3 +         20            <TAB>             /

Planning ahead, the calculation would have gone like this:

     1:  20         2:  20         3:  20         2:  20         1:  4
         .          1:  2          2:  2          1:  5              .
                        .          1:  3              .
                                       .

       20 <RET>         2 <RET>            3              +              /

   A related stack command is `M-<TAB>' (hold <META> and type <TAB>).
It rotates the top three elements of the stack upward, bringing the
object in level 3 to the top.

     1:  10         2:  10         3:  10         3:  20         3:  30
         .          1:  20         2:  20         2:  30         2:  10
                        .          1:  30         1:  10         1:  20
                                       .              .              .

       10 <RET>         20 <RET>         30 <RET>         M-<TAB>          M-<TAB>

   (*) *Exercise 3.* Suppose the numbers 10, 20, and 30 are on the
stack.  Figure out how to add one to the number in level 2 without
affecting the rest of the stack.  Also figure out how to add one to the
number in level 3.  *Note 3: RPN Answer 3. (*)

   Operations like `+', `-', `*', `/', and `^' pop two arguments from
the stack and push a result.  Operations like `n' and `Q' (square root)
pop a single number and push the result.  You can think of them as
simply operating on the top element of the stack.

     1:  3          1:  9          2:  9          1:  25         1:  5
         .              .          1:  16             .              .
                                       .

       3 <RET>          <RET> *        4 <RET> <RET> *        +              Q

(Note that capital `Q' means to hold down the Shift key while typing
`q'.  Remember, plain unshifted `q' is the Quit command.)

   Here we've used the Pythagorean Theorem to determine the hypotenuse
of a right triangle.  Calc actually has a built-in command for that
called `f h', but let's suppose we can't remember the necessary
keystrokes.  We can still enter it by its full name using `M-x'
notation:

     1:  3          2:  3          1:  5
         .          1:  4              .
                        .

       3 <RET>          4 <RET>      M-x calc-hypot

   All Calculator commands begin with the word `calc-'.  Since it gets
tiring to type this, Calc provides an `x' key which is just like the
regular Emacs `M-x' key except that it types the `calc-' prefix for you:

     1:  3          2:  3          1:  5
         .          1:  4              .
                        .

       3 <RET>          4 <RET>         x hypot

   What happens if you take the square root of a negative number?

     1:  4          1:  -4         1:  (0, 2)
         .              .              .

       4 <RET>            n              Q

The notation `(a, b)' represents a complex number.  Complex numbers are
more traditionally written `a + b i'; Calc can display in this format,
too, but for now we'll stick to the `(a, b)' notation.

   If you don't know how complex numbers work, you can safely ignore
this feature.  Complex numbers only arise from operations that would be
errors in a calculator that didn't have complex numbers.  (For example,
taking the square root or logarithm of a negative number produces a
complex result.)

   Complex numbers are entered in the notation shown.  The `(' and `,'
and `)' keys manipulate "incomplete complex numbers."

     1:  ( ...      2:  ( ...      1:  (2, ...    1:  (2, ...    1:  (2, 3)
         .          1:  2              .              3              .
                        .                             .

         (              2              ,              3              )

   You can perform calculations while entering parts of incomplete
objects.  However, an incomplete object cannot actually participate in
a calculation:

     1:  ( ...      2:  ( ...      3:  ( ...      1:  ( ...      1:  ( ...
         .          1:  2          2:  2              5              5
                        .          1:  3              .              .
                                       .
                                                                  (error)
         (             2 <RET>           3              +              +

Adding 5 to an incomplete object makes no sense, so the last command
produces an error message and leaves the stack the same.

   Incomplete objects can't participate in arithmetic, but they can be
moved around by the regular stack commands.

     2:  2          3:  2          3:  3          1:  ( ...      1:  (2, 3)
     1:  3          2:  3          2:  ( ...          2              .
         .          1:  ( ...      1:  2              3
                        .              .              .

     2 <RET> 3 <RET>        (            M-<TAB>          M-<TAB>            )

Note that the `,' (comma) key did not have to be used here.  When you
press `)' all the stack entries between the incomplete entry and the
top are collected, so there's never really a reason to use the comma.
It's up to you.

   (*) *Exercise 4.*  To enter the complex number `(2, 3)', your friend
Joe typed `( 2 , <SPC> 3 )'.  What happened?  (Joe thought of a clever
way to correct his mistake in only two keystrokes, but it didn't quite
work.  Try it to find out why.)  *Note 4: RPN Answer 4. (*)

   Vectors are entered the same way as complex numbers, but with square
brackets in place of parentheses.  We'll meet vectors again later in
the tutorial.

   Any Emacs command can be given a "numeric prefix argument" by typing
a series of <META>-digits beforehand.  If <META> is awkward for you,
you can instead type `C-u' followed by the necessary digits.  Numeric
prefix arguments can be negative, as in `M-- M-3 M-5' or `C-u - 3 5'.
Calc commands use numeric prefix arguments in a variety of ways.  For
example, a numeric prefix on the `+' operator adds any number of stack
entries at once:

     1:  10         2:  10         3:  10         3:  10         1:  60
         .          1:  20         2:  20         2:  20             .
                        .          1:  30         1:  30
                                       .              .

       10 <RET>         20 <RET>         30 <RET>         C-u 3            +

   For stack manipulation commands like <RET>, a positive numeric
prefix argument operates on the top N stack entries at once.  A
negative argument operates on the entry in level N only.  An argument
of zero operates on the entire stack.  In this example, we copy the
second-to-top element of the stack:

     1:  10         2:  10         3:  10         3:  10         4:  10
         .          1:  20         2:  20         2:  20         3:  20
                        .          1:  30         1:  30         2:  30
                                       .              .          1:  20
                                                                     .

       10 <RET>         20 <RET>         30 <RET>         C-u -2          <RET>

   Another common idiom is `M-0 <DEL>', which clears the stack.  (The
`M-0' numeric prefix tells <DEL> to operate on the entire stack.)

File: calc,  Node: Algebraic Tutorial,  Next: Undo Tutorial,  Prev: RPN Tutorial,  Up: Basic Tutorial

4.1.2 Algebraic-Style Calculations
----------------------------------

If you are not used to RPN notation, you may prefer to operate the
Calculator in Algebraic mode, which is closer to the way non-RPN
calculators work.  In Algebraic mode, you enter formulas in traditional
`2+3' notation.

   *Notice:* Calc gives `/' lower precedence than `*', so that `a/b*c'
is interpreted as `a/(b*c)'; this is not standard across all computer
languages.  See below for details.

   You don't really need any special "mode" to enter algebraic formulas.
You can enter a formula at any time by pressing the apostrophe (`'')
key.  Answer the prompt with the desired formula, then press <RET>.
The formula is evaluated and the result is pushed onto the RPN stack.
If you don't want to think in RPN at all, you can enter your whole
computation as a formula, read the result from the stack, then press
<DEL> to delete it from the stack.

   Try pressing the apostrophe key, then `2+3+4', then <RET>.  The
result should be the number 9.

   Algebraic formulas use the operators `+', `-', `*', `/', and `^'.
You can use parentheses to make the order of evaluation clear.  In the
absence of parentheses, `^' is evaluated first, then `*', then `/',
then finally `+' and `-'.  For example, the expression

     2 + 3*4*5 / 6*7^8 - 9

is equivalent to

     2 + ((3*4*5) / (6*(7^8)) - 9

or, in large mathematical notation,

         3 * 4 * 5
     2 + --------- - 9
               8
          6 * 7

The result of this expression will be the number -6.99999826533.

   Calc's order of evaluation is the same as for most computer
languages, except that `*' binds more strongly than `/', as the above
example shows.  As in normal mathematical notation, the `*' symbol can
often be omitted:  `2 a' is the same as `2*a'.

   Operators at the same level are evaluated from left to right, except
that `^' is evaluated from right to left.  Thus, `2-3-4' is equivalent
to `(2-3)-4' or -5, whereas `2^3^4' is equivalent to `2^(3^4)' (a very
large integer; try it!).

   If you tire of typing the apostrophe all the time, there is
Algebraic mode, where Calc automatically senses when you are about to
type an algebraic expression.  To enter this mode, press the two
letters `m a'.  (An `Alg' indicator should appear in the Calc window's
mode line.)

   Press `m a', then `2+3+4' with no apostrophe, then <RET>.

   In Algebraic mode, when you press any key that would normally begin
entering a number (such as a digit, a decimal point, or the `_' key),
or if you press `(' or `[', Calc automatically begins an algebraic
entry.

   Functions which do not have operator symbols like `+' and `*' must
be entered in formulas using function-call notation.  For example, the
function name corresponding to the square-root key `Q' is `sqrt'.  To
compute a square root in a formula, you would use the notation
`sqrt(X)'.

   Press the apostrophe, then type `sqrt(5*2) - 3'.  The result should
be `0.16227766017'.

   Note that if the formula begins with a function name, you need to use
the apostrophe even if you are in Algebraic mode.  If you type `arcsin'
out of the blue, the `a r' will be taken as an Algebraic Rewrite
command, and the `csin' will be taken as the name of the rewrite rule
to use!

   Some people prefer to enter complex numbers and vectors in algebraic
form because they find RPN entry with incomplete objects to be too
distracting, even though they otherwise use Calc as an RPN calculator.

   Still in Algebraic mode, type:

     1:  (2, 3)     2:  (2, 3)     1:  (8, -1)    2:  (8, -1)    1:  (9, -1)
         .          1:  (1, -2)        .          1:  1              .
                        .                             .

      (2,3) <RET>      (1,-2) <RET>        *              1 <RET>          +

   Algebraic mode allows us to enter complex numbers without pressing
an apostrophe first, but it also means we need to press <RET> after
every entry, even for a simple number like `1'.

   (You can type `C-u m a' to enable a special Incomplete Algebraic
mode in which the `(' and `[' keys use algebraic entry even though
regular numeric keys still use RPN numeric entry.  There is also Total
Algebraic mode, started by typing `m t', in which all normal keys begin
algebraic entry.  You must then use the <META> key to type Calc
commands:  `M-m t' to get back out of Total Algebraic mode, `M-q' to
quit, etc.)

   If you're still in Algebraic mode, press `m a' again to turn it off.

   Actual non-RPN calculators use a mixture of algebraic and RPN styles.
In general, operators of two numbers (like `+' and `*') use algebraic
form, but operators of one number (like `n' and `Q') use RPN form.
Also, a non-RPN calculator allows you to see the intermediate results
of a calculation as you go along.  You can accomplish this in Calc by
performing your calculation as a series of algebraic entries, using the
`$' sign to tie them together.  In an algebraic formula, `$' represents
the number on the top of the stack.  Here, we perform the calculation
`sqrt(2*4+1)', which on a traditional calculator would be done by
pressing `2 * 4 + 1 =' and then the square-root key.

     1:  8          1:  9          1:  3
         .              .              .

       ' 2*4 <RET>        $+1 <RET>        Q

Notice that we didn't need to press an apostrophe for the `$+1',
because the dollar sign always begins an algebraic entry.

   (*) *Exercise 1.*  How could you get the same effect as pressing `Q'
but using an algebraic entry instead?  How about if the `Q' key on your
keyboard were broken?  *Note 1: Algebraic Answer 1. (*)

   The notations `$$', `$$$', and so on stand for higher stack entries.
For example, `' $$+$ <RET>' is just like typing `+'.

   Algebraic formulas can include "variables".  To store in a variable,
press `s s', then type the variable name, then press <RET>.  (There are
actually two flavors of store command: `s s' stores a number in a
variable but also leaves the number on the stack, while `s t' removes a
number from the stack and stores it in the variable.)  A variable name
should consist of one or more letters or digits, beginning with a
letter.

     1:  17             .          1:  a + a^2    1:  306
         .                             .              .

         17          s t a <RET>      ' a+a^2 <RET>       =

The `=' key "evaluates" a formula by replacing all its variables by the
values that were stored in them.

   For RPN calculations, you can recall a variable's value on the stack
either by entering its name as a formula and pressing `=', or by using
the `s r' command.

     1:  17         2:  17         3:  17         2:  17         1:  306
         .          1:  17         2:  17         1:  289            .
                        .          1:  2              .
                                       .

       s r a <RET>     ' a <RET> =         2              ^              +

   If you press a single digit for a variable name (as in `s t 3', you
get one of ten "quick variables" `q0' through `q9'.  They are "quick"
simply because you don't have to type the letter `q' or the <RET> after
their names.  In fact, you can type simply `s 3' as a shorthand for `s
s 3', and likewise for `t 3' and `r 3'.

   Any variables in an algebraic formula for which you have not stored
values are left alone, even when you evaluate the formula.

     1:  2 a + 2 b     1:  34 + 2 b
         .                 .

      ' 2a+2b <RET>          =

   Calls to function names which are undefined in Calc are also left
alone, as are calls for which the value is undefined.

     1:  2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
         .

      ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) <RET>

In this example, the first call to `log10' works, but the other calls
are not evaluated.  In the second call, the logarithm is undefined for
that value of the argument; in the third, the argument is symbolic, and
in the fourth, there are too many arguments.  In the fifth case, there
is no function called `foo'.  You will see a "Wrong number of
arguments" message referring to `log10(5,6)'.  Press the `w' ("why")
key to see any other messages that may have arisen from the last
calculation.  In this case you will get "logarithm of zero," then
"number expected: `x'".  Calc automatically displays the first message
only if the message is sufficiently important; for example, Calc
considers "wrong number of arguments" and "logarithm of zero" to be
important enough to report automatically, while a message like "number
expected: `x'" will only show up if you explicitly press the `w' key.

   (*) *Exercise 2.*  Joe entered the formula `2 x y', stored 5 in `x',
pressed `=', and got the expected result, `10 y'.  He then tried the
same for the formula `2 x (1+y)', expecting `10 (1+y)', but it didn't
work.  Why not?  *Note 2: Algebraic Answer 2. (*)

   (*) *Exercise 3.*  What result would you expect `1 <RET> 0 /' to
give?  What if you then type `0 *'?  *Note 3: Algebraic Answer 3. (*)

   One interesting way to work with variables is to use the
"evaluates-to" (`=>') operator.  It works like this: Enter a formula
algebraically in the usual way, but follow the formula with an `=>'
symbol.  (There is also an `s =' command which builds an `=>' formula
using the stack.)  On the stack, you will see two copies of the formula
with an `=>' between them.  The lefthand formula is exactly like you
typed it; the righthand formula has been evaluated as if by typing `='.

     2:  2 + 3 => 5                     2:  2 + 3 => 5
     1:  2 a + 2 b => 34 + 2 b          1:  2 a + 2 b => 20 + 2 b
         .                                  .

     ' 2+3 => <RET>  ' 2a+2b <RET> s =          10 s t a <RET>

Notice that the instant we stored a new value in `a', all `=>'
operators already on the stack that referred to `a' were updated to use
the new value.  With `=>', you can push a set of formulas on the stack,
then change the variables experimentally to see the effects on the
formulas' values.

   You can also "unstore" a variable when you are through with it:

     2:  2 + 5 => 5
     1:  2 a + 2 b => 2 a + 2 b
         .

         s u a <RET>

   We will encounter formulas involving variables and functions again
when we discuss the algebra and calculus features of the Calculator.

File: calc,  Node: Undo Tutorial,  Next: Modes Tutorial,  Prev: Algebraic Tutorial,  Up: Basic Tutorial

4.1.3 Undo and Redo
-------------------

If you make a mistake, you can usually correct it by pressing shift-`U',
the "undo" command.  First, clear the stack (`M-0 <DEL>') and exit and
restart Calc (`C-x * * C-x * *') to make sure things start off with a
clean slate.  Now:

     1:  2          2:  2          1:  8          2:  2          1:  6
         .          1:  3              .          1:  3              .
                        .                             .

        2 <RET>           3              ^              U              *

   You can undo any number of times.  Calc keeps a complete record of
all you have done since you last opened the Calc window.  After the
above example, you could type:

     1:  6          2:  2          1:  2              .              .
         .          1:  3              .
                        .
                                                                  (error)
                        U              U              U              U

   You can also type `D' to "redo" a command that you have undone
mistakenly.

         .          1:  2          2:  2          1:  6          1:  6
                        .          1:  3              .              .
                                       .
                                                                  (error)
                        D              D              D              D

It was not possible to redo past the `6', since that was placed there
by something other than an undo command.

   You can think of undo and redo as a sort of "time machine."  Press
`U' to go backward in time, `D' to go forward.  If you go backward and
do something (like `*') then, as any science fiction reader knows, you
have changed your future and you cannot go forward again.  Thus, the
inability to redo past the `6' even though there was an earlier undo
command.

   You can always recall an earlier result using the Trail.  We've
ignored the trail so far, but it has been faithfully recording
everything we did since we loaded the Calculator.  If the Trail is not
displayed, press `t d' now to turn it on.

   Let's try grabbing an earlier result.  The `8' we computed was
undone by a `U' command, and was lost even to Redo when we pressed `*',
but it's still there in the trail.  There should be a little `>' arrow
(the "trail pointer") resting on the last trail entry.  If there isn't,
press `t ]' to reset the trail pointer.  Now, press `t p' to move the
arrow onto the line containing `8', and press `t y' to "yank" that
number back onto the stack.

   If you press `t ]' again, you will see that even our Yank command
went into the trail.

   Let's go further back in time.  Earlier in the tutorial we computed
a huge integer using the formula `2^3^4'.  We don't remember what it
was, but the first digits were "241".  Press `t r' (which stands for
trail-search-reverse), then type `241'.  The trail cursor will jump
back to the next previous occurrence of the string "241" in the trail.
This is just a regular Emacs incremental search; you can now press
`C-s' or `C-r' to continue the search forwards or backwards as you like.

   To finish the search, press <RET>.  This halts the incremental
search and leaves the trail pointer at the thing we found.  Now we can
type `t y' to yank that number onto the stack.  If we hadn't remembered
the "241", we could simply have searched for `2^3^4', then pressed
`<RET> t n' to halt and then move to the next item.

   You may have noticed that all the trail-related commands begin with
the letter `t'.  (The store-and-recall commands, on the other hand, all
began with `s'.)  Calc has so many commands that there aren't enough
keys for all of them, so various commands are grouped into two-letter
sequences where the first letter is called the "prefix" key.  If you
type a prefix key by accident, you can press `C-g' to cancel it.  (In
fact, you can press `C-g' to cancel almost anything in Emacs.)  To get
help on a prefix key, press that key followed by `?'.  Some prefixes
have several lines of help, so you need to press `?' repeatedly to see
them all.  You can also type `h h' to see all the help at once.

   Try pressing `t ?' now.  You will see a line of the form,

     trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank:  [MORE]  t-

The word "trail" indicates that the `t' prefix key contains
trail-related commands.  Each entry on the line shows one command, with
a single capital letter showing which letter you press to get that
command.  We have used `t n', `t p', `t ]', and `t y' so far.  The
`[MORE]' means you can press `?' again to see more `t'-prefix commands.
Notice that the commands are roughly divided (by semicolons) into
related groups.

   When you are in the help display for a prefix key, the prefix is
still active.  If you press another key, like `y' for example, it will
be interpreted as a `t y' command.  If all you wanted was to look at
the help messages, press `C-g' afterwards to cancel the prefix.

   One more way to correct an error is by editing the stack entries.
The actual Stack buffer is marked read-only and must not be edited
directly, but you can press ``' (the backquote or accent grave) to edit
a stack entry.

   Try entering `3.141439' now.  If this is supposed to represent `pi',
it's got several errors.  Press ``' to edit this number.  Now use the
normal Emacs cursor motion and editing keys to change the second 4 to a
5, and to transpose the 3 and the 9.  When you press <RET>, the number
on the stack will be replaced by your new number.  This works for
formulas, vectors, and all other types of values you can put on the
stack.  The ``' key also works during entry of a number or algebraic
formula.

File: calc,  Node: Modes Tutorial,  Prev: Undo Tutorial,  Up: Basic Tutorial

4.1.4 Mode-Setting Commands
---------------------------

Calc has many types of "modes" that affect the way it interprets your
commands or the way it displays data.  We have already seen one mode,
namely Algebraic mode.  There are many others, too; we'll try some of
the most common ones here.

   Perhaps the most fundamental mode in Calc is the current "precision".
Notice the `12' on the Calc window's mode line:

     --%*-Calc: 12 Deg       (Calculator)----All------

Most of the symbols there are Emacs things you don't need to worry
about, but the `12' and the `Deg' are mode indicators.  The `12' means
that calculations should always be carried to 12 significant figures.
That is why, when we type `1 <RET> 7 /', we get `0.142857142857' with
exactly 12 digits, not counting leading and trailing zeros.

   You can set the precision to anything you like by pressing `p', then
entering a suitable number.  Try pressing `p 30 <RET>', then doing `1
<RET> 7 /' again:

     1:  0.142857142857
     2:  0.142857142857142857142857142857
         .

   Although the precision can be set arbitrarily high, Calc always has
to have _some_ value for the current precision.  After all, the true
value `1/7' is an infinitely repeating decimal; Calc has to stop
somewhere.

   Of course, calculations are slower the more digits you request.
Press `p 12' now to set the precision back down to the default.

   Calculations always use the current precision.  For example, even
though we have a 30-digit value for `1/7' on the stack, if we use it in
a calculation in 12-digit mode it will be rounded down to 12 digits
before it is used.  Try it; press <RET> to duplicate the number, then
`1 +'.  Notice that the <RET> key didn't round the number, because it
doesn't do any calculation.  But the instant we pressed `+', the number
was rounded down.

     1:  0.142857142857
     2:  0.142857142857142857142857142857
     3:  1.14285714286
         .

In fact, since we added a digit on the left, we had to lose one digit
on the right from even the 12-digit value of `1/7'.

   How did we get more than 12 digits when we computed `2^3^4'?  The
answer is that Calc makes a distinction between "integers" and
"floating-point" numbers, or "floats".  An integer is a number that
does not contain a decimal point.  There is no such thing as an
"infinitely repeating fraction integer," so Calc doesn't have to limit
itself.  If you asked for `2^10000' (don't try this!), you would have
to wait a long time but you would eventually get an exact answer.  If
you ask for `2.^10000', you will quickly get an answer which is correct
only to 12 places.  The decimal point tells Calc that it should use
floating-point arithmetic to get the answer, not exact integer
arithmetic.

   You can use the `F' (`calc-floor') command to convert a
floating-point value to an integer, and `c f' (`calc-float') to convert
an integer to floating-point form.

   Let's try entering that last calculation:

     1:  2.         2:  2.         1:  1.99506311689e3010
         .          1:  10000          .
                        .

       2.0 <RET>          10000 <RET>      ^

Notice the letter `e' in there.  It represents "times ten to the power
of," and is used by Calc automatically whenever writing the number out
fully would introduce more extra zeros than you probably want to see.
You can enter numbers in this notation, too.

     1:  2.         2:  2.         1:  1.99506311678e3010
         .          1:  10000.         .
                        .

       2.0 <RET>          1e4 <RET>        ^

Hey, the answer is different!  Look closely at the middle columns of
the two examples.  In the first, the stack contained the exact integer
`10000', but in the second it contained a floating-point value with a
decimal point.  When you raise a number to an integer power, Calc uses
repeated squaring and multiplication to get the answer.  When you use a
floating-point power, Calc uses logarithms and exponentials.  As you
can see, a slight error crept in during one of these methods.  Which
one should we trust?  Let's raise the precision a bit and find out:

         .          1:  2.         2:  2.         1:  1.995063116880828e3010
                        .          1:  10000.         .
                                       .

      p 16 <RET>        2. <RET>           1e4            ^    p 12 <RET>

Presumably, it doesn't matter whether we do this higher-precision
calculation using an integer or floating-point power, since we have
added enough "guard digits" to trust the first 12 digits no matter
what.  And the verdict is...  Integer powers were more accurate; in
fact, the result was only off by one unit in the last place.

   Calc does many of its internal calculations to a slightly higher
precision, but it doesn't always bump the precision up enough.  In each
case, Calc added about two digits of precision during its calculation
and then rounded back down to 12 digits afterward.  In one case, it was
enough; in the other, it wasn't.  If you really need X digits of
precision, it never hurts to do the calculation with a few extra guard
digits.

   What if we want guard digits but don't want to look at them?  We can
set the "float format".  Calc supports four major formats for
floating-point numbers, called "normal", "fixed-point", "scientific
notation", and "engineering notation".  You get them by pressing `d n',
`d f', `d s', and `d e', respectively.  In each case, you can supply a
numeric prefix argument which says how many digits should be displayed.
As an example, let's put a few numbers onto the stack and try some
different display modes.  First, use `M-0 <DEL>' to clear the stack,
then enter the four numbers shown here:

     4:  12345      4:  12345      4:  12345      4:  12345      4:  12345
     3:  12345.     3:  12300.     3:  1.2345e4   3:  1.23e4     3:  12345.000
     2:  123.45     2:  123.       2:  1.2345e2   2:  1.23e2     2:  123.450
     1:  12.345     1:  12.3       1:  1.2345e1   1:  1.23e1     1:  12.345
         .              .              .              .              .

        d n          M-3 d n          d s          M-3 d s        M-3 d f

Notice that when we typed `M-3 d n', the numbers were rounded down to
three significant digits, but then when we typed `d s' all five
significant figures reappeared.  The float format does not affect how
numbers are stored, it only affects how they are displayed.  Only the
current precision governs the actual rounding of numbers in the
Calculator's memory.

   Engineering notation, not shown here, is like scientific notation
except the exponent (the power-of-ten part) is always adjusted to be a
multiple of three (as in "kilo," "micro," etc.).  As a result there
will be one, two, or three digits before the decimal point.

   Whenever you change a display-related mode, Calc redraws everything
in the stack.  This may be slow if there are many things on the stack,
so Calc allows you to type shift-`H' before any mode command to prevent
it from updating the stack.  Anything Calc displays after the
mode-changing command will appear in the new format.

     4:  12345      4:  12345      4:  12345      4:  12345      4:  12345
     3:  12345.000  3:  12345.000  3:  12345.000  3:  1.2345e4   3:  12345.
     2:  123.450    2:  123.450    2:  1.2345e1   2:  1.2345e1   2:  123.45
     1:  12.345     1:  1.2345e1   1:  1.2345e2   1:  1.2345e2   1:  12.345
         .              .              .              .              .

         H d s          <DEL> U          <TAB>            d <SPC>          d n

Here the `H d s' command changes to scientific notation but without
updating the screen.  Deleting the top stack entry and undoing it back
causes it to show up in the new format; swapping the top two stack
entries reformats both entries.  The `d <SPC>' command refreshes the
whole stack.  The `d n' command changes back to the normal float
format; since it doesn't have an `H' prefix, it also updates all the
stack entries to be in `d n' format.

   Notice that the integer `12345' was not affected by any of the float
formats.  Integers are integers, and are always displayed exactly.

   Large integers have their own problems.  Let's look back at the
result of `2^3^4'.

     2417851639229258349412352

Quick--how many digits does this have?  Try typing `d g':

     2,417,851,639,229,258,349,412,352

Now how many digits does this have?  It's much easier to tell!  We can
actually group digits into clumps of any size.  Some people prefer `M-5
d g':

     24178,51639,22925,83494,12352

   Let's see what happens to floating-point numbers when they are
grouped.  First, type `p 25 <RET>' to make sure we have enough precision
to get ourselves into trouble.  Now, type `1e13 /':

     24,17851,63922.9258349412352

The integer part is grouped but the fractional part isn't.  Now try
`M-- M-5 d g' (that's meta-minus-sign, meta-five):

     24,17851,63922.92583,49412,352

   If you find it hard to tell the decimal point from the commas, try
changing the grouping character to a space with `d , <SPC>':

     24 17851 63922.92583 49412 352

   Type `d , ,' to restore the normal grouping character, then `d g'
again to turn grouping off.  Also, press `p 12' to restore the default
precision.

   Press `U' enough times to get the original big integer back.
(Notice that `U' does not undo each mode-setting command; if you want
to undo a mode-setting command, you have to do it yourself.)  Now, type
`d r 16 <RET>':

     16#200000000000000000000

The number is now displayed in "hexadecimal", or "base-16" form.
Suddenly it looks pretty simple; this should be no surprise, since we
got this number by computing a power of two, and 16 is a power of 2.
In fact, we can use `d r 2 <RET>' to see it in actual binary form:

     2#1000000000000000000000000000000000000000000000000000000 ...

We don't have enough space here to show all the zeros!  They won't fit
on a typical screen, either, so you will have to use horizontal
scrolling to see them all.  Press `<' and `>' to scroll the stack
window left and right by half its width.  Another way to view something
large is to press ``' (back-quote) to edit the top of stack in a
separate window.  (Press `C-c C-c' when you are done.)

   You can enter non-decimal numbers using the `#' symbol, too.  Let's
see what the hexadecimal number `5FE' looks like in binary.  Type
`16#5FE' (the letters can be typed in upper or lower case; they will
always appear in upper case).  It will also help to turn grouping on
with `d g':

     2#101,1111,1110

   Notice that `d g' groups by fours by default if the display radix is
binary or hexadecimal, but by threes if it is decimal, octal, or any
other radix.

   Now let's see that number in decimal; type `d r 10':

     1,534

   Numbers are not _stored_ with any particular radix attached.  They're
just numbers; they can be entered in any radix, and are always displayed
in whatever radix you've chosen with `d r'.  The current radix applies
to integers, fractions, and floats.

   (*) *Exercise 1.*  Your friend Joe tried to enter one-third as
`3#0.1' in `d r 3' mode with a precision of 12.  He got
`3#0.0222222...' (with 25 2's) in the display.  When he multiplied that
by three, he got `3#0.222222...' instead of the expected `3#1'.  Next,
Joe entered `3#0.2' and, to his great relief, saw `3#0.2' on the
screen.  But when he typed `2 /', he got `3#0.10000001' (some zeros
omitted).  What's going on here?  *Note 1: Modes Answer 1. (*)

   (*) *Exercise 2.*  Scientific notation works in non-decimal modes in
the natural way (the exponent is a power of the radix instead of a
power of ten, although the exponent itself is always written in
decimal).  Thus `8#1.23e3 = 8#1230.0'.  Suppose we have the hexadecimal
number `f.e8f' times 16 to the 15th power:  We write `16#f.e8fe15'.
What is wrong with this picture?  What could we write instead that would
work better?  *Note 2: Modes Answer 2. (*)

   The `m' prefix key has another set of modes, relating to the way
Calc interprets your inputs and does computations.  Whereas `d'-prefix
modes generally affect the way things look, `m'-prefix modes affect the
way they are actually computed.

   The most popular `m'-prefix mode is the "angular mode".  Notice the
`Deg' indicator in the mode line.  This means that if you use a command
that interprets a number as an angle, it will assume the angle is
measured in degrees.  For example,

     1:  45         1:  0.707106781187   1:  0.500000000001    1:  0.5
         .              .                    .                     .

         45             S                    2 ^                   c 1

The shift-`S' command computes the sine of an angle.  The sine of 45
degrees is `sqrt(2)/2'; squaring this yields `2/4 = 0.5'.  However,
there has been a slight roundoff error because the representation of
`sqrt(2)/2' wasn't exact.  The `c 1' command is a handy way to clean up
numbers in this case; it temporarily reduces the precision by one digit
while it re-rounds the number on the top of the stack.

   (*) *Exercise 3.*  Your friend Joe computed the sine of 45 degrees
as shown above, then, hoping to avoid an inexact result, he increased
the precision to 16 digits before squaring.  What happened?  *Note 3:
Modes Answer 3. (*)

   To do this calculation in radians, we would type `m r' first.  (The
indicator changes to `Rad'.)  45 degrees corresponds to `pi/4' radians.
To get `pi', press the `P' key.  (Once again, this is a shifted capital
`P'.  Remember, unshifted `p' sets the precision.)

     1:  3.14159265359   1:  0.785398163398   1:  0.707106781187
         .                   .                .

         P                   4 /       m r    S

   Likewise, inverse trigonometric functions generate results in either
radians or degrees, depending on the current angular mode.

     1:  0.707106781187   1:  0.785398163398   1:  45.
         .                    .                    .

         .5 Q        m r      I S        m d       U I S

Here we compute the Inverse Sine of `sqrt(0.5)', first in radians, then
in degrees.

   Use `c d' and `c r' to convert a number from radians to degrees and
vice-versa.

     1:  45         1:  0.785398163397     1:  45.
         .              .                      .

         45             c r                    c d

   Another interesting mode is "Fraction mode".  Normally, dividing two
integers produces a floating-point result if the quotient can't be
expressed as an exact integer.  Fraction mode causes integer division
to produce a fraction, i.e., a rational number, instead.

     2:  12         1:  1.33333333333    1:  4:3
     1:  9              .                    .
         .

      12 <RET> 9          /          m f       U /      m f

In the first case, we get an approximate floating-point result.  In the
second case, we get an exact fractional result (four-thirds).

   You can enter a fraction at any time using `:' notation.  (Calc uses
`:' instead of `/' as the fraction separator because `/' is already
used to divide the top two stack elements.)  Calculations involving
fractions will always produce exact fractional results; Fraction mode
only says what to do when dividing two integers.

   (*) *Exercise 4.*  If fractional arithmetic is exact, why would you
ever use floating-point numbers instead?  *Note 4: Modes Answer 4. (*)

   Typing `m f' doesn't change any existing values in the stack.  In
the above example, we had to Undo the division and do it over again
when we changed to Fraction mode.  But if you use the evaluates-to
operator you can get commands like `m f' to recompute for you.

     1:  12 / 9 => 1.33333333333    1:  12 / 9 => 1.333    1:  12 / 9 => 4:3
         .                              .                      .

        ' 12/9 => <RET>                   p 4 <RET>                m f

In this example, the righthand side of the `=>' operator on the stack
is recomputed when we change the precision, then again when we change
to Fraction mode.  All `=>' expressions on the stack are recomputed
every time you change any mode that might affect their values.

File: calc,  Node: Arithmetic Tutorial,  Next: Vector/Matrix Tutorial,  Prev: Basic Tutorial,  Up: Tutorial

4.2 Arithmetic Tutorial
=======================

In this section, we explore the arithmetic and scientific functions
available in the Calculator.

   The standard arithmetic commands are `+', `-', `*', `/', and `^'.
Each normally takes two numbers from the top of the stack and pushes
back a result.  The `n' and `&' keys perform change-sign and reciprocal
operations, respectively.

     1:  5          1:  0.2        1:  5.         1:  -5.        1:  5.
         .              .              .              .              .

         5              &              &              n              n

   You can apply a "binary operator" like `+' across any number of
stack entries by giving it a numeric prefix.  You can also apply it
pairwise to several stack elements along with the top one if you use a
negative prefix.

     3:  2          1:  9          3:  2          4:  2          3:  12
     2:  3              .          2:  3          3:  3          2:  13
     1:  4                         1:  4          2:  4          1:  14
         .                             .          1:  10             .
                                                      .

     2 <RET> 3 <RET> 4     M-3 +           U              10          M-- M-3 +

   You can apply a "unary operator" like `&' to the top N stack entries
with a numeric prefix, too.

     3:  2          3:  0.5                3:  0.5
     2:  3          2:  0.333333333333     2:  3.
     1:  4          1:  0.25               1:  4.
         .              .                      .

     2 <RET> 3 <RET> 4      M-3 &                  M-2 &

   Notice that the results here are left in floating-point form.  We
can convert them back to integers by pressing `F', the "floor"
function.  This function rounds down to the next lower integer.  There
is also `R', which rounds to the nearest integer.

     7:  2.         7:  2          7:  2
     6:  2.4        6:  2          6:  2
     5:  2.5        5:  2          5:  3
     4:  2.6        4:  2          4:  3
     3:  -2.        3:  -2         3:  -2
     2:  -2.4       2:  -3         2:  -2
     1:  -2.6       1:  -3         1:  -3
         .              .              .

                       M-7 F        U M-7 R

   Since dividing-and-flooring (i.e., "integer quotient") is such a
common operation, Calc provides a special command for that purpose, the
backslash `\'.  Another common arithmetic operator is `%', which
computes the remainder that would arise from a `\' operation, i.e., the
"modulo" of two numbers.  For example,

     2:  1234       1:  12         2:  1234       1:  34
     1:  100            .          1:  100            .
         .                             .

     1234 <RET> 100       \              U              %

   These commands actually work for any real numbers, not just integers.

     2:  3.1415     1:  3          2:  3.1415     1:  0.1415
     1:  1              .          1:  1              .
         .                             .

     3.1415 <RET> 1       \              U              %

   (*) *Exercise 1.*  The `\' command would appear to be a frill, since
you could always do the same thing with `/ F'.  Think of a situation
where this is not true--`/ F' would be inadequate.  Now think of a way
you could get around the problem if Calc didn't provide a `\' command.
*Note 1: Arithmetic Answer 1. (*)

   We've already seen the `Q' (square root) and `S' (sine) commands.
Other commands along those lines are `C' (cosine), `T' (tangent), `E'
(`e^x') and `L' (natural logarithm).  These can be modified by the `I'
(inverse) and `H' (hyperbolic) prefix keys.

   Let's compute the sine and cosine of an angle, and verify the
identity `sin(x)^2 + cos(x)^2 = 1'.  We'll arbitrarily pick -64 degrees
as a good value for `x'.  With the angular mode set to degrees (type
`m d'), do:

     2:  -64        2:  -64        2:  -0.89879   2:  -0.89879   1:  1.
     1:  -64        1:  -0.89879   1:  -64        1:  0.43837        .
         .              .              .              .

      64 n <RET> <RET>      S              <TAB>            C              f h

(For brevity, we're showing only five digits of the results here.  You
can of course do these calculations to any precision you like.)

   Remember, `f h' is the `calc-hypot', or square-root of sum of
squares, command.

   Another identity is `tan(x) = sin(x) / cos(x)'.

     2:  -0.89879   1:  -2.0503    1:  -64.
     1:  0.43837        .              .
         .

         U              /              I T

   A physical interpretation of this calculation is that if you move
`0.89879' units downward and `0.43837' units to the right, your
direction of motion is -64 degrees from horizontal.  Suppose we move in
the opposite direction, up and to the left:

     2:  -0.89879   2:  0.89879    1:  -2.0503    1:  -64.
     1:  0.43837    1:  -0.43837       .              .
         .              .

         U U            M-2 n          /              I T

How can the angle be the same?  The answer is that the `/' operation
loses information about the signs of its inputs.  Because the quotient
is negative, we know exactly one of the inputs was negative, but we
can't tell which one.  There is an `f T' [`arctan2'] function which
computes the inverse tangent of the quotient of a pair of numbers.
Since you feed it the two original numbers, it has enough information
to give you a full 360-degree answer.

     2:  0.89879    1:  116.       3:  116.       2:  116.       1:  180.
     1:  -0.43837       .          2:  -0.89879   1:  -64.           .
         .                         1:  0.43837        .
                                       .

         U U            f T         M-<RET> M-2 n       f T            -

The resulting angles differ by 180 degrees; in other words, they point
in opposite directions, just as we would expect.

   The <META>-<RET> we used in the third step is the "last-arguments"
command.  It is sort of like Undo, except that it restores the
arguments of the last command to the stack without removing the
command's result.  It is useful in situations like this one, where we
need to do several operations on the same inputs.  We could have
accomplished the same thing by using `M-2 <RET>' to duplicate the top
two stack elements right after the `U U', then a pair of `M-<TAB>'
commands to cycle the 116 up around the duplicates.

   A similar identity is supposed to hold for hyperbolic sines and
cosines, except that it is the _difference_ `cosh(x)^2 - sinh(x)^2'
that always equals one.  Let's try to verify this identity.

     2:  -64        2:  -64        2:  -64        2:  9.7192e54  2:  9.7192e54
     1:  -64        1:  -3.1175e27 1:  9.7192e54  1:  -64        1:  9.7192e54
         .              .              .              .              .

      64 n <RET> <RET>      H C            2 ^            <TAB>            H S 2 ^

Something's obviously wrong, because when we subtract these numbers the
answer will clearly be zero!  But if you think about it, if these
numbers _did_ differ by one, it would be in the 55th decimal place.
The difference we seek has been lost entirely to roundoff error.

   We could verify this hypothesis by doing the actual calculation with,
say, 60 decimal places of precision.  This will be slow, but not
enormously so.  Try it if you wish; sure enough, the answer is 0.99999,
reasonably close to 1.

   Of course, a more reasonable way to verify the identity is to use a
more reasonable value for `x'!

   Some Calculator commands use the Hyperbolic prefix for other
purposes.  The logarithm and exponential functions, for example, work
to the base `e' normally but use base-10 instead if you use the
Hyperbolic prefix.

     1:  1000       1:  6.9077     1:  1000       1:  3
         .              .              .              .

         1000           L              U              H L

First, we mistakenly compute a natural logarithm.  Then we undo and
compute a common logarithm instead.

   The `B' key computes a general base-B logarithm for any value of B.

     2:  1000       1:  3          1:  1000.      2:  1000.      1:  6.9077
     1:  10             .              .          1:  2.71828        .
         .                                            .

      1000 <RET> 10       B              H E            H P            B

Here we first use `B' to compute the base-10 logarithm, then use the
"hyperbolic" exponential as a cheap hack to recover the number 1000,
then use `B' again to compute the natural logarithm.  Note that `P'
with the hyperbolic prefix pushes the constant `e' onto the stack.

   You may have noticed that both times we took the base-10 logarithm
of 1000, we got an exact integer result.  Calc always tries to give an
exact rational result for calculations involving rational numbers where
possible.  But when we used `H E', the result was a floating-point
number for no apparent reason.  In fact, if we had computed `10 <RET> 3
^' we _would_ have gotten an exact integer 1000.  But the `H E' command
is rigged to generate a floating-point result all of the time so that
`1000 H E' will not waste time computing a thousand-digit integer when
all you probably wanted was `1e1000'.

   (*) *Exercise 2.*  Find a pair of integer inputs to the `B' command
for which Calc could find an exact rational result but doesn't.  *Note
2: Arithmetic Answer 2. (*)

   The Calculator also has a set of functions relating to combinatorics
and statistics.  You may be familiar with the "factorial" function,
which computes the product of all the integers up to a given number.

     1:  100        1:  93326215443...    1:  100.       1:  9.3326e157
         .              .                     .              .

         100            !                     U c f          !

Recall, the `c f' command converts the integer or fraction at the top
of the stack to floating-point format.  If you take the factorial of a
floating-point number, you get a floating-point result accurate to the
current precision.  But if you give `!' an exact integer, you get an
exact integer result (158 digits long in this case).

   If you take the factorial of a non-integer, Calc uses a generalized
factorial function defined in terms of Euler's Gamma function `gamma(n)'
(which is itself available as the `f g' command).

     3:  4.         3:  24.               1:  5.5        1:  52.342777847
     2:  4.5        2:  52.3427777847         .              .
     1:  5.         1:  120.
         .              .

                        M-3 !              M-0 <DEL> 5.5       f g

Here we verify the identity `N! = gamma(N+1)'.

   The binomial coefficient N-choose-M is defined by `n! / m! (n-m)!'
for all reals `n' and `m'.  The intermediate results in this formula
can become quite large even if the final result is small; the `k c'
command computes a binomial coefficient in a way that avoids large
intermediate values.

   The `k' prefix key defines several common functions out of
combinatorics and number theory.  Here we compute the binomial
coefficient 30-choose-20, then determine its prime factorization.

     2:  30         1:  30045015   1:  [3, 3, 5, 7, 11, 13, 23, 29]
     1:  20             .              .
         .

      30 <RET> 20         k c            k f

You can verify these prime factors by using `V R *' to multiply
together the elements of this vector.  The result is the original
number, 30045015.

   Suppose a program you are writing needs a hash table with at least
10000 entries.  It's best to use a prime number as the actual size of a
hash table.  Calc can compute the next prime number after 10000:

     1:  10000      1:  10007      1:  9973
         .              .              .

         10000          k n            I k n

Just for kicks we've also computed the next prime _less_ than 10000.

   *Note Financial Functions::, for a description of the Calculator
commands that deal with business and financial calculations (functions
like `pv', `rate', and `sln').

   *Note Binary Functions::, to read about the commands for operating
on binary numbers (like `and', `xor', and `lsh').

File: calc,  Node: Vector/Matrix Tutorial,  Next: Types Tutorial,  Prev: Arithmetic Tutorial,  Up: Tutorial

4.3 Vector/Matrix Tutorial
==========================

A "vector" is a list of numbers or other Calc data objects.  Calc
provides a large set of commands that operate on vectors.  Some are
familiar operations from vector analysis.  Others simply treat a vector
as a list of objects.

* Menu:

* Vector Analysis Tutorial::
* Matrix Tutorial::
* List Tutorial::

File: calc,  Node: Vector Analysis Tutorial,  Next: Matrix Tutorial,  Prev: Vector/Matrix Tutorial,  Up: Vector/Matrix Tutorial

4.3.1 Vector Analysis
---------------------

If you add two vectors, the result is a vector of the sums of the
elements, taken pairwise.

     1:  [1, 2, 3]     2:  [1, 2, 3]     1:  [8, 8, 3]
         .             1:  [7, 6, 0]         .
                           .

         [1,2,3]  s 1      [7 6 0]  s 2      +

Note that we can separate the vector elements with either commas or
spaces.  This is true whether we are using incomplete vectors or
algebraic entry.  The `s 1' and `s 2' commands save these vectors so we
can easily reuse them later.

   If you multiply two vectors, the result is the sum of the products
of the elements taken pairwise.  This is called the "dot product" of
the vectors.

     2:  [1, 2, 3]     1:  19
     1:  [7, 6, 0]         .
         .

         r 1 r 2           *

   The dot product of two vectors is equal to the product of their
lengths times the cosine of the angle between them.  (Here the vector
is interpreted as a line from the origin `(0,0,0)' to the specified
point in three-dimensional space.)  The `A' (absolute value) command
can be used to compute the length of a vector.

     3:  19            3:  19          1:  0.550782    1:  56.579
     2:  [1, 2, 3]     2:  3.741657        .               .
     1:  [7, 6, 0]     1:  9.219544
         .                 .

         M-<RET>             M-2 A          * /             I C

First we recall the arguments to the dot product command, then we
compute the absolute values of the top two stack entries to obtain the
lengths of the vectors, then we divide the dot product by the product
of the lengths to get the cosine of the angle.  The inverse cosine
finds that the angle between the vectors is about 56 degrees.

   The "cross product" of two vectors is a vector whose length is the
product of the lengths of the inputs times the sine of the angle
between them, and whose direction is perpendicular to both input
vectors.  Unlike the dot product, the cross product is defined only for
three-dimensional vectors.  Let's double-check our computation of the
angle using the cross product.

     2:  [1, 2, 3]  3:  [-18, 21, -8]  1:  [-0.52, 0.61, -0.23]  1:  56.579
     1:  [7, 6, 0]  2:  [1, 2, 3]          .                         .
         .          1:  [7, 6, 0]
                        .

         r 1 r 2        V C  s 3  M-<RET>    M-2 A * /                 A I S

First we recall the original vectors and compute their cross product,
which we also store for later reference.  Now we divide the vector by
the product of the lengths of the original vectors.  The length of this
vector should be the sine of the angle; sure enough, it is!

   Vector-related commands generally begin with the `v' prefix key.
Some are uppercase letters and some are lowercase.  To make it easier
to type these commands, the shift-`V' prefix key acts the same as the
`v' key.  (*Note General Mode Commands::, for a way to make all prefix
keys have this property.)

   If we take the dot product of two perpendicular vectors we expect to
get zero, since the cosine of 90 degrees is zero.  Let's check that the
cross product is indeed perpendicular to both inputs:

     2:  [1, 2, 3]      1:  0          2:  [7, 6, 0]      1:  0
     1:  [-18, 21, -8]      .          1:  [-18, 21, -8]      .
         .                                 .

         r 1 r 3            *          <DEL> r 2 r 3            *

   (*) *Exercise 1.*  Given a vector on the top of the stack, what
keystrokes would you use to "normalize" the vector, i.e., to reduce its
length to one without changing its direction?  *Note 1: Vector Answer
1. (*)

   (*) *Exercise 2.*  Suppose a certain particle can be at any of
several positions along a ruler.  You have a list of those positions in
the form of a vector, and another list of the probabilities for the
particle to be at the corresponding positions.  Find the average
position of the particle.  *Note 2: Vector Answer 2. (*)

File: calc,  Node: Matrix Tutorial,  Next: List Tutorial,  Prev: Vector Analysis Tutorial,  Up: Vector/Matrix Tutorial

4.3.2 Matrices
--------------

A "matrix" is just a vector of vectors, all the same length.  This
means you can enter a matrix using nested brackets.  You can also use
the semicolon character to enter a matrix.  We'll show both methods
here:

     1:  [ [ 1, 2, 3 ]             1:  [ [ 1, 2, 3 ]
           [ 4, 5, 6 ] ]                 [ 4, 5, 6 ] ]
         .                             .

       [[1 2 3] [4 5 6]]             ' [1 2 3; 4 5 6] <RET>

We'll be using this matrix again, so type `s 4' to save it now.

   Note that semicolons work with incomplete vectors, but they work
better in algebraic entry.  That's why we use the apostrophe in the
second example.

   When two matrices are multiplied, the lefthand matrix must have the
same number of columns as the righthand matrix has rows.  Row `i',
column `j' of the result is effectively the dot product of row `i' of
the left matrix by column `j' of the right matrix.

   If we try to duplicate this matrix and multiply it by itself, the
dimensions are wrong and the multiplication cannot take place:

     1:  [ [ 1, 2, 3 ]   * [ [ 1, 2, 3 ]
           [ 4, 5, 6 ] ]     [ 4, 5, 6 ] ]
         .

         <RET> *

Though rather hard to read, this is a formula which shows the product
of two matrices.  The `*' function, having invalid arguments, has been
left in symbolic form.

   We can multiply the matrices if we "transpose" one of them first.

     2:  [ [ 1, 2, 3 ]       1:  [ [ 14, 32 ]      1:  [ [ 17, 22, 27 ]
           [ 4, 5, 6 ] ]           [ 32, 77 ] ]          [ 22, 29, 36 ]
     1:  [ [ 1, 4 ]              .                       [ 27, 36, 45 ] ]
           [ 2, 5 ]                                    .
           [ 3, 6 ] ]
         .

         U v t                   *                     U <TAB> *

   Matrix multiplication is not commutative; indeed, switching the
order of the operands can even change the dimensions of the result
matrix, as happened here!

   If you multiply a plain vector by a matrix, it is treated as a
single row or column depending on which side of the matrix it is on.
The result is a plain vector which should also be interpreted as a row
or column as appropriate.

     2:  [ [ 1, 2, 3 ]      1:  [14, 32]
           [ 4, 5, 6 ] ]        .
     1:  [1, 2, 3]
         .

         r 4 r 1                *

   Multiplying in the other order wouldn't work because the number of
rows in the matrix is different from the number of elements in the
vector.

   (*) *Exercise 1.*  Use `*' to sum along the rows of the above 2x3
matrix to get `[6, 15]'.  Now use `*' to sum along the columns to get
`[5, 7, 9]'.  *Note 1: Matrix Answer 1. (*)

   An "identity matrix" is a square matrix with ones along the diagonal
and zeros elsewhere.  It has the property that multiplication by an
identity matrix, on the left or on the right, always produces the
original matrix.

     1:  [ [ 1, 2, 3 ]      2:  [ [ 1, 2, 3 ]      1:  [ [ 1, 2, 3 ]
           [ 4, 5, 6 ] ]          [ 4, 5, 6 ] ]          [ 4, 5, 6 ] ]
         .                  1:  [ [ 1, 0, 0 ]          .
                                  [ 0, 1, 0 ]
                                  [ 0, 0, 1 ] ]
                                .

         r 4                    v i 3 <RET>              *

   If a matrix is square, it is often possible to find its "inverse",
that is, a matrix which, when multiplied by the original matrix, yields
an identity matrix.  The `&' (reciprocal) key also computes the inverse
of a matrix.

     1:  [ [ 1, 2, 3 ]      1:  [ [   -2.4,     1.2,   -0.2 ]
           [ 4, 5, 6 ]            [    2.8,    -1.4,    0.4 ]
           [ 7, 6, 0 ] ]          [ -0.73333, 0.53333, -0.2 ] ]
         .                      .

         r 4 r 2 |  s 5         &

The vertical bar `|' "concatenates" numbers, vectors, and matrices
together.  Here we have used it to add a new row onto our matrix to
make it square.

   We can multiply these two matrices in either order to get an
identity.

     1:  [ [ 1., 0., 0. ]      1:  [ [ 1., 0., 0. ]
           [ 0., 1., 0. ]            [ 0., 1., 0. ]
           [ 0., 0., 1. ] ]          [ 0., 0., 1. ] ]
         .                         .

         M-<RET>  *                  U <TAB> *

   Matrix inverses are related to systems of linear equations in
algebra.  Suppose we had the following set of equations:

         a + 2b + 3c = 6
        4a + 5b + 6c = 2
        7a + 6b      = 3

This can be cast into the matrix equation,

        [ [ 1, 2, 3 ]     [ [ a ]     [ [ 6 ]
          [ 4, 5, 6 ]   *   [ b ]   =   [ 2 ]
          [ 7, 6, 0 ] ]     [ c ] ]     [ 3 ] ]

   We can solve this system of equations by multiplying both sides by
the inverse of the matrix.  Calc can do this all in one step:

     2:  [6, 2, 3]          1:  [-12.6, 15.2, -3.93333]
     1:  [ [ 1, 2, 3 ]          .
           [ 4, 5, 6 ]
           [ 7, 6, 0 ] ]
         .

         [6,2,3] r 5            /

The result is the `[a, b, c]' vector that solves the equations.
(Dividing by a square matrix is equivalent to multiplying by its
inverse.)

   Let's verify this solution:

     2:  [ [ 1, 2, 3 ]                1:  [6., 2., 3.]
           [ 4, 5, 6 ]                    .
           [ 7, 6, 0 ] ]
     1:  [-12.6, 15.2, -3.93333]
         .

         r 5  <TAB>                         *

Note that we had to be careful about the order in which we multiplied
the matrix and vector.  If we multiplied in the other order, Calc would
assume the vector was a row vector in order to make the dimensions come
out right, and the answer would be incorrect.  If you don't feel safe
letting Calc take either interpretation of your vectors, use explicit
Nx1 or 1xN matrices instead.  In this case, you would enter the
original column vector as `[[6], [2], [3]]' or `[6; 2; 3]'.

   (*) *Exercise 2.*  Algebraic entry allows you to make vectors and
matrices that include variables.  Solve the following system of
equations to get expressions for `x' and `y' in terms of `a' and `b'.

        x + a y = 6
        x + b y = 10

*Note 2: Matrix Answer 2. (*)

   (*) *Exercise 3.*  A system of equations is "over-determined" if it
has more equations than variables.  It is often the case that there are
no values for the variables that will satisfy all the equations at
once, but it is still useful to find a set of values which "nearly"
satisfy all the equations.  In terms of matrix equations, you can't
solve `A X = B' directly because the matrix `A' is not square for an
over-determined system.  Matrix inversion works only for square
matrices.  One common trick is to multiply both sides on the left by
the transpose of `A': `trn(A)*A*X = trn(A)*B'.  Now `trn(A)*A' is a
square matrix so a solution is possible.  It turns out that the `X'
vector you compute in this way will be a "least-squares" solution,
which can be regarded as the "closest" solution to the set of
equations.  Use Calc to solve the following over-determined system:

         a + 2b + 3c = 6
        4a + 5b + 6c = 2
        7a + 6b      = 3
        2a + 4b + 6c = 11

*Note 3: Matrix Answer 3. (*)

File: calc,  Node: List Tutorial,  Prev: Matrix Tutorial,  Up: Vector/Matrix Tutorial

4.3.3 Vectors as Lists
----------------------

Although Calc has a number of features for manipulating vectors and
matrices as mathematical objects, you can also treat vectors as simple
lists of values.  For example, we saw that the `k f' command returns a
vector which is a list of the prime factors of a number.

   You can pack and unpack stack entries into vectors:

     3:  10         1:  [10, 20, 30]     3:  10
     2:  20             .                2:  20
     1:  30                              1:  30
         .                                   .

                        M-3 v p              v u

   You can also build vectors out of consecutive integers, or out of
many copies of a given value:

     1:  [1, 2, 3, 4]    2:  [1, 2, 3, 4]    2:  [1, 2, 3, 4]
         .               1:  17              1:  [17, 17, 17, 17]
                             .                   .

         v x 4 <RET>           17                  v b 4 <RET>

   You can apply an operator to every element of a vector using the
"map" command.

     1:  [17, 34, 51, 68]   1:  [289, 1156, 2601, 4624]  1:  [17, 34, 51, 68]
         .                      .                            .

         V M *                  2 V M ^                      V M Q

In the first step, we multiply the vector of integers by the vector of
17's elementwise.  In the second step, we raise each element to the
power two.  (The general rule is that both operands must be vectors of
the same length, or else one must be a vector and the other a plain
number.)  In the final step, we take the square root of each element.

   (*) *Exercise 1.*  Compute a vector of powers of two from `2^-4' to
`2^4'.  *Note 1: List Answer 1. (*)

   You can also "reduce" a binary operator across a vector.  For
example, reducing `*' computes the product of all the elements in the
vector:

     1:  123123     1:  [3, 7, 11, 13, 41]      1:  123123
         .              .                           .

         123123         k f                         V R *

In this example, we decompose 123123 into its prime factors, then
multiply those factors together again to yield the original number.

   We could compute a dot product "by hand" using mapping and reduction:

     2:  [1, 2, 3]     1:  [7, 12, 0]     1:  19
     1:  [7, 6, 0]         .                  .
         .

         r 1 r 2           V M *              V R +

Recalling two vectors from the previous section, we compute the sum of
pairwise products of the elements to get the same answer for the dot
product as before.

   A slight variant of vector reduction is the "accumulate" operation,
`V U'.  This produces a vector of the intermediate results from a
corresponding reduction.  Here we compute a table of factorials:

     1:  [1, 2, 3, 4, 5, 6]    1:  [1, 2, 6, 24, 120, 720]
         .                         .

         v x 6 <RET>                 V U *

   Calc allows vectors to grow as large as you like, although it gets
rather slow if vectors have more than about a hundred elements.
Actually, most of the time is spent formatting these large vectors for
display, not calculating on them.  Try the following experiment (if
your computer is very fast you may need to substitute a larger vector
size).

     1:  [1, 2, 3, 4, ...      1:  [2, 3, 4, 5, ...
         .                         .

         v x 500 <RET>               1 V M +

   Now press `v .' (the letter `v', then a period) and try the
experiment again.  In `v .' mode, long vectors are displayed
"abbreviated" like this:

     1:  [1, 2, 3, ..., 500]   1:  [2, 3, 4, ..., 501]
         .                         .

         v x 500 <RET>               1 V M +

(where now the `...' is actually part of the Calc display).  You will
find both operations are now much faster.  But notice that even in
`v .' mode, the full vectors are still shown in the Trail.  Type `t .'
to cause the trail to abbreviate as well, and try the experiment one
more time.  Operations on long vectors are now quite fast!  (But of
course if you use `t .' you will lose the ability to get old vectors
back using the `t y' command.)

   An easy way to view a full vector when `v .' mode is active is to
press ``' (back-quote) to edit the vector; editing always works with
the full, unabbreviated value.

   As a larger example, let's try to fit a straight line to some data,
using the method of least squares.  (Calc has a built-in command for
least-squares curve fitting, but we'll do it by hand here just to
practice working with vectors.)  Suppose we have the following list of
values in a file we have loaded into Emacs:

       x        y
      ---      ---
      1.34    0.234
      1.41    0.298
      1.49    0.402
      1.56    0.412
      1.64    0.466
      1.73    0.473
      1.82    0.601
      1.91    0.519
      2.01    0.603
      2.11    0.637
      2.22    0.645
      2.33    0.705
      2.45    0.917
      2.58    1.009
      2.71    0.971
      2.85    1.062
      3.00    1.148
      3.15    1.157
      3.32    1.354

If you are reading this tutorial in printed form, you will find it
easiest to press `C-x * i' to enter the on-line Info version of the
manual and find this table there.  (Press `g', then type `List
Tutorial', to jump straight to this section.)

   Position the cursor at the upper-left corner of this table, just to
the left of the `1.34'.  Press `C-@' to set the mark.  (On your system
this may be `C-2', `C-<SPC>', or `NUL'.)  Now position the cursor to
the lower-right, just after the `1.354'.  You have now defined this
region as an Emacs "rectangle."  Still in the Info buffer, type `C-x *
r'.  This command (`calc-grab-rectangle') will pop you back into the
Calculator, with the contents of the rectangle you specified in the
form of a matrix.

     1:  [ [ 1.34, 0.234 ]
           [ 1.41, 0.298 ]
           ...

(You may wish to use `v .' mode to abbreviate the display of this large
matrix.)

   We want to treat this as a pair of lists.  The first step is to
transpose this matrix into a pair of rows.  Remember, a matrix is just
a vector of vectors.  So we can unpack the matrix into a pair of row
vectors on the stack.

     1:  [ [ 1.34,  1.41,  1.49,  ... ]     2:  [1.34, 1.41, 1.49, ... ]
           [ 0.234, 0.298, 0.402, ... ] ]   1:  [0.234, 0.298, 0.402, ... ]
         .                                      .

         v t                                    v u

Let's store these in quick variables 1 and 2, respectively.

     1:  [1.34, 1.41, 1.49, ... ]        .
         .

         t 2                             t 1

(Recall that `t 2' is a variant of `s 2' that removes the stored value
from the stack.)

   In a least squares fit, the slope `m' is given by the formula

     m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)

where `sum(x)' represents the sum of all the values of `x'.  While
there is an actual `sum' function in Calc, it's easier to sum a vector
using a simple reduction.  First, let's compute the four different sums
that this formula uses.

     1:  41.63                 1:  98.0003
         .                         .

      r 1 V R +   t 3           r 1 2 V M ^ V R +   t 4

     1:  13.613                1:  33.36554
         .                         .

      r 2 V R +   t 5           r 1 r 2 V M * V R +   t 6

These are `sum(x)', `sum(x^2)', `sum(y)', and `sum(x y)', respectively.
(We could have used `*' to compute `sum(x^2)' and `sum(x y)'.)

   Finally, we also need `N', the number of data points.  This is just
the length of either of our lists.

     1:  19
         .

      r 1 v l   t 7

(That's `v' followed by a lower-case `l'.)

   Now we grind through the formula:

     1:  633.94526  2:  633.94526  1:  67.23607
         .          1:  566.70919      .
                        .

      r 7 r 6 *      r 3 r 5 *         -

     2:  67.23607   3:  67.23607   2:  67.23607   1:  0.52141679
     1:  1862.0057  2:  1862.0057  1:  128.9488       .
         .          1:  1733.0569      .
                        .

      r 7 r 4 *      r 3 2 ^           -              /   t 8

   That gives us the slope `m'.  The y-intercept `b' can now be found
with the simple formula,

     b = (sum(y) - m sum(x)) / N

     1:  13.613     2:  13.613     1:  -8.09358   1:  -0.425978
         .          1:  21.70658       .              .
                        .

        r 5            r 8 r 3 *       -              r 7 /   t 9

   Let's "plot" this straight line approximation, `m x + b', and
compare it with the original data.

     1:  [0.699, 0.735, ... ]    1:  [0.273, 0.309, ... ]
         .                           .

         r 1 r 8 *                   r 9 +    s 0

Notice that multiplying a vector by a constant, and adding a constant
to a vector, can be done without mapping commands since these are
common operations from vector algebra.  As far as Calc is concerned,
we've just been doing geometry in 19-dimensional space!

   We can subtract this vector from our original `y' vector to get a
feel for the error of our fit.  Let's find the maximum error:

     1:  [0.0387, 0.0112, ... ]   1:  [0.0387, 0.0112, ... ]   1:  0.0897
         .                            .                            .

         r 2 -                        V M A                        V R X

First we compute a vector of differences, then we take the absolute
values of these differences, then we reduce the `max' function across
the vector.  (The `max' function is on the two-key sequence `f x';
because it is so common to use `max' in a vector operation, the letters
`X' and `N' are also accepted for `max' and `min' in this context.  In
general, you answer the `V M' or `V R' prompt with the actual key
sequence that invokes the function you want.  You could have typed `V R
f x' or even `V R x max <RET>' if you had preferred.)

   If your system has the GNUPLOT program, you can see graphs of your
data and your straight line to see how well they match.  (If you have
GNUPLOT 3.0 or higher, the following instructions will work regardless
of the kind of display you have.  Some GNUPLOT 2.0, non-X-windows
systems may require additional steps to view the graphs.)

   Let's start by plotting the original data.  Recall the "X" and "Y"
vectors onto the stack and press `g f'.  This "fast" graphing command
does everything you need to do for simple, straightforward plotting of
data.

     2:  [1.34, 1.41, 1.49, ... ]
     1:  [0.234, 0.298, 0.402, ... ]
         .

         r 1 r 2    g f

   If all goes well, you will shortly get a new window containing a
graph of the data.  (If not, contact your GNUPLOT or Calc installer to
find out what went wrong.)  In the X window system, this will be a
separate graphics window.  For other kinds of displays, the default is
to display the graph in Emacs itself using rough character graphics.
Press `q' when you are done viewing the character graphics.

   Next, let's add the line we got from our least-squares fit.  (If you
are reading this tutorial on-line while running Calc, typing `g a' may
cause the tutorial to disappear from its window and be replaced by a
buffer named `*Gnuplot Commands*'.  The tutorial will reappear when you
terminate GNUPLOT by typing `g q'.)

     2:  [1.34, 1.41, 1.49, ... ]
     1:  [0.273, 0.309, 0.351, ... ]
         .

         <DEL> r 0    g a  g p

   It's not very useful to get symbols to mark the data points on this
second curve; you can type `g S g p' to remove them.  Type `g q' when
you are done to remove the X graphics window and terminate GNUPLOT.

   (*) *Exercise 2.*  An earlier exercise showed how to do least
squares fitting to a general system of equations.  Our 19 data points
are really 19 equations of the form `y_i = m x_i + b' for different
pairs of `(x_i,y_i)'.  Use the matrix-transpose method to solve for `m'
and `b', duplicating the above result.  *Note 2: List Answer 2. (*)

   (*) *Exercise 3.*  If the input data do not form a rectangle, you
can use `C-x * g' (`calc-grab-region') to grab the data the way Emacs
normally works with regions--it reads left-to-right, top-to-bottom,
treating line breaks the same as spaces.  Use this command to find the
geometric mean of the following numbers.  (The geometric mean is the
Nth root of the product of N numbers.)

     2.3  6  22  15.1  7
       15  14  7.5
       2.5

The `C-x * g' command accepts numbers separated by spaces or commas,
with or without surrounding vector brackets.  *Note 3: List Answer 3.
(*)

   As another example, a theorem about binomial coefficients tells us
that the alternating sum of binomial coefficients N-choose-0 minus
N-choose-1 plus N-choose-2, and so on up to N-choose-N, always comes
out to zero.  Let's verify this for `n=6'.

     1:  [1, 2, 3, 4, 5, 6, 7]     1:  [0, 1, 2, 3, 4, 5, 6]
         .                             .

         v x 7 <RET>                     1 -

     1:  [1, -6, 15, -20, 15, -6, 1]          1:  0
         .                                        .

         V M ' (-1)^$ choose(6,$) <RET>             V R +

   The `V M '' command prompts you to enter any algebraic expression to
define the function to map over the vector.  The symbol `$' inside this
expression represents the argument to the function.  The Calculator
applies this formula to each element of the vector, substituting each
element's value for the `$' sign(s) in turn.

   To define a two-argument function, use `$$' for the first argument
and `$' for the second:  `V M ' $$-$ <RET>' is equivalent to `V M -'.
This is analogous to regular algebraic entry, where `$$' would refer to
the next-to-top stack entry and `$' would refer to the top stack entry,
and `' $$-$ <RET>' would act exactly like `-'.

   Notice that the `V M '' command has recorded two things in the
trail:  The result, as usual, and also a funny-looking thing marked
`oper' that represents the operator function you typed in.  The
function is enclosed in `< >' brackets, and the argument is denoted by
a `#' sign.  If there were several arguments, they would be shown as
`#1', `#2', and so on.  (For example, `V M ' $$-$' will put the
function `<#1 - #2>' on the trail.)  This object is a "nameless
function"; you can use nameless `< >' notation to answer the `V M ''
prompt if you like.  Nameless function notation has the interesting,
occasionally useful property that a nameless function is not actually
evaluated until it is used.  For example, `V M ' $+random(2.0)'
evaluates `random(2.0)' once and adds that random number to all elements
of the vector, but `V M ' <#+random(2.0)>' evaluates the `random(2.0)'
separately for each vector element.

   Another group of operators that are often useful with `V M' are the
relational operators:  `a =', for example, compares two numbers and
gives the result 1 if they are equal, or 0 if not.  Similarly, `a <'
checks for one number being less than another.

   Other useful vector operations include `v v', to reverse a vector
end-for-end; `V S', to sort the elements of a vector into increasing
order; and `v r' and `v c', to extract one row or column of a matrix,
or (in both cases) to extract one element of a plain vector.  With a
negative argument, `v r' and `v c' instead delete one row, column, or
vector element.

   (*) *Exercise 4.*  The `k'th "divisor function" is the sum of the
`k'th powers of all the divisors of an integer `n'.  Figure out a
method for computing the divisor function for reasonably small values
of `n'.  As a test, the 0th and 1st divisor functions of 30 are 8 and
72, respectively.  *Note 4: List Answer 4. (*)

   (*) *Exercise 5.*  The `k f' command produces a list of prime
factors for a number.  Sometimes it is important to know that a number
is "square-free", i.e., that no prime occurs more than once in its list
of prime factors.  Find a sequence of keystrokes to tell if a number is
square-free; your method should leave 1 on the stack if it is, or 0 if
it isn't.  *Note 5: List Answer 5. (*)

   (*) *Exercise 6.*  Build a list of lists that looks like the
following diagram.  (You may wish to use the `v /' command to enable
multi-line display of vectors.)

     1:  [ [1],
           [1, 2],
           [1, 2, 3],
           [1, 2, 3, 4],
           [1, 2, 3, 4, 5],
           [1, 2, 3, 4, 5, 6] ]

*Note 6: List Answer 6. (*)

   (*) *Exercise 7.*  Build the following list of lists.

     1:  [ [0],
           [1, 2],
           [3, 4, 5],
           [6, 7, 8, 9],
           [10, 11, 12, 13, 14],
           [15, 16, 17, 18, 19, 20] ]

*Note 7: List Answer 7. (*)

   (*) *Exercise 8.*  Compute a list of values of Bessel's `J1'
function `besJ(1,x)' for `x' from 0 to 5 in steps of 0.25.  Find the
value of `x' (from among the above set of values) for which `besJ(1,x)'
is a maximum.  Use an "automatic" method, i.e., just reading along the
list by hand to find the largest value is not allowed!  (There is an `a
X' command which does this kind of thing automatically; *note Numerical
Solutions::.)  *Note 8: List Answer 8. (*)

   (*) *Exercise 9.*  You are given an integer in the range `0 <= N <
10^m' for `m=12' (i.e., an integer of less than twelve digits).
Convert this integer into a vector of `m' digits, each in the range
from 0 to 9.  In vector-of-digits notation, add one to this integer to
produce a vector of `m+1' digits (since there could be a carry out of
the most significant digit).  Convert this vector back into a regular
integer.  A good integer to try is 25129925999.  *Note 9: List Answer
9. (*)

   (*) *Exercise 10.*  Your friend Joe tried to use `V R a =' to test
if all numbers in a list were equal.  What happened?  How would you do
this test?  *Note 10: List Answer 10. (*)

   (*) *Exercise 11.*  The area of a circle of radius one is `pi'.  The
area of the 2x2 square that encloses that circle is 4.  So if we throw
N darts at random points in the square, about `pi/4' of them will land
inside the circle.  This gives us an entertaining way to estimate the
value of `pi'.  The `k r' command picks a random number between zero
and the value on the stack.  We could get a random floating-point
number between -1 and 1 by typing `2.0 k r 1 -'.  Build a vector of 100
random `(x,y)' points in this square, then use vector mapping and
reduction to count how many points lie inside the unit circle.  Hint:
Use the `v b' command.  *Note 11: List Answer 11. (*)

   (*) *Exercise 12.*  The "matchstick problem" provides another way to
calculate `pi'.  Say you have an infinite field of vertical lines with
a spacing of one inch.  Toss a one-inch matchstick onto the field.  The
probability that the matchstick will land crossing a line turns out to
be `2/pi'.  Toss 100 matchsticks to estimate `pi'.  (If you want still
more fun, the probability that the GCD (`k g') of two large integers is
one turns out to be `6/pi^2'.  That provides yet another way to
estimate `pi'.)  *Note 12: List Answer 12. (*)

   (*) *Exercise 13.*  An algebraic entry of a string in double-quote
marks, `"hello"', creates a vector of the numerical (ASCII) codes of
the characters (here, `[104, 101, 108, 108, 111]').  Sometimes it is
convenient to compute a "hash code" of a string, which is just an
integer that represents the value of that string.  Two equal strings
have the same hash code; two different strings "probably" have
different hash codes.  (For example, Calc has over 400 function names,
but Emacs can quickly find the definition for any given name because it
has sorted the functions into "buckets" by their hash codes.  Sometimes
a few names will hash into the same bucket, but it is easier to search
among a few names than among all the names.)  One popular hash function
is computed as follows:  First set `h = 0'.  Then, for each character
from the string in turn, set `h = 3h + c_i' where `c_i' is the
character's ASCII code.  If we have 511 buckets, we then take the hash
code modulo 511 to get the bucket number.  Develop a simple command or
commands for converting string vectors into hash codes.  The hash code
for `"Testing, 1, 2, 3"' is 1960915098, which modulo 511 is 121.  *Note
13: List Answer 13. (*)

   (*) *Exercise 14.*  The `H V R' and `H V U' commands do nested
function evaluations.  `H V U' takes a starting value and a number of
steps N from the stack; it then applies the function you give to the
starting value 0, 1, 2, up to N times and returns a vector of the
results.  Use this command to create a "random walk" of 50 steps.
Start with the two-dimensional point `(0,0)'; then take one step a
random distance between -1 and 1 in both `x' and `y'; then take another
step, and so on.  Use the `g f' command to display this random walk.
Now modify your random walk to walk a unit distance, but in a random
direction, at each step.  (Hint:  The `sincos' function returns a
vector of the cosine and sine of an angle.)  *Note 14: List Answer 14.
(*)

File: calc,  Node: Types Tutorial,  Next: Algebra Tutorial,  Prev: Vector/Matrix Tutorial,  Up: Tutorial

4.4 Types Tutorial
==================

Calc understands a variety of data types as well as simple numbers.  In
this section, we'll experiment with each of these types in turn.

   The numbers we've been using so far have mainly been either
"integers" or "floats".  We saw that floats are usually a good
approximation to the mathematical concept of real numbers, but they are
only approximations and are susceptible to roundoff error.  Calc also
supports "fractions", which can exactly represent any rational number.

     1:  3628800    2:  3628800    1:  518400:7   1:  518414:7   1:  7:518414
         .          1:  49             .              .              .
                        .

         10 !           49 <RET>         :              2 +            &

The `:' command divides two integers to get a fraction; `/' would
normally divide integers to get a floating-point result.  Notice we had
to type <RET> between the `49' and the `:' since the `:' would
otherwise be interpreted as part of a fraction beginning with 49.

   You can convert between floating-point and fractional format using
`c f' and `c F':

     1:  1.35027217629e-5    1:  7:518414
         .                       .

         c f                     c F

   The `c F' command replaces a floating-point number with the
"simplest" fraction whose floating-point representation is the same, to
within the current precision.

     1:  3.14159265359   1:  1146408:364913   1:  3.1416   1:  355:113
         .                   .                    .            .

         P                   c F      <DEL>       p 5 <RET> P      c F

   (*) *Exercise 1.*  A calculation has produced the result
1.26508260337.  You suspect it is the square root of the product of
`pi' and some rational number.  Is it?  (Be sure to allow for roundoff
error!)  *Note 1: Types Answer 1. (*)

   "Complex numbers" can be stored in both rectangular and polar form.

     1:  -9     1:  (0, 3)    1:  (3; 90.)   1:  (6; 90.)   1:  (2.4495; 45.)
         .          .             .              .              .

         9 n        Q             c p            2 *            Q

The square root of -9 is by default rendered in rectangular form
(`0 + 3i'), but we can convert it to polar form (3 with a phase angle
of 90 degrees).  All the usual arithmetic and scientific operations are
defined on both types of complex numbers.

   Another generalized kind of number is "infinity".  Infinity isn't
really a number, but it can sometimes be treated like one.  Calc uses
the symbol `inf' to represent positive infinity, i.e., a value greater
than any real number.  Naturally, you can also write `-inf' for minus
infinity, a value less than any real number.  The word `inf' can only
be input using algebraic entry.

     2:  inf        2:  -inf       2:  -inf       2:  -inf       1:  nan
     1:  -17        1:  -inf       1:  -inf       1:  inf            .
         .              .              .              .

     ' inf <RET> 17 n     *  <RET>         72 +           A              +

Since infinity is infinitely large, multiplying it by any finite number
(like -17) has no effect, except that since -17 is negative, it changes
a plus infinity to a minus infinity.  ("A huge positive number,
multiplied by -17, yields a huge negative number.")  Adding any finite
number to infinity also leaves it unchanged.  Taking an absolute value
gives us plus infinity again.  Finally, we add this plus infinity to
the minus infinity we had earlier.  If you work it out, you might expect
the answer to be -72 for this.  But the 72 has been completely lost
next to the infinities; by the time we compute `inf - inf' the finite
difference between them, if any, is undetectable.  So we say the result
is "indeterminate", which Calc writes with the symbol `nan' (for Not A
Number).

   Dividing by zero is normally treated as an error, but you can get
Calc to write an answer in terms of infinity by pressing `m i' to turn
on Infinite mode.

     3:  nan        2:  nan        2:  nan        2:  nan        1:  nan
     2:  1          1:  1 / 0      1:  uinf       1:  uinf           .
     1:  0              .              .              .
         .

       1 <RET> 0          /       m i    U /            17 n *         +

Dividing by zero normally is left unevaluated, but after `m i' it
instead gives an infinite result.  The answer is actually `uinf',
"undirected infinity."  If you look at a graph of `1 / x' around
`x = 0', you'll see that it goes toward plus infinity as you approach
zero from above, but toward minus infinity as you approach from below.
Since we said only `1 / 0', Calc knows that the answer is infinite but
not in which direction.  That's what `uinf' means.  Notice that
multiplying `uinf' by a negative number still leaves plain `uinf';
there's no point in saying `-uinf' because the sign of `uinf' is
unknown anyway.  Finally, we add `uinf' to our `nan', yielding `nan'
again.  It's easy to see that, because `nan' means "totally unknown"
while `uinf' means "unknown sign but known to be infinite," the more
mysterious `nan' wins out when it is combined with `uinf', or, for that
matter, with anything else.

   (*) *Exercise 2.*  Predict what Calc will answer for each of these
formulas:  `inf / inf', `exp(inf)', `exp(-inf)', `sqrt(-inf)',
`sqrt(uinf)', `abs(uinf)', `ln(0)'.  *Note 2: Types Answer 2. (*)

   (*) *Exercise 3.*  We saw that `inf - inf = nan', which stands for
an unknown value.  Can `nan' stand for a complex number?  Can it stand
for infinity?  *Note 3: Types Answer 3. (*)

   "HMS forms" represent a value in terms of hours, minutes, and
seconds.

     1:  2@ 30' 0"     1:  3@ 30' 0"     2:  3@ 30' 0"     1:  2.
         .                 .             1:  1@ 45' 0."        .
                                             .

       2@ 30' <RET>          1 +               <RET> 2 /           /

   HMS forms can also be used to hold angles in degrees, minutes, and
seconds.

     1:  0.5        1:  26.56505   1:  26@ 33' 54.18"    1:  0.44721
         .              .              .                     .

         0.5            I T            c h                   S

First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
form, then we take the sine of that angle.  Note that the trigonometric
functions will accept HMS forms directly as input.

   (*) *Exercise 4.*  The Beatles' _Abbey Road_ is 47 minutes and 26
seconds long, and contains 17 songs.  What is the average length of a
song on _Abbey Road_?  If the Extended Disco Version of _Abbey Road_
added 20 seconds to the length of each song, how long would the album
be?  *Note 4: Types Answer 4. (*)

   A "date form" represents a date, or a date and time.  Dates must be
entered using algebraic entry.  Date forms are surrounded by `< >'
symbols; most standard formats for dates are recognized.

     2:  <Sun Jan 13, 1991>                    1:  2.25
     1:  <6:00pm Thu Jan 10, 1991>                 .
         .

     ' <13 Jan 1991>, <1/10/91, 6pm> <RET>           -

In this example, we enter two dates, then subtract to find the number
of days between them.  It is also possible to add an HMS form or a
number (of days) to a date form to get another date form.

     1:  <4:45:59pm Mon Jan 14, 1991>     1:  <2:50:59am Thu Jan 17, 1991>
         .                                    .

         t N                                  2 + 10@ 5' +

The `t N' ("now") command pushes the current date and time on the
stack; then we add two days, ten hours and five minutes to the date and
time.  Other date-and-time related commands include `t J', which does
Julian day conversions, `t W', which finds the beginning of the week in
which a date form lies, and `t I', which increments a date by one or
several months.  *Note Date Arithmetic::, for more.

   (*) *Exercise 5.*  How many days until the next Friday the 13th?
*Note 5: Types Answer 5. (*)

   (*) *Exercise 6.*  How many leap years will there be between now and
the year 10001 A.D.?  *Note 6: Types Answer 6. (*)

   An "error form" represents a mean value with an attached standard
deviation, or error estimate.  Suppose our measurements indicate that a
certain telephone pole is about 30 meters away, with an estimated error
of 1 meter, and 8 meters tall, with an estimated error of 0.2 meters.
What is the slope of a line from here to the top of the pole, and what
is the equivalent angle in degrees?

     1:  8 +/- 0.2    2:  8 +/- 0.2   1:  0.266 +/- 0.011   1:  14.93 +/- 0.594
         .            1:  30 +/- 1        .                     .
                          .

         8 p .2 <RET>       30 p 1          /                     I T

This means that the angle is about 15 degrees, and, assuming our
original error estimates were valid standard deviations, there is about
a 60% chance that the result is correct within 0.59 degrees.

   (*) *Exercise 7.*  The volume of a torus (a donut shape) is
`2 pi^2 R r^2' where `R' is the radius of the circle that defines the
center of the tube and `r' is the radius of the tube itself.  Suppose
`R' is 20 cm and `r' is 4 cm, each known to within 5 percent.  What is
the volume and the relative uncertainty of the volume?  *Note 7: Types
Answer 7. (*)

   An "interval form" represents a range of values.  While an error
form is best for making statistical estimates, intervals give you exact
bounds on an answer.  Suppose we additionally know that our telephone
pole is definitely between 28 and 31 meters away, and that it is
between 7.7 and 8.1 meters tall.

     1:  [7.7 .. 8.1]  2:  [7.7 .. 8.1]  1:  [0.24 .. 0.28]  1:  [13.9 .. 16.1]
         .             1:  [28 .. 31]        .                   .
                           .

       [ 7.7 .. 8.1 ]    [ 28 .. 31 ]        /                   I T

If our bounds were correct, then the angle to the top of the pole is
sure to lie in the range shown.

   The square brackets around these intervals indicate that the
endpoints themselves are allowable values.  In other words, the
distance to the telephone pole is between 28 and 31, _inclusive_.  You
can also make an interval that is exclusive of its endpoints by writing
parentheses instead of square brackets.  You can even make an interval
which is inclusive ("closed") on one end and exclusive ("open") on the
other.

     1:  [1 .. 10)    1:  (0.1 .. 1]   2:  (0.1 .. 1]   1:  (0.2 .. 3)
         .                .            1:  [2 .. 3)         .
                                           .

       [ 1 .. 10 )        &              [ 2 .. 3 )         *

The Calculator automatically keeps track of which end values should be
open and which should be closed.  You can also make infinite or
semi-infinite intervals by using `-inf' or `inf' for one or both
endpoints.

   (*) *Exercise 8.*  What answer would you expect from `1 /
(0 .. 10)'?  What about `1 / (-10 .. 0)'?  What about `1 / [0 .. 10]'
(where the interval actually includes zero)?  What about `1 /
(-10 .. 10)'?  *Note 8: Types Answer 8. (*)

   (*) *Exercise 9.*  Two easy ways of squaring a number are `<RET> *'
and `2 ^'.  Normally these produce the same answer.  Would you expect
this still to hold true for interval forms?  If not, which of these
will result in a larger interval?  *Note 9: Types Answer 9. (*)

   A "modulo form" is used for performing arithmetic modulo M.  For
example, arithmetic involving time is generally done modulo 12 or 24
hours.

     1:  17 mod 24    1:  3 mod 24     1:  21 mod 24    1:  9 mod 24
         .                .                .                .

         17 M 24 <RET>      10 +             n                5 /

In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
new number which, when multiplied by 5 modulo 24, produces the original
number, 21.  If M is prime and the divisor is not a multiple of M, it
is always possible to find such a number.  For non-prime M like 24, it
is only sometimes possible.

     1:  10 mod 24    1:  16 mod 24    1:  1000000...   1:  16
         .                .                .                .

         10 M 24 <RET>      100 ^            10 <RET> 100 ^     24 %

These two calculations get the same answer, but the first one is much
more efficient because it avoids the huge intermediate value that
arises in the second one.

   (*) *Exercise 10.*  A theorem of Pierre de Fermat says that `x^(n-1)
mod n = 1' if `n' is a prime number and `x' is an integer less than
`n'.  If `n' is _not_ a prime number, this will _not_ be true for most
values of `x'.  Thus we can test informally if a number is prime by
trying this formula for several values of `x'.  Use this test to tell
whether the following numbers are prime: 811749613, 15485863.  *Note
10: Types Answer 10. (*)

   It is possible to use HMS forms as parts of error forms, intervals,
modulo forms, or as the phase part of a polar complex number.  For
example, the `calc-time' command pushes the current time of day on the
stack as an HMS/modulo form.

     1:  17@ 34' 45" mod 24@ 0' 0"     1:  6@ 22' 15" mod 24@ 0' 0"
         .                                 .

         x time <RET>                        n

This calculation tells me it is six hours and 22 minutes until midnight.

   (*) *Exercise 11.*  A rule of thumb is that one year is about
`pi * 10^7' seconds.  What time will it be that many seconds from right
now?  *Note 11: Types Answer 11. (*)

   (*) *Exercise 12.*  You are preparing to order packaging for the CD
release of the Extended Disco Version of _Abbey Road_.  You are told
that the songs will actually be anywhere from 20 to 60 seconds longer
than the originals.  One CD can hold about 75 minutes of music.  Should
you order single or double packages?  *Note 12: Types Answer 12. (*)

   Another kind of data the Calculator can manipulate is numbers with
"units".  This isn't strictly a new data type; it's simply an
application of algebraic expressions, where we use variables with
suggestive names like `cm' and `in' to represent units like centimeters
and inches.

     1:  2 in        1:  5.08 cm      1:  0.027778 fath   1:  0.0508 m
         .               .                .                   .

         ' 2in <RET>       u c cm <RET>       u c fath <RET>        u b

We enter the quantity "2 inches" (actually an algebraic expression
which means two times the variable `in'), then we convert it first to
centimeters, then to fathoms, then finally to "base" units, which in
this case means meters.

     1:  9 acre     1:  3 sqrt(acre)   1:  190.84 m   1:  190.84 m + 30 cm
         .              .                  .              .

      ' 9 acre <RET>      Q                  u s            ' $+30 cm <RET>

     1:  191.14 m     1:  36536.3046 m^2    1:  365363046 cm^2
         .                .                     .

         u s              2 ^                   u c cgs

Since units expressions are really just formulas, taking the square
root of `acre' is undefined.  After all, `acre' might be an algebraic
variable that you will someday assign a value.  We use the
"units-simplify" command to simplify the expression with variables
being interpreted as unit names.

   In the final step, we have converted not to a particular unit, but
to a units system.  The "cgs" system uses centimeters instead of meters
as its standard unit of length.

   There is a wide variety of units defined in the Calculator.

     1:  55 mph     1:  88.5139 kph   1:   88.5139 km / hr   1:  8.201407e-8 c
         .              .                  .                     .

      ' 55 mph <RET>      u c kph <RET>        u c km/hr <RET>         u c c <RET>

We express a speed first in miles per hour, then in kilometers per
hour, then again using a slightly more explicit notation, then finally
in terms of fractions of the speed of light.

   Temperature conversions are a bit more tricky.  There are two ways to
interpret "20 degrees Fahrenheit"--it could mean an actual temperature,
or it could mean a change in temperature.  For normal units there is no
difference, but temperature units have an offset as well as a scale
factor and so there must be two explicit commands for them.

     1:  20 degF       1:  11.1111 degC     1:  -20:3 degC    1:  -6.666 degC
         .                 .                    .                 .

       ' 20 degF <RET>       u c degC <RET>         U u t degC <RET>    c f

First we convert a change of 20 degrees Fahrenheit into an equivalent
change in degrees Celsius (or Centigrade).  Then, we convert the
absolute temperature 20 degrees Fahrenheit into Celsius.  Since this
comes out as an exact fraction, we then convert to floating-point for
easier comparison with the other result.

   For simple unit conversions, you can put a plain number on the stack.
Then `u c' and `u t' will prompt for both old and new units.  When you
use this method, you're responsible for remembering which numbers are
in which units:

     1:  55         1:  88.5139              1:  8.201407e-8
         .              .                        .

         55             u c mph <RET> kph <RET>      u c km/hr <RET> c <RET>

   To see a complete list of built-in units, type `u v'.  Press
`C-x * c' again to re-enter the Calculator when you're done looking at
the units table.

   (*) *Exercise 13.*  How many seconds are there really in a year?
*Note 13: Types Answer 13. (*)

   (*) *Exercise 14.*  Supercomputer designs are limited by the speed
of light (and of electricity, which is nearly as fast).  Suppose a
computer has a 4.1 ns (nanosecond) clock cycle, and its cabinet is one
meter across.  Is speed of light going to be a significant factor in
its design?  *Note 14: Types Answer 14. (*)

   (*) *Exercise 15.*  Sam the Slug normally travels about five yards
in an hour.  He has obtained a supply of Power Pills; each Power Pill
he eats doubles his speed.  How many Power Pills can he swallow and
still travel legally on most US highways?  *Note 15: Types Answer 15.
(*)

File: calc,  Node: Algebra Tutorial,  Next: Programming Tutorial,  Prev: Types Tutorial,  Up: Tutorial

4.5 Algebra and Calculus Tutorial
=================================

This section shows how to use Calc's algebra facilities to solve
equations, do simple calculus problems, and manipulate algebraic
formulas.

* Menu:

* Basic Algebra Tutorial::
* Rewrites Tutorial::

File: calc,  Node: Basic Algebra Tutorial,  Next: Rewrites Tutorial,  Prev: Algebra Tutorial,  Up: Algebra Tutorial

4.5.1 Basic Algebra
-------------------

If you enter a formula in Algebraic mode that refers to variables, the
formula itself is pushed onto the stack.  You can manipulate formulas
as regular data objects.

     1:  2 x^2 - 6       1:  6 - 2 x^2       1:  (6 - 2 x^2) (3 x^2 + y)
         .                   .                   .

         ' 2x^2-6 <RET>        n                   ' 3x^2+y <RET> *

   (*) *Exercise 1.*  Do `' x <RET> Q 2 ^' and `' x <RET> 2 ^ Q' both
wind up with the same result (`x')?  Why or why not?  *Note 1: Algebra
Answer 1. (*)

   There are also commands for doing common algebraic operations on
formulas.  Continuing with the formula from the last example,

     1:  18 x^2 + 6 y - 6 x^4 - 2 x^2 y    1:  (18 - 2 y) x^2 - 6 x^4 + 6 y
         .                                     .

         a x                                   a c x <RET>

First we "expand" using the distributive law, then we "collect" terms
involving like powers of `x'.

   Let's find the value of this expression when `x' is 2 and `y' is
one-half.

     1:  17 x^2 - 6 x^4 + 3      1:  -25
         .                           .

         1:2 s l y <RET>               2 s l x <RET>

The `s l' command means "let"; it takes a number from the top of the
stack and temporarily assigns it as the value of the variable you
specify.  It then evaluates (as if by the `=' key) the next expression
on the stack.  After this command, the variable goes back to its
original value, if any.

   (An earlier exercise in this tutorial involved storing a value in the
variable `x'; if this value is still there, you will have to unstore it
with `s u x <RET>' before the above example will work properly.)

   Let's find the maximum value of our original expression when `y' is
one-half and `x' ranges over all possible values.  We can do this by
taking the derivative with respect to `x' and examining values of `x'
for which the derivative is zero.  If the second derivative of the
function at that value of `x' is negative, the function has a local
maximum there.

     1:  17 x^2 - 6 x^4 + 3      1:  34 x - 24 x^3
         .                           .

         U <DEL>  s 1                  a d x <RET>   s 2

Well, the derivative is clearly zero when `x' is zero.  To find the
other root(s), let's divide through by `x' and then solve:

     1:  (34 x - 24 x^3) / x    1:  34 x / x - 24 x^3 / x    1:  34 - 24 x^2
         .                          .                            .

         ' x <RET> /                  a x                          a s

     1:  34 - 24 x^2 = 0        1:  x = 1.19023
         .                          .

         0 a =  s 3                 a S x <RET>

Notice the use of `a s' to "simplify" the formula.  When the default
algebraic simplifications don't do enough, you can use `a s' to tell
Calc to spend more time on the job.

   Now we compute the second derivative and plug in our values of `x':

     1:  1.19023        2:  1.19023         2:  1.19023
         .              1:  34 x - 24 x^3   1:  34 - 72 x^2
                            .                   .

         a .                r 2                 a d x <RET> s 4

(The `a .' command extracts just the righthand side of an equation.
Another method would have been to use `v u' to unpack the equation
`x = 1.19' to `x' and `1.19', then use `M-- M-2 <DEL>' to delete the
`x'.)

     2:  34 - 72 x^2   1:  -68.         2:  34 - 72 x^2     1:  34
     1:  1.19023           .            1:  0                   .
         .                                  .

         <TAB>               s l x <RET>        U <DEL> 0             s l x <RET>

The first of these second derivatives is negative, so we know the
function has a maximum value at `x = 1.19023'.  (The function also has a
local _minimum_ at `x = 0'.)

   When we solved for `x', we got only one value even though `34 - 24
x^2 = 0' is a quadratic equation that ought to have two solutions.  The
reason is that `a S' normally returns a single "principal" solution.
If it needs to come up with an arbitrary sign (as occurs in the
quadratic formula) it picks `+'.  If it needs an arbitrary integer, it
picks zero.  We can get a full solution by pressing `H' (the Hyperbolic
flag) before `a S'.

     1:  34 - 24 x^2 = 0    1:  x = 1.19023 s1      1:  x = -1.19023
         .                      .                       .

         r 3                    H a S x <RET>  s 5        1 n  s l s1 <RET>

Calc has invented the variable `s1' to represent an unknown sign; it is
supposed to be either +1 or -1.  Here we have used the "let" command to
evaluate the expression when the sign is negative.  If we plugged this
into our second derivative we would get the same, negative, answer, so
`x = -1.19023' is also a maximum.

   To find the actual maximum value, we must plug our two values of `x'
into the original formula.

     2:  17 x^2 - 6 x^4 + 3    1:  24.08333 s1^2 - 12.04166 s1^4 + 3
     1:  x = 1.19023 s1            .
         .

         r 1 r 5                   s l <RET>

(Here we see another way to use `s l'; if its input is an equation with
a variable on the lefthand side, then `s l' treats the equation like an
assignment to that variable if you don't give a variable name.)

   It's clear that this will have the same value for either sign of
`s1', but let's work it out anyway, just for the exercise:

     2:  [-1, 1]              1:  [15.04166, 15.04166]
     1:  24.08333 s1^2 ...        .
         .

       [ 1 n , 1 ] <TAB>            V M $ <RET>

Here we have used a vector mapping operation to evaluate the function
at several values of `s1' at once.  `V M $' is like `V M '' except that
it takes the formula from the top of the stack.  The formula is
interpreted as a function to apply across the vector at the next-to-top
stack level.  Since a formula on the stack can't contain `$' signs,
Calc assumes the variables in the formula stand for different
arguments.  It prompts you for an "argument list", giving the list of
all variables in the formula in alphabetical order as the default list.
In this case the default is `(s1)', which is just what we want so we
simply press <RET> at the prompt.

   If there had been several different values, we could have used
`V R X' to find the global maximum.

   Calc has a built-in `a P' command that solves an equation using
`H a S' and returns a vector of all the solutions.  It simply automates
the job we just did by hand.  Applied to our original cubic polynomial,
it would produce the vector of solutions `[1.19023, -1.19023, 0]'.
(There is also an `a X' command which finds a local maximum of a
function.  It uses a numerical search method rather than examining the
derivatives, and thus requires you to provide some kind of initial
guess to show it where to look.)

   (*) *Exercise 2.*  Given a vector of the roots of a polynomial (such
as the output of an `a P' command), what sequence of commands would you
use to reconstruct the original polynomial?  (The answer will be unique
to within a constant multiple; choose the solution where the leading
coefficient is one.)  *Note 2: Algebra Answer 2. (*)

   The `m s' command enables Symbolic mode, in which formulas like
`sqrt(5)' that can't be evaluated exactly are left in symbolic form
rather than giving a floating-point approximate answer.  Fraction mode
(`m f') is also useful when doing algebra.

     2:  34 x - 24 x^3        2:  34 x - 24 x^3
     1:  34 x - 24 x^3        1:  [sqrt(51) / 6, sqrt(51) / -6, 0]
         .                        .

         r 2  <RET>     m s  m f    a P x <RET>

   One more mode that makes reading formulas easier is Big mode.

                    3
     2:  34 x - 24 x

           ____   ____
          V 51   V 51
     1:  [-----, -----, 0]
            6     -6

         .

         d B

   Here things like powers, square roots, and quotients and fractions
are displayed in a two-dimensional pictorial form.  Calc has other
language modes as well, such as C mode, FORTRAN mode, TeX mode and
LaTeX mode.

     2:  34*x - 24*pow(x, 3)               2:  34*x - 24*x**3
     1:  {sqrt(51) / 6, sqrt(51) / -6, 0}  1:  /sqrt(51) / 6, sqrt(51) / -6, 0/
         .                                     .

         d C                                   d F

     3:  34 x - 24 x^3
     2:  [{\sqrt{51} \over 6}, {\sqrt{51} \over -6}, 0]
     1:  {2 \over 3} \sqrt{5}
         .

         d T   ' 2 \sqrt{5} \over 3 <RET>

As you can see, language modes affect both entry and display of
formulas.  They affect such things as the names used for built-in
functions, the set of arithmetic operators and their precedences, and
notations for vectors and matrices.

   Notice that `sqrt(51)' may cause problems with older implementations
of C and FORTRAN, which would require something more like `sqrt(51.0)'.
It is always wise to check over the formulas produced by the various
language modes to make sure they are fully correct.

   Type `m s', `m f', and `d N' to reset these modes.  (You may prefer
to remain in Big mode, but all the examples in the tutorial are shown
in normal mode.)

   What is the area under the portion of this curve from `x = 1' to `2'?
This is simply the integral of the function:

     1:  17 x^2 - 6 x^4 + 3     1:  5.6666 x^3 - 1.2 x^5 + 3 x
         .                          .

         r 1                        a i x

We want to evaluate this at our two values for `x' and subtract.  One
way to do it is again with vector mapping and reduction:

     2:  [2, 1]            1:  [12.93333, 7.46666]    1:  5.46666
     1:  5.6666 x^3 ...        .                          .

        [ 2 , 1 ] <TAB>          V M $ <RET>                  V R -

   (*) *Exercise 3.*  Find the integral from 1 to `y' of `x sin(pi x)'
(where the sine is calculated in radians).  Find the values of the
integral for integers `y' from 1 to 5.  *Note 3: Algebra Answer 3. (*)

   Calc's integrator can do many simple integrals symbolically, but many
others are beyond its capabilities.  Suppose we wish to find the area
under the curve `sin(x) ln(x)' over the same range of `x'.  If you
entered this formula and typed `a i x <RET>' (don't bother to try
this), Calc would work for a long time but would be unable to find a
solution.  In fact, there is no closed-form solution to this integral.
Now what do we do?

   One approach would be to do the integral numerically.  It is not hard
to do this by hand using vector mapping and reduction.  It is rather
slow, though, since the sine and logarithm functions take a long time.
We can save some time by reducing the working precision.

     3:  10                  1:  [1, 1.1, 1.2,  ...  , 1.8, 1.9]
     2:  1                       .
     1:  0.1
         .

      10 <RET> 1 <RET> .1 <RET>        C-u v x

(Note that we have used the extended version of `v x'; we could also
have used plain `v x' as follows:  `v x 10 <RET> 9 + .1 *'.)

     2:  [1, 1.1, ... ]              1:  [0., 0.084941, 0.16993, ... ]
     1:  sin(x) ln(x)                    .
         .

         ' sin(x) ln(x) <RET>  s 1    m r  p 5 <RET>   V M $ <RET>

     1:  3.4195     0.34195
         .          .

         V R +      0.1 *

(If you got wildly different results, did you remember to switch to
Radians mode?)

   Here we have divided the curve into ten segments of equal width;
approximating these segments as rectangular boxes (i.e., assuming the
curve is nearly flat at that resolution), we compute the areas of the
boxes (height times width), then sum the areas.  (It is faster to sum
first, then multiply by the width, since the width is the same for
every box.)

   The true value of this integral turns out to be about 0.374, so
we're not doing too well.  Let's try another approach.

     1:  sin(x) ln(x)    1:  0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
         .                   .

         r 1                 a t x=1 <RET> 4 <RET>

Here we have computed the Taylor series expansion of the function about
the point `x=1'.  We can now integrate this polynomial approximation,
since polynomials are easy to integrate.

     1:  0.42074 x^2 + ...    1:  [-0.0446, -0.42073]      1:  0.3761
         .                        .                            .

         a i x <RET>            [ 2 , 1 ] <TAB>  V M $ <RET>         V R -

Better!  By increasing the precision and/or asking for more terms in
the Taylor series, we can get a result as accurate as we like.  (Taylor
series converge better away from singularities in the function such as
the one at `ln(0)', so it would also help to expand the series about
the points `x=2' or `x=1.5' instead of `x=1'.)

   (*) *Exercise 4.*  Our first method approximated the curve by
stairsteps of width 0.1; the total area was then the sum of the areas
of the rectangles under these stairsteps.  Our second method
approximated the function by a polynomial, which turned out to be a
better approximation than stairsteps.  A third method is "Simpson's
rule", which is like the stairstep method except that the steps are not
required to be flat.  Simpson's rule boils down to the formula,

     (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
                   + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))

where `n' (which must be even) is the number of slices and `h' is the
width of each slice.  These are 10 and 0.1 in our example.  For
reference, here is the corresponding formula for the stairstep method:

     h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
               + f(a+(n-2)*h) + f(a+(n-1)*h))

   Compute the integral from 1 to 2 of `sin(x) ln(x)' using Simpson's
rule with 10 slices.  *Note 4: Algebra Answer 4. (*)

   Calc has a built-in `a I' command for doing numerical integration.
It uses "Romberg's method", which is a more sophisticated cousin of
Simpson's rule.  In particular, it knows how to keep refining the
result until the current precision is satisfied.

   Aside from the commands we've seen so far, Calc also provides a
large set of commands for operating on parts of formulas.  You indicate
the desired sub-formula by placing the cursor on any part of the
formula before giving a "selection" command.  Selections won't be
covered in the tutorial; *note Selecting Subformulas::, for details and
examples.

File: calc,  Node: Rewrites Tutorial,  Prev: Basic Algebra Tutorial,  Up: Algebra Tutorial

4.5.2 Rewrite Rules
-------------------

No matter how many built-in commands Calc provided for doing algebra,
there would always be something you wanted to do that Calc didn't have
in its repertoire.  So Calc also provides a "rewrite rule" system that
you can use to define your own algebraic manipulations.

   Suppose we want to simplify this trigonometric formula:

     1:  1 / cos(x) - sin(x) tan(x)
         .

         ' 1/cos(x) - sin(x) tan(x) <RET>   s 1

If we were simplifying this by hand, we'd probably replace the `tan'
with a `sin/cos' first, then combine over a common denominator.  There
is no Calc command to do the former; the `a n' algebra command will do
the latter but we'll do both with rewrite rules just for practice.

   Rewrite rules are written with the `:=' symbol.

     1:  1 / cos(x) - sin(x)^2 / cos(x)
         .

         a r tan(a) := sin(a)/cos(a) <RET>

(The "assignment operator" `:=' has several uses in Calc.  All by
itself the formula `tan(a) := sin(a)/cos(a)' doesn't do anything, but
when it is given to the `a r' command, that command interprets it as a
rewrite rule.)

   The lefthand side, `tan(a)', is called the "pattern" of the rewrite
rule.  Calc searches the formula on the stack for parts that match the
pattern.  Variables in a rewrite pattern are called "meta-variables",
and when matching the pattern each meta-variable can match any
sub-formula.  Here, the meta-variable `a' matched the actual variable
`x'.

   When the pattern part of a rewrite rule matches a part of the
formula, that part is replaced by the righthand side with all the
meta-variables substituted with the things they matched.  So the result
is `sin(x) / cos(x)'.  Calc's normal algebraic simplifications then mix
this in with the rest of the original formula.

   To merge over a common denominator, we can use another simple rule:

     1:  (1 - sin(x)^2) / cos(x)
         .

         a r a/x + b/x := (a+b)/x <RET>

   This rule points out several interesting features of rewrite
patterns.  First, if a meta-variable appears several times in a
pattern, it must match the same thing everywhere.  This rule detects
common denominators because the same meta-variable `x' is used in both
of the denominators.

   Second, meta-variable names are independent from variables in the
target formula.  Notice that the meta-variable `x' here matches the
subformula `cos(x)'; Calc never confuses the two meanings of `x'.

   And third, rewrite patterns know a little bit about the algebraic
properties of formulas.  The pattern called for a sum of two quotients;
Calc was able to match a difference of two quotients by matching `a =
1', `b = -sin(x)^2', and `x = cos(x)'.

   We could just as easily have written `a/x - b/x := (a-b)/x' for the
rule.  It would have worked just the same in all cases.  (If we really
wanted the rule to apply only to `+' or only to `-', we could have used
the `plain' symbol.  *Note Algebraic Properties of Rewrite Rules::, for
some examples of this.)

   One more rewrite will complete the job.  We want to use the identity
`sin(x)^2 + cos(x)^2 = 1', but of course we must first rearrange the
identity in a way that matches our formula.  The obvious rule would be
`1 - sin(x)^2 := cos(x)^2', but a little thought shows that the rule
`sin(x)^2 := 1 - cos(x)^2' will also work.  The latter rule has a more
general pattern so it will work in many other situations, too.

     1:  (1 + cos(x)^2 - 1) / cos(x)           1:  cos(x)
         .                                         .

         a r sin(x)^2 := 1 - cos(x)^2 <RET>          a s

   You may ask, what's the point of using the most general rule if you
have to type it in every time anyway?  The answer is that Calc allows
you to store a rewrite rule in a variable, then give the variable name
in the `a r' command.  In fact, this is the preferred way to use
rewrites.  For one, if you need a rule once you'll most likely need it
again later.  Also, if the rule doesn't work quite right you can simply
Undo, edit the variable, and run the rule again without having to
retype it.

     ' tan(x) := sin(x)/cos(x) <RET>      s t tsc <RET>
     ' a/x + b/x := (a+b)/x <RET>         s t merge <RET>
     ' sin(x)^2 := 1 - cos(x)^2 <RET>     s t sinsqr <RET>

     1:  1 / cos(x) - sin(x) tan(x)     1:  cos(x)
         .                                  .

         r 1                a r tsc <RET>  a r merge <RET>  a r sinsqr <RET>  a s

   To edit a variable, type `s e' and the variable name, use regular
Emacs editing commands as necessary, then type `C-c C-c' to store the
edited value back into the variable.  You can also use `s e' to create
a new variable if you wish.

   Notice that the first time you use each rule, Calc puts up a
"compiling" message briefly.  The pattern matcher converts rules into a
special optimized pattern-matching language rather than using them
directly.  This allows `a r' to apply even rather complicated rules very
efficiently.  If the rule is stored in a variable, Calc compiles it
only once and stores the compiled form along with the variable.  That's
another good reason to store your rules in variables rather than
entering them on the fly.

   (*) *Exercise 1.*  Type `m s' to get Symbolic mode, then enter the
formula `(2 + sqrt(2)) / (1 + sqrt(2))'.  Using a rewrite rule,
simplify this formula by multiplying the top and bottom by the
conjugate `1 - sqrt(2)'.  The result will have to be expanded by the
distributive law; do this with another rewrite.  *Note 1: Rewrites
Answer 1. (*)

   The `a r' command can also accept a vector of rewrite rules, or a
variable containing a vector of rules.

     1:  [tsc, merge, sinsqr]          1:  [tan(x) := sin(x) / cos(x), ... ]
         .                                 .

         ' [tsc,merge,sinsqr] <RET>          =

     1:  1 / cos(x) - sin(x) tan(x)    1:  cos(x)
         .                                 .

         s t trig <RET>  r 1                 a r trig <RET>  a s

   Calc tries all the rules you give against all parts of the formula,
repeating until no further change is possible.  (The exact order in
which things are tried is rather complex, but for simple rules like the
ones we've used here the order doesn't really matter.  *Note Nested
Formulas with Rewrite Rules::.)

   Calc actually repeats only up to 100 times, just in case your rule
set has gotten into an infinite loop.  You can give a numeric prefix
argument to `a r' to specify any limit.  In particular, `M-1 a r' does
only one rewrite at a time.

     1:  1 / cos(x) - sin(x)^2 / cos(x)    1:  (1 - sin(x)^2) / cos(x)
         .                                     .

         r 1  M-1 a r trig <RET>                 M-1 a r trig <RET>

   You can type `M-0 a r' if you want no limit at all on the number of
rewrites that occur.

   Rewrite rules can also be "conditional".  Simply follow the rule
with a `::' symbol and the desired condition.  For example,

     1:  exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
         .

         ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) <RET>

     1:  1 + exp(3 pi i) + 1
         .

         a r exp(k pi i) := 1 :: k % 2 = 0 <RET>

(Recall, `k % 2' is the remainder from dividing `k' by 2, which will be
zero only when `k' is an even integer.)

   An interesting point is that the variables `pi' and `i' were matched
literally rather than acting as meta-variables.  This is because they
are special-constant variables.  The special constants `e', `phi', and
so on also match literally.  A common error with rewrite rules is to
write, say, `f(a,b,c,d,e) := g(a+b+c+d+e)', expecting to match any `f'
with five arguments but in fact matching only when the fifth argument
is literally `e'!

   Rewrite rules provide an interesting way to define your own
functions.  Suppose we want to define `fib(n)' to produce the Nth
Fibonacci number.  The first two Fibonacci numbers are each 1; later
numbers are formed by summing the two preceding numbers in the
sequence.  This is easy to express in a set of three rules:

     ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] <RET>  s t fib

     1:  fib(7)               1:  13
         .                        .

         ' fib(7) <RET>             a r fib <RET>

   One thing that is guaranteed about the order that rewrites are tried
is that, for any given subformula, earlier rules in the rule set will
be tried for that subformula before later ones.  So even though the
first and third rules both match `fib(1)', we know the first will be
used preferentially.

   This rule set has one dangerous bug:  Suppose we apply it to the
formula `fib(x)'?  (Don't actually try this.)  The third rule will
match `fib(x)' and replace it with `fib(x-1) + fib(x-2)'.  Each of
these will then be replaced to get `fib(x-2) + 2 fib(x-3) + fib(x-4)',
and so on, expanding forever.  What we really want is to apply the
third rule only when `n' is an integer greater than two.  Type
`s e fib <RET>', then edit the third rule to:

     fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2

Now:

     1:  fib(6) + fib(x) + fib(0)      1:  8 + fib(x) + fib(0)
         .                                 .

         ' fib(6)+fib(x)+fib(0) <RET>        a r fib <RET>

We've created a new function, `fib', and a new command,
`a r fib <RET>', which means "evaluate all `fib' calls in this
formula."  To make things easier still, we can tell Calc to apply these
rules automatically by storing them in the special variable `EvalRules'.

     1:  [fib(1) := ...]    .                1:  [8, 13]
         .                                       .

         s r fib <RET>        s t EvalRules <RET>    ' [fib(6), fib(7)] <RET>

   It turns out that this rule set has the problem that it does far
more work than it needs to when `n' is large.  Consider the first few
steps of the computation of `fib(6)':

     fib(6) =
     fib(5)              +               fib(4) =
     fib(4)     +      fib(3)     +      fib(3)     +      fib(2) =
     fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...

Note that `fib(3)' appears three times here.  Unless Calc's algebraic
simplifier notices the multiple `fib(3)'s and combines them (and, as it
happens, it doesn't), this rule set does lots of needless
recomputation.  To cure the problem, type `s e EvalRules' to edit the
rules (or just `s E', a shorthand command for editing `EvalRules') and
add another condition:

     fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember

If a `:: remember' condition appears anywhere in a rule, then if that
rule succeeds Calc will add another rule that describes that match to
the front of the rule set.  (Remembering works in any rule set, but for
technical reasons it is most effective in `EvalRules'.)  For example,
if the rule rewrites `fib(7)' to something that evaluates to 13, then
the rule `fib(7) := 13' will be added to the rule set.

   Type `' fib(8) <RET>' to compute the eighth Fibonacci number, then
type `s E' again to see what has happened to the rule set.

   With the `remember' feature, our rule set can now compute `fib(N)'
in just N steps.  In the process it builds up a table of all Fibonacci
numbers up to N.  After we have computed the result for a particular N,
we can get it back (and the results for all smaller N) later in just
one step.

   All Calc operations will run somewhat slower whenever `EvalRules'
contains any rules.  You should type `s u EvalRules <RET>' now to
un-store the variable.

   (*) *Exercise 2.*  Sometimes it is possible to reformulate a problem
to reduce the amount of recursion necessary to solve it.  Create a rule
that, in about N simple steps and without recourse to the `remember'
option, replaces `fib(N, 1, 1)' with `fib(1, X, Y)' where X and Y are
the Nth and N+1st Fibonacci numbers, respectively.  This rule is rather
clunky to use, so add a couple more rules to make the "user interface"
the same as for our first version: enter `fib(N)', get back a plain
number.  *Note 2: Rewrites Answer 2. (*)

   There are many more things that rewrites can do.  For example, there
are `&&&' and `|||' pattern operators that create "and" and "or"
combinations of rules.  As one really simple example, we could combine
our first two Fibonacci rules thusly:

     [fib(1 ||| 2) := 1, fib(n) := ... ]

That means "`fib' of something matching either 1 or 2 rewrites to 1."

   You can also make meta-variables optional by enclosing them in `opt'.
For example, the pattern `a + b x' matches `2 + 3 x' but not `2 + x' or
`3 x' or `x'.  The pattern `opt(a) + opt(b) x' matches all of these
forms, filling in a default of zero for `a' and one for `b'.

   (*) *Exercise 3.*  Your friend Joe had `2 + 3 x' on the stack and
tried to use the rule `opt(a) + opt(b) x := f(a, b, x)'.  What happened?
*Note 3: Rewrites Answer 3. (*)

   (*) *Exercise 4.*  Starting with a positive integer `a', divide `a'
by two if it is even, otherwise compute `3 a + 1'.  Now repeat this
step over and over.  A famous unproved conjecture is that for any
starting `a', the sequence always eventually reaches 1.  Given the
formula `seq(A, 0)', write a set of rules that convert this into
`seq(1, N)' where N is the number of steps it took the sequence to
reach the value 1.  Now enhance the rules to accept `seq(A)' as a
starting configuration, and to stop with just the number N by itself.
Now make the result be a vector of values in the sequence, from A to 1.
(The formula `X|Y' appends the vectors X and Y.)  For example,
rewriting `seq(6)' should yield the vector `[6, 3, 10, 5, 16, 8, 4, 2,
1]'.  *Note 4: Rewrites Answer 4. (*)

   (*) *Exercise 5.*  Define, using rewrite rules, a function
`nterms(X)' that returns the number of terms in the sum X, or 1 if X is
not a sum.  (A "sum" for our purposes is one or more non-sum terms
separated by `+' or `-' signs, so that `2 - 3 (x + y) + x y' is a sum
of three terms.)  *Note 5: Rewrites Answer 5. (*)

   (*) *Exercise 6.*  A Taylor series for a function is an infinite
series that exactly equals the value of that function at values of `x'
near zero.

     cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...

   The `a t' command produces a "truncated Taylor series" which is
obtained by dropping all the terms higher than, say, `x^2'.  Calc
represents the truncated Taylor series as a polynomial in `x'.
Mathematicians often write a truncated series using a "big-O" notation
that records what was the lowest term that was truncated.

     cos(x) = 1 - x^2 / 2! + O(x^3)

The meaning of `O(x^3)' is "a quantity which is negligibly small if
`x^3' is considered negligibly small as `x' goes to zero."

   The exercise is to create rewrite rules that simplify sums and
products of power series represented as `POLYNOMIAL + O(VAR^N)'.  For
example, given `1 - x^2 / 2 + O(x^3)' and `x - x^3 / 6 + O(x^4)' on the
stack, we want to be able to type `*' and get the result `x - 2:3 x^3 +
O(x^4)'.  Don't worry if the terms of the sum are rearranged or if `a
s' needs to be typed after rewriting.  (This one is rather tricky; the
solution at the end of this chapter uses 6 rewrite rules.  Hint:  The
`constant(x)' condition tests whether `x' is a number.)  *Note 6:
Rewrites Answer 6. (*)

   Just for kicks, try adding the rule `2+3 := 6' to `EvalRules'.  What
happens?  (Be sure to remove this rule afterward, or you might get a
nasty surprise when you use Calc to balance your checkbook!)

   *Note Rewrite Rules::, for the whole story on rewrite rules.

File: calc,  Node: Programming Tutorial,  Next: Answers to Exercises,  Prev: Algebra Tutorial,  Up: Tutorial

4.6 Programming Tutorial
========================

The Calculator is written entirely in Emacs Lisp, a highly extensible
language.  If you know Lisp, you can program the Calculator to do
anything you like.  Rewrite rules also work as a powerful programming
system.  But Lisp and rewrite rules take a while to master, and often
all you want to do is define a new function or repeat a command a few
times.  Calc has features that allow you to do these things easily.

   One very limited form of programming is defining your own functions.
Calc's `Z F' command allows you to define a function name and key
sequence to correspond to any formula.  Programming commands use the
shift-`Z' prefix; the user commands they create use the lower case `z'
prefix.

     1:  1 + x + x^2 / 2 + x^3 / 6         1:  1 + x + x^2 / 2 + x^3 / 6
         .                                     .

         ' 1 + x + x^2/2! + x^3/3! <RET>         Z F e myexp <RET> <RET> <RET> y

   This polynomial is a Taylor series approximation to `exp(x)'.  The
`Z F' command asks a number of questions.  The above answers say that
the key sequence for our function should be `z e'; the `M-x' equivalent
should be `calc-myexp'; the name of the function in algebraic formulas
should also be `myexp'; the default argument list `(x)' is acceptable;
and finally `y' answers the question "leave it in symbolic form for
non-constant arguments?"

     1:  1.3495     2:  1.3495     3:  1.3495
         .          1:  1.34986    2:  1.34986
                        .          1:  myexp(a + 1)
                                       .

         .3 z e         .3 E           ' a+1 <RET> z e

First we call our new `exp' approximation with 0.3 as an argument, and
compare it with the true `exp' function.  Then we note that, as
requested, if we try to give `z e' an argument that isn't a plain
number, it leaves the `myexp' function call in symbolic form.  If we
had answered `n' to the final question, `myexp(a + 1)' would have
evaluated by plugging in `a + 1' for `x' in the defining formula.

   (*) *Exercise 1.*  The "sine integral" function `Si(x)' is defined
as the integral of `sin(t)/t' for `t = 0' to `x' in radians.  (It was
invented because this integral has no solution in terms of basic
functions; if you give it to Calc's `a i' command, it will ponder it
for a long time and then give up.)  We can use the numerical
integration command, however, which in algebraic notation is written
like `ninteg(f(t), t, 0, x)' with any integrand `f(t)'.  Define a `z s'
command and `Si' function that implement this.  You will need to edit
the default argument list a bit.  As a test, `Si(1)' should return
0.946083. (If you don't get this answer, you might want to check that
Calc is in Radians mode.  Also, `ninteg' will run a lot faster if you
reduce the precision to, say, six digits beforehand.)  *Note 1:
Programming Answer 1. (*)

   The simplest way to do real "programming" of Emacs is to define a
"keyboard macro".  A keyboard macro is simply a sequence of keystrokes
which Emacs has stored away and can play back on demand.  For example,
if you find yourself typing `H a S x <RET>' often, you may wish to
program a keyboard macro to type this for you.

     1:  y = sqrt(x)          1:  x = y^2
         .                        .

         ' y=sqrt(x) <RET>       C-x ( H a S x <RET> C-x )

     1:  y = cos(x)           1:  x = s1 arccos(y) + 2 pi n1
         .                        .

         ' y=cos(x) <RET>           X

When you type `C-x (', Emacs begins recording.  But it is also still
ready to execute your keystrokes, so you're really "training" Emacs by
walking it through the procedure once.  When you type `C-x )', the
macro is recorded.  You can now type `X' to re-execute the same
keystrokes.

   You can give a name to your macro by typing `Z K'.

     1:  .              1:  y = x^4         1:  x = s2 sqrt(s1 sqrt(y))
                            .                   .

       Z K x <RET>            ' y=x^4 <RET>         z x

Notice that we use shift-`Z' to define the command, and lower-case `z'
to call it up.

   Keyboard macros can call other macros.

     1:  abs(x)        1:  x = s1 y                1:  2 / x    1:  x = 2 / y
         .                 .                           .            .

      ' abs(x) <RET>   C-x ( ' y <RET> a = z x C-x )    ' 2/x <RET>       X

   (*) *Exercise 2.*  Define a keyboard macro to negate the item in
level 3 of the stack, without disturbing the rest of the stack.  *Note
2: Programming Answer 2. (*)

   (*) *Exercise 3.*  Define keyboard macros to compute the following
functions:

  1. Compute `sin(x) / x', where `x' is the number on the top of the
     stack.

  2. Compute the base-`b' logarithm, just like the `B' key except the
     arguments are taken in the opposite order.

  3. Produce a vector of integers from 1 to the integer on the top of
     the stack.
        *Note 3: Programming Answer 3. (*)

   (*) *Exercise 4.*  Define a keyboard macro to compute the average
(mean) value of a list of numbers.  *Note 4: Programming Answer 4. (*)

   In many programs, some of the steps must execute several times.
Calc has "looping" commands that allow this.  Loops are useful inside
keyboard macros, but actually work at any time.

     1:  x^6          2:  x^6        1: 360 x^2
         .            1:  4             .
                          .

       ' x^6 <RET>          4         Z < a d x <RET> Z >

Here we have computed the fourth derivative of `x^6' by enclosing a
derivative command in a "repeat loop" structure.  This structure pops a
repeat count from the stack, then executes the body of the loop that
many times.

   If you make a mistake while entering the body of the loop, type
`Z C-g' to cancel the loop command.

   Here's another example:

     3:  1               2:  10946
     2:  1               1:  17711
     1:  20                  .
         .

     1 <RET> <RET> 20       Z < <TAB> C-j + Z >

The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
numbers, respectively.  (To see what's going on, try a few repetitions
of the loop body by hand; `C-j', also on the Line-Feed or <LFD> key if
you have one, makes a copy of the number in level 2.)

   A fascinating property of the Fibonacci numbers is that the `n'th
Fibonacci number can be found directly by computing `phi^n / sqrt(5)'
and then rounding to the nearest integer, where `phi', the "golden
ratio," is `(1 + sqrt(5)) / 2'.  (For convenience, this constant is
available from the `phi' variable, or the `I H P' command.)

     1:  1.61803         1:  24476.0000409    1:  10945.9999817    1:  10946
         .                   .                    .                    .

         I H P               21 ^                 5 Q /                R

   (*) *Exercise 5.*  The "continued fraction" representation of `phi'
is `1 + 1/(1 + 1/(1 + 1/( ... )))'.  We can compute an approximate
value by carrying this however far and then replacing the innermost
`1/( ... )' by 1.  Approximate `phi' using a twenty-term continued
fraction.  *Note 5: Programming Answer 5. (*)

   (*) *Exercise 6.*  Linear recurrences like the one for Fibonacci
numbers can be expressed in terms of matrices.  Given a vector `[a, b]'
determine a matrix which, when multiplied by this vector, produces the
vector `[b, c]', where `a', `b' and `c' are three successive Fibonacci
numbers.  Now write a program that, given an integer `n', computes the
`n'th Fibonacci number using matrix arithmetic.  *Note 6: Programming
Answer 6. (*)

   A more sophisticated kind of loop is the "for" loop.  Suppose we
wish to compute the 20th "harmonic" number, which is equal to the sum
of the reciprocals of the integers from 1 to 20.

     3:  0               1:  3.597739
     2:  1                   .
     1:  20
         .

     0 <RET> 1 <RET> 20         Z ( & + 1 Z )

The "for" loop pops two numbers, the lower and upper limits, then
repeats the body of the loop as an internal counter increases from the
lower limit to the upper one.  Just before executing the loop body, it
pushes the current loop counter.  When the loop body finishes, it pops
the "step," i.e., the amount by which to increment the loop counter.
As you can see, our loop always uses a step of one.

   This harmonic number function uses the stack to hold the running
total as well as for the various loop housekeeping functions.  If you
find this disorienting, you can sum in a variable instead:

     1:  0         2:  1                  .            1:  3.597739
         .         1:  20                                  .
                       .

         0 t 7       1 <RET> 20      Z ( & s + 7 1 Z )       r 7

The `s +' command adds the top-of-stack into the value in a variable
(and removes that value from the stack).

   It's worth noting that many jobs that call for a "for" loop can also
be done more easily by Calc's high-level operations.  Two other ways to
compute harmonic numbers are to use vector mapping and reduction (`v x
20', then `V M &', then `V R +'), or to use the summation command `a
+'.  Both of these are probably easier than using loops.  However,
there are some situations where loops really are the way to go:

   (*) *Exercise 7.*  Use a "for" loop to find the first harmonic
number which is greater than 4.0.  *Note 7: Programming Answer 7. (*)

   Of course, if we're going to be using variables in our programs, we
have to worry about the programs clobbering values that the caller was
keeping in those same variables.  This is easy to fix, though:

         .        1:  0.6667       1:  0.6667     3:  0.6667
                      .                .          2:  3.597739
                                                  1:  0.6667
                                                      .

        Z `    p 4 <RET> 2 <RET> 3 /   s 7 s s a <RET>    Z '  r 7 s r a <RET>

When we type `Z `' (that's a back-quote character), Calc saves its mode
settings and the contents of the ten "quick variables" for later
reference.  When we type `Z '' (that's an apostrophe now), Calc
restores those saved values.  Thus the `p 4' and `s 7' commands have no
effect outside this sequence.  Wrapping this around the body of a
keyboard macro ensures that it doesn't interfere with what the user of
the macro was doing.  Notice that the contents of the stack, and the
values of named variables, survive past the `Z '' command.

   The "Bernoulli numbers" are a sequence with the interesting property
that all of the odd Bernoulli numbers are zero, and the even ones,
while difficult to compute, can be roughly approximated by the formula
`2 n! / (2 pi)^n'.  Let's write a keyboard macro to compute
(approximate) Bernoulli numbers.  (Calc has a command, `k b', to
compute exact Bernoulli numbers, but this command is very slow for
large `n' since the higher Bernoulli numbers are very large fractions.)

     1:  10               1:  0.0756823
         .                    .

         10     C-x ( <RET> 2 % Z [ <DEL> 0 Z : ' 2 $! / (2 pi)^$ <RET> = Z ] C-x )

You can read `Z [' as "then," `Z :' as "else," and `Z ]' as "end-if."
There is no need for an explicit "if" command.  For the purposes of
`Z [', the condition is "true" if the value it pops from the stack is a
nonzero number, or "false" if it pops zero or something that is not a
number (like a formula).  Here we take our integer argument modulo 2;
this will be nonzero if we're asking for an odd Bernoulli number.

   The actual tenth Bernoulli number is `5/66'.

     3:  0.0756823    1:  0          1:  0.25305    1:  0          1:  1.16659
     2:  5:66             .              .              .              .
     1:  0.0757575
         .

     10 k b <RET> c f   M-0 <DEL> 11 X   <DEL> 12 X       <DEL> 13 X       <DEL> 14 X

   Just to exercise loops a bit more, let's compute a table of even
Bernoulli numbers.

     3:  []             1:  [0.10132, 0.03079, 0.02340, 0.033197, ...]
     2:  2                  .
     1:  30
         .

      [ ] 2 <RET> 30          Z ( X | 2 Z )

The vertical-bar `|' is the vector-concatenation command.  When we
execute it, the list we are building will be in stack level 2
(initially this is an empty list), and the next Bernoulli number will
be in level 1.  The effect is to append the Bernoulli number onto the
end of the list.  (To create a table of exact fractional Bernoulli
numbers, just replace `X' with `k b' in the above sequence of
keystrokes.)

   With loops and conditionals, you can program essentially anything in
Calc.  One other command that makes looping easier is `Z /', which
takes a condition from the stack and breaks out of the enclosing loop
if the condition is true (non-zero).  You can use this to make "while"
and "until" style loops.

   If you make a mistake when entering a keyboard macro, you can edit
it using `Z E'.  First, you must attach it to a key with `Z K'.  One
technique is to enter a throwaway dummy definition for the macro, then
enter the real one in the edit command.

     1:  3                   1:  3           Calc Macro Edit Mode.
         .                       .           Original keys: 1 <return> 2 +

                                             1                          ;; calc digits
                                             RET                        ;; calc-enter
                                             2                          ;; calc digits
                                             +                          ;; calc-plus

     C-x ( 1 <RET> 2 + C-x )    Z K h <RET>      Z E h

A keyboard macro is stored as a pure keystroke sequence.  The `edmacro'
package (invoked by `Z E') scans along the macro and tries to decode it
back into human-readable steps.  Descriptions of the keystrokes are
given as comments, which begin with `;;', and which are ignored when
the edited macro is saved.  Spaces and line breaks are also ignored
when the edited macro is saved.  To enter a space into the macro, type
`SPC'.  All the special characters `RET', `LFD', `TAB', `SPC', `DEL',
and `NUL' must be written in all uppercase, as must the prefixes `C-'
and `M-'.

   Let's edit in a new definition, for computing harmonic numbers.
First, erase the four lines of the old definition.  Then, type in the
new definition (or use Emacs `M-w' and `C-y' commands to copy it from
this page of the Info file; you can of course skip typing the comments,
which begin with `;;').

     Z`                      ;; calc-kbd-push     (Save local values)
     0                       ;; calc digits       (Push a zero onto the stack)
     st                      ;; calc-store-into   (Store it in the following variable)
     1                       ;; calc quick variable  (Quick variable q1)
     1                       ;; calc digits       (Initial value for the loop)
     TAB                     ;; calc-roll-down    (Swap initial and final)
     Z(                      ;; calc-kbd-for      (Begin the "for" loop)
     &                       ;; calc-inv          (Take the reciprocal)
     s+                      ;; calc-store-plus   (Add to the following variable)
     1                       ;; calc quick variable  (Quick variable q1)
     1                       ;; calc digits       (The loop step is 1)
     Z)                      ;; calc-kbd-end-for  (End the "for" loop)
     sr                      ;; calc-recall       (Recall the final accumulated value)
     1                       ;; calc quick variable (Quick variable q1)
     Z'                      ;; calc-kbd-pop      (Restore values)

Press `C-c C-c' to finish editing and return to the Calculator.

     1:  20         1:  3.597739
         .              .

         20             z h

   The `edmacro' package defines a handy `read-kbd-macro' command which
reads the current region of the current buffer as a sequence of
keystroke names, and defines that sequence on the `X' (and `C-x e')
key.  Because this is so useful, Calc puts this command on the `C-x *
m' key.  Try reading in this macro in the following form:  Press `C-@'
(or `C-<SPC>') at one end of the text below, then type `C-x * m' at the
other.

     Z ` 0 t 1
         1 TAB
         Z (  & s + 1  1 Z )
         r 1
     Z '

   (*) *Exercise 8.*  A general algorithm for solving equations
numerically is "Newton's Method".  Given the equation `f(x) = 0' for
any function `f', and an initial guess `x_0' which is reasonably close
to the desired solution, apply this formula over and over:

     new_x = x - f(x)/f'(x)

where `f'(x)' is the derivative of `f'.  The `x' values will quickly
converge to a solution, i.e., eventually `new_x' and `x' will be equal
to within the limits of the current precision.  Write a program which
takes a formula involving the variable `x', and an initial guess `x_0',
on the stack, and produces a value of `x' for which the formula is
zero.  Use it to find a solution of `sin(cos(x)) = 0.5' near `x = 4.5'.
(Use angles measured in radians.)  Note that the built-in `a R'
(`calc-find-root') command uses Newton's method when it is able.  *Note
8: Programming Answer 8. (*)

   (*) *Exercise 9.*  The "digamma" function `psi(z)' is defined as the
derivative of `ln(gamma(z))'.  For large values of `z', it can be
approximated by the infinite sum

     psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)

where `sum' represents the sum over `n' from 1 to infinity (or to some
limit high enough to give the desired accuracy), and the `bern'
function produces (exact) Bernoulli numbers.  While this sum is not
guaranteed to converge, in practice it is safe.  An interesting
mathematical constant is Euler's gamma, which is equal to about 0.5772.
One way to compute it is by the formula, `gamma = -psi(1)'.
Unfortunately, 1 isn't a large enough argument for the above formula to
work (5 is a much safer value for `z').  Fortunately, we can compute
`psi(1)' from `psi(5)' using the recurrence `psi(z+1) = psi(z) + 1/z'.
Your task:  Develop a program to compute `psi(z)'; it should "pump up"
`z' if necessary to be greater than 5, then use the above summation
formula.  Use looping commands to compute the sum.  Use your function
to compute `gamma' to twelve decimal places.  (Calc has a built-in
command for Euler's constant, `I P', which you can use to check your
answer.)  *Note 9: Programming Answer 9. (*)

   (*) *Exercise 10.*  Given a polynomial in `x' and a number `m' on
the stack, where the polynomial is of degree `m' or less (i.e., does
not have any terms higher than `x^m'), write a program to convert the
polynomial into a list-of-coefficients notation.  For example, `5 x^4 +
(x + 1)^2' with `m = 6' should produce the list `[1, 2, 1, 0, 5, 0,
0]'.  Also develop a way to convert from this form back to the standard
algebraic form.  *Note 10: Programming Answer 10. (*)

   (*) *Exercise 11.*  The "Stirling numbers of the first kind" are
defined by the recurrences,

     s(n,n) = 1   for n >= 0,
     s(n,0) = 0   for n > 0,
     s(n+1,m) = s(n,m-1) - n s(n,m)   for n >= m >= 1.

   This can be implemented using a "recursive" program in Calc; the
program must invoke itself in order to calculate the two righthand
terms in the general formula.  Since it always invokes itself with
"simpler" arguments, it's easy to see that it must eventually finish
the computation.  Recursion is a little difficult with Emacs keyboard
macros since the macro is executed before its definition is complete.
So here's the recommended strategy:  Create a "dummy macro" and assign
it to a key with, e.g., `Z K s'.  Now enter the true definition, using
the `z s' command to call itself recursively, then assign it to the
same key with `Z K s'.  Now the `z s' command will run the complete
recursive program.  (Another way is to use `Z E' or `C-x * m'
(`read-kbd-macro') to read the whole macro at once, thus avoiding the
"training" phase.)  The task:  Write a program that computes Stirling
numbers of the first kind, given `n' and `m' on the stack.  Test it
with _small_ inputs like `s(4,2)'.  (There is a built-in command for
Stirling numbers, `k s', which you can use to check your answers.)
*Note 11: Programming Answer 11. (*)

   The programming commands we've seen in this part of the tutorial are
low-level, general-purpose operations.  Often you will find that a
higher-level function, such as vector mapping or rewrite rules, will do
the job much more easily than a detailed, step-by-step program can:

   (*) *Exercise 12.*  Write another program for computing Stirling
numbers of the first kind, this time using rewrite rules.  Once again,
`n' and `m' should be taken from the stack.  *Note 12: Programming
Answer 12. (*)


   This ends the tutorial section of the Calc manual.  Now you know
enough about Calc to use it effectively for many kinds of calculations.
But Calc has many features that were not even touched upon in this
tutorial.  The rest of this manual tells the whole story.

File: calc,  Node: Answers to Exercises,  Prev: Programming Tutorial,  Up: Tutorial

4.7 Answers to Exercises
========================

This section includes answers to all the exercises in the Calc tutorial.

* Menu:

* RPN Answer 1::           1 <RET> 2 <RET> 3 <RET> 4 + * -
* RPN Answer 2::           2*4 + 7*9.5 + 5/4
* RPN Answer 3::           Operating on levels 2 and 3
* RPN Answer 4::           Joe's complex problems
* Algebraic Answer 1::     Simulating Q command
* Algebraic Answer 2::     Joe's algebraic woes
* Algebraic Answer 3::     1 / 0
* Modes Answer 1::         3#0.1 = 3#0.0222222?
* Modes Answer 2::         16#f.e8fe15
* Modes Answer 3::         Joe's rounding bug
* Modes Answer 4::         Why floating point?
* Arithmetic Answer 1::    Why the \ command?
* Arithmetic Answer 2::    Tripping up the B command
* Vector Answer 1::        Normalizing a vector
* Vector Answer 2::        Average position
* Matrix Answer 1::        Row and column sums
* Matrix Answer 2::        Symbolic system of equations
* Matrix Answer 3::        Over-determined system
* List Answer 1::          Powers of two
* List Answer 2::          Least-squares fit with matrices
* List Answer 3::          Geometric mean
* List Answer 4::          Divisor function
* List Answer 5::          Duplicate factors
* List Answer 6::          Triangular list
* List Answer 7::          Another triangular list
* List Answer 8::          Maximum of Bessel function
* List Answer 9::          Integers the hard way
* List Answer 10::         All elements equal
* List Answer 11::         Estimating pi with darts
* List Answer 12::         Estimating pi with matchsticks
* List Answer 13::         Hash codes
* List Answer 14::         Random walk
* Types Answer 1::         Square root of pi times rational
* Types Answer 2::         Infinities
* Types Answer 3::         What can "nan" be?
* Types Answer 4::         Abbey Road
* Types Answer 5::         Friday the 13th
* Types Answer 6::         Leap years
* Types Answer 7::         Erroneous donut
* Types Answer 8::         Dividing intervals
* Types Answer 9::         Squaring intervals
* Types Answer 10::        Fermat's primality test
* Types Answer 11::        pi * 10^7 seconds
* Types Answer 12::        Abbey Road on CD
* Types Answer 13::        Not quite pi * 10^7 seconds
* Types Answer 14::        Supercomputers and c
* Types Answer 15::        Sam the Slug
* Algebra Answer 1::       Squares and square roots
* Algebra Answer 2::       Building polynomial from roots
* Algebra Answer 3::       Integral of x sin(pi x)
* Algebra Answer 4::       Simpson's rule
* Rewrites Answer 1::      Multiplying by conjugate
* Rewrites Answer 2::      Alternative fib rule
* Rewrites Answer 3::      Rewriting opt(a) + opt(b) x
* Rewrites Answer 4::      Sequence of integers
* Rewrites Answer 5::      Number of terms in sum
* Rewrites Answer 6::      Truncated Taylor series
* Programming Answer 1::   Fresnel's C(x)
* Programming Answer 2::   Negate third stack element
* Programming Answer 3::   Compute sin(x) / x, etc.
* Programming Answer 4::   Average value of a list
* Programming Answer 5::   Continued fraction phi
* Programming Answer 6::   Matrix Fibonacci numbers
* Programming Answer 7::   Harmonic number greater than 4
* Programming Answer 8::   Newton's method
* Programming Answer 9::   Digamma function
* Programming Answer 10::  Unpacking a polynomial
* Programming Answer 11::  Recursive Stirling numbers
* Programming Answer 12::  Stirling numbers with rewrites

File: calc,  Node: RPN Answer 1,  Next: RPN Answer 2,  Prev: Answers to Exercises,  Up: Answers to Exercises

4.7.1 RPN Tutorial Exercise 1
-----------------------------

`1 <RET> 2 <RET> 3 <RET> 4 + * -'

   The result is `1 - (2 * (3 + 4)) = -13'.

File: calc,  Node: RPN Answer 2,  Next: RPN Answer 3,  Prev: RPN Answer 1,  Up: Answers to Exercises

4.7.2 RPN Tutorial Exercise 2
-----------------------------

`2*4 + 7*9.5 + 5/4 = 75.75'

   After computing the intermediate term `2*4 = 8', you can leave that
result on the stack while you compute the second term.  With both of
these results waiting on the stack you can then compute the final term,
then press `+ +' to add everything up.

     2:  2          1:  8          3:  8          2:  8
     1:  4              .          2:  7          1:  66.5
         .                         1:  9.5            .
                                       .

       2 <RET> 4          *          7 <RET> 9.5          *

     4:  8          3:  8          2:  8          1:  75.75
     3:  66.5       2:  66.5       1:  67.75          .
     2:  5          1:  1.25           .
     1:  4              .
         .

       5 <RET> 4          /              +              +

   Alternatively, you could add the first two terms before going on
with the third term.

     2:  8          1:  74.5       3:  74.5       2:  74.5       1:  75.75
     1:  66.5           .          2:  5          1:  1.25           .
         .                         1:  4              .
                                       .

        ...             +            5 <RET> 4          /              +

   On an old-style RPN calculator this second method would have the
advantage of using only three stack levels.  But since Calc's stack can
grow arbitrarily large this isn't really an issue.  Which method you
choose is purely a matter of taste.

File: calc,  Node: RPN Answer 3,  Next: RPN Answer 4,  Prev: RPN Answer 2,  Up: Answers to Exercises

4.7.3 RPN Tutorial Exercise 3
-----------------------------

The <TAB> key provides a way to operate on the number in level 2.

     3:  10         3:  10         4:  10         3:  10         3:  10
     2:  20         2:  30         3:  30         2:  30         2:  21
     1:  30         1:  20         2:  20         1:  21         1:  30
         .              .          1:  1              .              .
                                       .

                       <TAB>             1              +             <TAB>

   Similarly, `M-<TAB>' gives you access to the number in level 3.

     3:  10         3:  21         3:  21         3:  30         3:  11
     2:  21         2:  30         2:  30         2:  11         2:  21
     1:  30         1:  10         1:  11         1:  21         1:  30
         .              .              .              .              .

                       M-<TAB>           1 +           M-<TAB>          M-<TAB>

File: calc,  Node: RPN Answer 4,  Next: Algebraic Answer 1,  Prev: RPN Answer 3,  Up: Answers to Exercises

4.7.4 RPN Tutorial Exercise 4
-----------------------------

Either `( 2 , 3 )' or `( 2 <SPC> 3 )' would have worked, but using both
the comma and the space at once yields:

     1:  ( ...      2:  ( ...      1:  (2, ...    2:  (2, ...    2:  (2, ...
         .          1:  2              .          1:  (2, ...    1:  (2, 3)
                        .                             .              .

         (              2              ,             <SPC>            3 )

   Joe probably tried to type `<TAB> <DEL>' to swap the extra
incomplete object to the top of the stack and delete it.  But a feature
of Calc is that <DEL> on an incomplete object deletes just one
component out of that object, so he had to press <DEL> twice to finish
the job.

     2:  (2, ...    2:  (2, 3)     2:  (2, 3)     1:  (2, 3)
     1:  (2, 3)     1:  (2, ...    1:  ( ...          .
         .              .              .

                       <TAB>            <DEL>            <DEL>

   (As it turns out, deleting the second-to-top stack entry happens
often enough that Calc provides a special key, `M-<DEL>', to do just
that.  `M-<DEL>' is just like `<TAB> <DEL>', except that it doesn't
exhibit the "feature" that tripped poor Joe.)

File: calc,  Node: Algebraic Answer 1,  Next: Algebraic Answer 2,  Prev: RPN Answer 4,  Up: Answers to Exercises

4.7.5 Algebraic Entry Tutorial Exercise 1
-----------------------------------------

Type `' sqrt($) <RET>'.

   If the `Q' key is broken, you could use `' $^0.5 <RET>'.  Or, RPN
style, `0.5 ^'.

   (Actually, `$^1:2', using the fraction one-half as the power, is a
closer equivalent, since `9^0.5' yields `3.0' whereas `sqrt(9)' and
`9^1:2' yield the exact integer `3'.)

File: calc,  Node: Algebraic Answer 2,  Next: Algebraic Answer 3,  Prev: Algebraic Answer 1,  Up: Answers to Exercises

4.7.6 Algebraic Entry Tutorial Exercise 2
-----------------------------------------

In the formula `2 x (1+y)', `x' was interpreted as a function name with
`1+y' as its argument.  Assigning a value to a variable has no relation
to a function by the same name.  Joe needed to use an explicit `*'
symbol here:  `2 x*(1+y)'.

File: calc,  Node: Algebraic Answer 3,  Next: Modes Answer 1,  Prev: Algebraic Answer 2,  Up: Answers to Exercises

4.7.7 Algebraic Entry Tutorial Exercise 3
-----------------------------------------

The result from `1 <RET> 0 /' will be the formula `1 / 0'.  The
"function" `/' cannot be evaluated when its second argument is zero, so
it is left in symbolic form.  When you now type `0 *', the result will
be zero because Calc uses the general rule that "zero times anything is
zero."

   The `m i' command enables an "Infinite mode" in which `1 / 0'
results in a special symbol that represents "infinity."  If you
multiply infinity by zero, Calc uses another special new symbol to show
that the answer is "indeterminate."  *Note Infinities::, for further
discussion of infinite and indeterminate values.

File: calc,  Node: Modes Answer 1,  Next: Modes Answer 2,  Prev: Algebraic Answer 3,  Up: Answers to Exercises

4.7.8 Modes Tutorial Exercise 1
-------------------------------

Calc always stores its numbers in decimal, so even though one-third has
an exact base-3 representation (`3#0.1'), it is still stored as
0.3333333 (chopped off after 12 or however many decimal digits) inside
the calculator's memory.  When this inexact number is converted back to
base 3 for display, it may still be slightly inexact.  When we multiply
this number by 3, we get 0.999999, also an inexact value.

   When Calc displays a number in base 3, it has to decide how many
digits to show.  If the current precision is 12 (decimal) digits, that
corresponds to `12 / log10(3) = 25.15' base-3 digits.  Because 25.15 is
not an exact integer, Calc shows only 25 digits, with the result that
stored numbers carry a little bit of extra information that may not
show up on the screen.  When Joe entered `3#0.2', the stored number
0.666666 happened to round to a pleasing value when it lost that last
0.15 of a digit, but it was still inexact in Calc's memory.  When he
divided by 2, he still got the dreaded inexact value 0.333333.
(Actually, he divided 0.666667 by 2 to get 0.333334, which is why he
got something a little higher than `3#0.1' instead of a little lower.)

   If Joe didn't want to be bothered with all this, he could have typed
`M-24 d n' to display with one less digit than the default.  (If you
give `d n' a negative argument, it uses default-minus-that, so `M-- d
n' would be an easier way to get the same effect.)  Those inexact
results would still be lurking there, but they would now be rounded to
nice, natural-looking values for display purposes.  (Remember,
`0.022222' in base 3 is like `0.099999' in base 10; rounding off one
digit will round the number up to `0.1'.)  Depending on the nature of
your work, this hiding of the inexactness may be a benefit or a danger.
With the `d n' command, Calc gives you the choice.

   Incidentally, another consequence of all this is that if you type
`M-30 d n' to display more digits than are "really there," you'll see
garbage digits at the end of the number.  (In decimal display mode,
with decimally-stored numbers, these garbage digits are always zero so
they vanish and you don't notice them.)  Because Calc rounds off that
0.15 digit, there is the danger that two numbers could be slightly
different internally but still look the same.  If you feel uneasy about
this, set the `d n' precision to be a little higher than normal; you'll
get ugly garbage digits, but you'll always be able to tell two distinct
numbers apart.

   An interesting side note is that most computers store their
floating-point numbers in binary, and convert to decimal for display.
Thus everyday programs have the same problem:  Decimal 0.1 cannot be
represented exactly in binary (try it: `0.1 d 2'), so `0.1 * 10' comes
out as an inexact approximation to 1 on some machines (though they
generally arrange to hide it from you by rounding off one digit as we
did above).  Because Calc works in decimal instead of binary, you can
be sure that numbers that look exact _are_ exact as long as you stay in
decimal display mode.

   It's not hard to show that any number that can be represented exactly
in binary, octal, or hexadecimal is also exact in decimal, so the kinds
of problems we saw in this exercise are likely to be severe only when
you use a relatively unusual radix like 3.

File: calc,  Node: Modes Answer 2,  Next: Modes Answer 3,  Prev: Modes Answer 1,  Up: Answers to Exercises

4.7.9 Modes Tutorial Exercise 2
-------------------------------

If the radix is 15 or higher, we can't use the letter `e' to mark the
exponent because `e' is interpreted as a digit.  When Calc needs to
display scientific notation in a high radix, it writes
`16#F.E8F*16.^15'.  You can enter a number like this as an algebraic
entry.  Also, pressing `e' without any digits before it normally types
`1e', but in a high radix it types `16.^' and puts you in algebraic
entry:  `16#f.e8f <RET> e 15 <RET> *' is another way to enter this
number.

   The reason Calc puts a decimal point in the `16.^' is to prevent
huge integers from being generated if the exponent is large (consider
`16#1.23*16^1000', where we compute `16^1000' as a giant exact integer
and then throw away most of the digits when we multiply it by the
floating-point `16#1.23').  While this wouldn't normally matter for
display purposes, it could give you a nasty surprise if you copied that
number into a file and later moved it back into Calc.

File: calc,  Node: Modes Answer 3,  Next: Modes Answer 4,  Prev: Modes Answer 2,  Up: Answers to Exercises

4.7.10 Modes Tutorial Exercise 3
--------------------------------

The answer he got was `0.5000000000006399'.

   The problem is not that the square operation is inexact, but that the
sine of 45 that was already on the stack was accurate to only 12 places.
Arbitrary-precision calculations still only give answers as good as
their inputs.

   The real problem is that there is no 12-digit number which, when
squared, comes out to 0.5 exactly.  The `f [' and `f ]' commands
decrease or increase a number by one unit in the last place (according
to the current precision).  They are useful for determining facts like
this.

     1:  0.707106781187      1:  0.500000000001
         .                       .

         45 S                    2 ^

     1:  0.707106781187      1:  0.707106781186      1:  0.499999999999
         .                       .                       .

         U  <DEL>                  f [                     2 ^

   A high-precision calculation must be carried out in high precision
all the way.  The only number in the original problem which was known
exactly was the quantity 45 degrees, so the precision must be raised
before anything is done after the number 45 has been entered in order
for the higher precision to be meaningful.

File: calc,  Node: Modes Answer 4,  Next: Arithmetic Answer 1,  Prev: Modes Answer 3,  Up: Answers to Exercises

4.7.11 Modes Tutorial Exercise 4
--------------------------------

Many calculations involve real-world quantities, like the width and
height of a piece of wood or the volume of a jar.  Such quantities
can't be measured exactly anyway, and if the data that is input to a
calculation is inexact, doing exact arithmetic on it is a waste of time.

   Fractions become unwieldy after too many calculations have been done
with them.  For example, the sum of the reciprocals of the integers
from 1 to 10 is 7381:2520.  The sum from 1 to 30 is
9304682830147:2329089562800.  After a point it will take a long time to
add even one more term to this sum, but a floating-point calculation of
the sum will not have this problem.

   Also, rational numbers cannot express the results of all
calculations.  There is no fractional form for the square root of two,
so if you type `2 Q', Calc has no choice but to give you a
floating-point answer.

File: calc,  Node: Arithmetic Answer 1,  Next: Arithmetic Answer 2,  Prev: Modes Answer 4,  Up: Answers to Exercises

4.7.12 Arithmetic Tutorial Exercise 1
-------------------------------------

Dividing two integers that are larger than the current precision may
give a floating-point result that is inaccurate even when rounded down
to an integer.  Consider `123456789 / 2' when the current precision is
6 digits.  The true answer is `61728394.5', but with a precision of 6
this will be rounded to `12345700. / 2. = 61728500.'.  The result, when
converted to an integer, will be off by 106.

   Here are two solutions:  Raise the precision enough that the
floating-point round-off error is strictly to the right of the decimal
point.  Or, convert to Fraction mode so that `123456789 / 2' produces
the exact fraction `123456789:2', which can be rounded down by the `F'
command without ever switching to floating-point format.

File: calc,  Node: Arithmetic Answer 2,  Next: Vector Answer 1,  Prev: Arithmetic Answer 1,  Up: Answers to Exercises

4.7.13 Arithmetic Tutorial Exercise 2
-------------------------------------

`27 <RET> 9 B' could give the exact result `3:2', but it does a
floating-point calculation instead and produces `1.5'.

   Calc will find an exact result for a logarithm if the result is an
integer or (when in Fraction mode) the reciprocal of an integer.  But
there is no efficient way to search the space of all possible rational
numbers for an exact answer, so Calc doesn't try.

File: calc,  Node: Vector Answer 1,  Next: Vector Answer 2,  Prev: Arithmetic Answer 2,  Up: Answers to Exercises

4.7.14 Vector Tutorial Exercise 1
---------------------------------

Duplicate the vector, compute its length, then divide the vector by its
length:  `<RET> A /'.

     1:  [1, 2, 3]  2:  [1, 2, 3]      1:  [0.27, 0.53, 0.80]  1:  1.
         .          1:  3.74165738677      .                       .
                        .

         r 1            <RET> A              /                       A

   The final `A' command shows that the normalized vector does indeed
have unit length.

File: calc,  Node: Vector Answer 2,  Next: Matrix Answer 1,  Prev: Vector Answer 1,  Up: Answers to Exercises

4.7.15 Vector Tutorial Exercise 2
---------------------------------

The average position is equal to the sum of the products of the
positions times their corresponding probabilities.  This is the
definition of the dot product operation.  So all you need to do is to
put the two vectors on the stack and press `*'.

File: calc,  Node: Matrix Answer 1,  Next: Matrix Answer 2,  Prev: Vector Answer 2,  Up: Answers to Exercises

4.7.16 Matrix Tutorial Exercise 1
---------------------------------

The trick is to multiply by a vector of ones.  Use `r 4 [1 1 1] *' to
get the row sum.  Similarly, use `[1 1] r 4 *' to get the column sum.

File: calc,  Node: Matrix Answer 2,  Next: Matrix Answer 3,  Prev: Matrix Answer 1,  Up: Answers to Exercises

4.7.17 Matrix Tutorial Exercise 2
---------------------------------

        x + a y = 6
        x + b y = 10

Just enter the righthand side vector, then divide by the lefthand side
matrix as usual.

     1:  [6, 10]    2:  [6, 10]         1:  [6 - 4 a / (b - a), 4 / (b - a) ]
         .          1:  [ [ 1, a ]          .
                          [ 1, b ] ]
                        .

     ' [6 10] <RET>     ' [1 a; 1 b] <RET>      /

   This can be made more readable using `d B' to enable Big display
mode:

               4 a     4
     1:  [6 - -----, -----]
              b - a  b - a

   Type `d N' to return to Normal display mode afterwards.

File: calc,  Node: Matrix Answer 3,  Next: List Answer 1,  Prev: Matrix Answer 2,  Up: Answers to Exercises

4.7.18 Matrix Tutorial Exercise 3
---------------------------------

To solve `trn(A) * A * X = trn(A) * B', first we compute `A2 = trn(A) *
A' and `B2 = trn(A) * B'; now, we have a system `A2 * X = B2' which we
can solve using Calc's `/' command.

         a + 2b + 3c = 6
        4a + 5b + 6c = 2
        7a + 6b      = 3
        2a + 4b + 6c = 11

   The first step is to enter the coefficient matrix.  We'll store it in
quick variable number 7 for later reference.  Next, we compute the `B2'
vector.

     1:  [ [ 1, 2, 3 ]             2:  [ [ 1, 4, 7, 2 ]     1:  [57, 84, 96]
           [ 4, 5, 6 ]                   [ 2, 5, 6, 4 ]         .
           [ 7, 6, 0 ]                   [ 3, 6, 0, 6 ] ]
           [ 2, 4, 6 ] ]           1:  [6, 2, 3, 11]
         .                             .

     ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] <RET>  s 7  v t  [6 2 3 11]   *

Now we compute the matrix `A2' and divide.

     2:  [57, 84, 96]          1:  [-11.64, 14.08, -3.64]
     1:  [ [ 70, 72, 39 ]          .
           [ 72, 81, 60 ]
           [ 39, 60, 81 ] ]
         .

         r 7 v t r 7 *             /

(The actual computed answer will be slightly inexact due to round-off
error.)

   Notice that the answers are similar to those for the 3x3 system
solved in the text.  That's because the fourth equation that was added
to the system is almost identical to the first one multiplied by two.
(If it were identical, we would have gotten the exact same answer since
the 4x3 system would be equivalent to the original 3x3 system.)

   Since the first and fourth equations aren't quite equivalent, they
can't both be satisfied at once.  Let's plug our answers back into the
original system of equations to see how well they match.

     2:  [-11.64, 14.08, -3.64]     1:  [5.6, 2., 3., 11.2]
     1:  [ [ 1, 2, 3 ]                  .
           [ 4, 5, 6 ]
           [ 7, 6, 0 ]
           [ 2, 4, 6 ] ]
         .

         r 7                            <TAB> *

This is reasonably close to our original `B' vector, `[6, 2, 3, 11]'.

File: calc,  Node: List Answer 1,  Next: List Answer 2,  Prev: Matrix Answer 3,  Up: Answers to Exercises

4.7.19 List Tutorial Exercise 1
-------------------------------

We can use `v x' to build a vector of integers.  This needs to be
adjusted to get the range of integers we desire.  Mapping `-' across
the vector will accomplish this, although it turns out the plain `-'
key will work just as well.

     2:  2                              2:  2
     1:  [1, 2, 3, 4, 5, 6, 7, 8, 9]    1:  [-4, -3, -2, -1, 0, 1, 2, 3, 4]
         .                                  .

         2  v x 9 <RET>                       5 V M -   or   5 -

Now we use `V M ^' to map the exponentiation operator across the vector.

     1:  [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
         .

         V M ^

File: calc,  Node: List Answer 2,  Next: List Answer 3,  Prev: List Answer 1,  Up: Answers to Exercises

4.7.20 List Tutorial Exercise 2
-------------------------------

Given `x' and `y' vectors in quick variables 1 and 2 as before, the
first job is to form the matrix that describes the problem.

        m*x + b*1 = y

   Thus we want a 19x2 matrix with our `x' vector as one column and
ones as the other column.  So, first we build the column of ones, then
we combine the two columns to form our `A' matrix.

     2:  [1.34, 1.41, 1.49, ... ]    1:  [ [ 1.34, 1 ]
     1:  [1, 1, 1, ...]                    [ 1.41, 1 ]
         .                                 [ 1.49, 1 ]
                                           ...

         r 1 1 v b 19 <RET>                M-2 v p v t   s 3

Now we compute `trn(A) * y' and `trn(A) * A' and divide.

     1:  [33.36554, 13.613]    2:  [33.36554, 13.613]
         .                     1:  [ [ 98.0003, 41.63 ]
                                     [  41.63,   19   ] ]
                                   .

      v t r 2 *                    r 3 v t r 3 *

(Hey, those numbers look familiar!)

     1:  [0.52141679, -0.425978]
         .

         /

   Since we were solving equations of the form `m*x + b*1 = y', these
numbers should be `m' and `b', respectively.  Sure enough, they agree
exactly with the result computed using `V M' and `V R'!

   The moral of this story:  `V M' and `V R' will probably solve your
problem, but there is often an easier way using the higher-level
arithmetic functions!

   In fact, there is a built-in `a F' command that does least-squares
fits.  *Note Curve Fitting::.

File: calc,  Node: List Answer 3,  Next: List Answer 4,  Prev: List Answer 2,  Up: Answers to Exercises

4.7.21 List Tutorial Exercise 3
-------------------------------

Move to one end of the list and press `C-@' (or `C-<SPC>' or whatever)
to set the mark, then move to the other end of the list and type
`C-x * g'.

     1:  [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
         .

   To make things interesting, let's assume we don't know at a glance
how many numbers are in this list.  Then we could type:

     2:  [2.3, 6, 22, ... ]     2:  [2.3, 6, 22, ... ]
     1:  [2.3, 6, 22, ... ]     1:  126356422.5
         .                          .

         <RET>                        V R *

     2:  126356422.5            2:  126356422.5     1:  7.94652913734
     1:  [2.3, 6, 22, ... ]     1:  9                   .
         .                          .

         <TAB>                        v l                 I ^

(The `I ^' command computes the Nth root of a number.  You could also
type `& ^' to take the reciprocal of 9 and then raise the number to
that power.)

File: calc,  Node: List Answer 4,  Next: List Answer 5,  Prev: List Answer 3,  Up: Answers to Exercises

4.7.22 List Tutorial Exercise 4
-------------------------------

A number `j' is a divisor of `n' if `n % j = 0'.  The first step is to
get a vector that identifies the divisors.

     2:  30                  2:  [0, 0, 0, 2, ...]    1:  [1, 1, 1, 0, ...]
     1:  [1, 2, 3, 4, ...]   1:  0                        .
         .                       .

      30 <RET> v x 30 <RET>   s 1    V M %  0                 V M a =  s 2

This vector has 1's marking divisors of 30 and 0's marking non-divisors.

   The zeroth divisor function is just the total number of divisors.
The first divisor function is the sum of the divisors.

     1:  8      3:  8                    2:  8                    2:  8
                2:  [1, 2, 3, 4, ...]    1:  [1, 2, 3, 0, ...]    1:  72
                1:  [1, 1, 1, 0, ...]        .                        .
                    .

        V R +       r 1 r 2                  V M *                  V R +

Once again, the last two steps just compute a dot product for which a
simple `*' would have worked equally well.

File: calc,  Node: List Answer 5,  Next: List Answer 6,  Prev: List Answer 4,  Up: Answers to Exercises

4.7.23 List Tutorial Exercise 5
-------------------------------

The obvious first step is to obtain the list of factors with `k f'.
This list will always be in sorted order, so if there are duplicates
they will be right next to each other.  A suitable method is to compare
the list with a copy of itself shifted over by one.

     1:  [3, 7, 7, 7, 19]   2:  [3, 7, 7, 7, 19]     2:  [3, 7, 7, 7, 19, 0]
         .                  1:  [3, 7, 7, 7, 19, 0]  1:  [0, 3, 7, 7, 7, 19]
                                .                        .

         19551 k f              <RET> 0 |                  <TAB> 0 <TAB> |

     1:  [0, 0, 1, 1, 0, 0]   1:  2          1:  0
         .                        .              .

         V M a =                  V R +          0 a =

Note that we have to arrange for both vectors to have the same length
so that the mapping operation works; no prime factor will ever be zero,
so adding zeros on the left and right is safe.  From then on the job is
pretty straightforward.

   Incidentally, Calc provides the "Moebius mu" function which is zero
if and only if its argument is square-free.  It would be a much more
convenient way to do the above test in practice.

File: calc,  Node: List Answer 6,  Next: List Answer 7,  Prev: List Answer 5,  Up: Answers to Exercises

4.7.24 List Tutorial Exercise 6
-------------------------------

First use `v x 6 <RET>' to get a list of integers, then `V M v x' to
get a list of lists of integers!

File: calc,  Node: List Answer 7,  Next: List Answer 8,  Prev: List Answer 6,  Up: Answers to Exercises

4.7.25 List Tutorial Exercise 7
-------------------------------

Here's one solution.  First, compute the triangular list from the
previous exercise and type `1 -' to subtract one from all the elements.

     1:  [ [0],
           [0, 1],
           [0, 1, 2],
           ...

         1 -

   The numbers down the lefthand edge of the list we desire are called
the "triangular numbers" (now you know why!).  The `n'th triangular
number is the sum of the integers from 1 to `n', and can be computed
directly by the formula `n * (n+1) / 2'.

     2:  [ [0], [0, 1], ... ]    2:  [ [0], [0, 1], ... ]
     1:  [0, 1, 2, 3, 4, 5]      1:  [0, 1, 3, 6, 10, 15]
         .                           .

         v x 6 <RET> 1 -               V M ' $ ($+1)/2 <RET>

Adding this list to the above list of lists produces the desired result:

     1:  [ [0],
           [1, 2],
           [3, 4, 5],
           [6, 7, 8, 9],
           [10, 11, 12, 13, 14],
           [15, 16, 17, 18, 19, 20] ]
           .

           V M +

   If we did not know the formula for triangular numbers, we could have
computed them using a `V U +' command.  We could also have gotten them
the hard way by mapping a reduction across the original triangular list.

     2:  [ [0], [0, 1], ... ]    2:  [ [0], [0, 1], ... ]
     1:  [ [0], [0, 1], ... ]    1:  [0, 1, 3, 6, 10, 15]
         .                           .

         <RET>                         V M V R +

(This means "map a `V R +' command across the vector," and since each
element of the main vector is itself a small vector, `V R +' computes
the sum of its elements.)

File: calc,  Node: List Answer 8,  Next: List Answer 9,  Prev: List Answer 7,  Up: Answers to Exercises

4.7.26 List Tutorial Exercise 8
-------------------------------

The first step is to build a list of values of `x'.

     1:  [1, 2, 3, ..., 21]  1:  [0, 1, 2, ..., 20]  1:  [0, 0.25, 0.5, ..., 5]
         .                       .                       .

         v x 21 <RET>              1 -                     4 /  s 1

   Next, we compute the Bessel function values.

     1:  [0., 0.124, 0.242, ..., -0.328]
         .

         V M ' besJ(1,$) <RET>

(Another way to do this would be `1 <TAB> V M f j'.)

   A way to isolate the maximum value is to compute the maximum using
`V R X', then compare all the Bessel values with that maximum.

     2:  [0., 0.124, 0.242, ... ]   1:  [0, 0, 0, ... ]    2:  [0, 0, 0, ... ]
     1:  0.5801562                      .                  1:  1
         .                                                     .

         <RET> V R X                      V M a =                <RET> V R +    <DEL>

It's a good idea to verify, as in the last step above, that only one
value is equal to the maximum.  (After all, a plot of `sin(x)' might
have many points all equal to the maximum value, 1.)

   The vector we have now has a single 1 in the position that indicates
the maximum value of `x'.  Now it is a simple matter to convert this
back into the corresponding value itself.

     2:  [0, 0, 0, ... ]         1:  [0, 0., 0., ... ]    1:  1.75
     1:  [0, 0.25, 0.5, ... ]        .                        .
         .

         r 1                         V M *                    V R +

   If `a =' had produced more than one `1' value, this method would
have given the sum of all maximum `x' values; not very useful!  In this
case we could have used `v m' (`calc-mask-vector') instead.  This
command deletes all elements of a "data" vector that correspond to
zeros in a "mask" vector, leaving us with, in this example, a vector of
maximum `x' values.

   The built-in `a X' command maximizes a function using more efficient
methods.  Just for illustration, let's use `a X' to maximize
`besJ(1,x)' over this same interval.

     2:  besJ(1, x)                 1:  [1.84115, 0.581865]
     1:  [0 .. 5]                       .
         .

     ' besJ(1,x), [0..5] <RET>            a X x <RET>

The output from `a X' is a vector containing the value of `x' that
maximizes the function, and the function's value at that maximum.  As
you can see, our simple search got quite close to the right answer.

File: calc,  Node: List Answer 9,  Next: List Answer 10,  Prev: List Answer 8,  Up: Answers to Exercises

4.7.27 List Tutorial Exercise 9
-------------------------------

Step one is to convert our integer into vector notation.

     1:  25129925999           3:  25129925999
         .                     2:  10
                               1:  [11, 10, 9, ..., 1, 0]
                                   .

         25129925999 <RET>           10 <RET> 12 <RET> v x 12 <RET> -

     1:  25129925999              1:  [0, 2, 25, 251, 2512, ... ]
     2:  [100000000000, ... ]         .
         .

         V M ^   s 1                  V M \

(Recall, the `\' command computes an integer quotient.)

     1:  [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
         .

         10 V M %   s 2

   Next we must increment this number.  This involves adding one to the
last digit, plus handling carries.  There is a carry to the left out of
a digit if that digit is a nine and all the digits to the right of it
are nines.

     1:  [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1]   1:  [1, 1, 1, 0, 0, 1, ... ]
         .                                          .

         9 V M a =                                  v v

     1:  [1, 1, 1, 0, 0, 0, ... ]   1:  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
         .                              .

         V U *                          v v 1 |

Accumulating `*' across a vector of ones and zeros will preserve only
the initial run of ones.  These are the carries into all digits except
the rightmost digit.  Concatenating a one on the right takes care of
aligning the carries properly, and also adding one to the rightmost
digit.

     2:  [0, 0, 0, 0, ... ]     1:  [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
     1:  [0, 0, 2, 5, ... ]         .
         .

         0 r 2 |                    V M +  10 V M %

Here we have concatenated 0 to the _left_ of the original number; this
takes care of shifting the carries by one with respect to the digits
that generated them.

   Finally, we must convert this list back into an integer.

     3:  [0, 0, 2, 5, ... ]        2:  [0, 0, 2, 5, ... ]
     2:  1000000000000             1:  [1000000000000, 100000000000, ... ]
     1:  [100000000000, ... ]          .
         .

         10 <RET> 12 ^  r 1              |

     1:  [0, 0, 20000000000, 5000000000, ... ]    1:  25129926000
         .                                            .

         V M *                                        V R +

Another way to do this final step would be to reduce the formula
`10 $$ + $' across the vector of digits.

     1:  [0, 0, 2, 5, ... ]        1:  25129926000
         .                             .

                                       V R ' 10 $$ + $ <RET>

File: calc,  Node: List Answer 10,  Next: List Answer 11,  Prev: List Answer 9,  Up: Answers to Exercises

4.7.28 List Tutorial Exercise 10
--------------------------------

For the list `[a, b, c, d]', the result is `((a = b) = c) = d', which
will compare `a' and `b' to produce a 1 or 0, which is then compared
with `c' to produce another 1 or 0, which is then compared with `d'.
This is not at all what Joe wanted.

   Here's a more correct method:

     1:  [7, 7, 7, 8, 7]      2:  [7, 7, 7, 8, 7]
         .                    1:  7
                                  .

       ' [7,7,7,8,7] <RET>          <RET> v r 1 <RET>

     1:  [1, 1, 1, 0, 1]      1:  0
         .                        .

         V M a =                  V R *

File: calc,  Node: List Answer 11,  Next: List Answer 12,  Prev: List Answer 10,  Up: Answers to Exercises

4.7.29 List Tutorial Exercise 11
--------------------------------

The circle of unit radius consists of those points `(x,y)' for which
`x^2 + y^2 < 1'.  We start by generating a vector of `x^2' and a vector
of `y^2'.

   We can make this go a bit faster by using the `v .' and `t .'
commands.

     2:  [2., 2., ..., 2.]          2:  [2., 2., ..., 2.]
     1:  [2., 2., ..., 2.]          1:  [1.16, 1.98, ..., 0.81]
         .                              .

      v . t .  2. v b 100 <RET> <RET>       V M k r

     2:  [2., 2., ..., 2.]          1:  [0.026, 0.96, ..., 0.036]
     1:  [0.026, 0.96, ..., 0.036]  2:  [0.53, 0.81, ..., 0.094]
         .                              .

         1 -  2 V M ^                   <TAB>  V M k r  1 -  2 V M ^

   Now we sum the `x^2' and `y^2' values, compare with 1 to get a
vector of 1/0 truth values, then sum the truth values.

     1:  [0.56, 1.78, ..., 0.13]    1:  [1, 0, ..., 1]    1:  84
         .                              .                     .

         +                              1 V M a <             V R +

The ratio `84/100' should approximate the ratio `pi/4'.

     1:  0.84       1:  3.36       2:  3.36       1:  1.0695
         .              .          1:  3.14159        .

         100 /          4 *            P              /

Our estimate, 3.36, is off by about 7%.  We could get a better estimate
by taking more points (say, 1000), but it's clear that this method is
not very efficient!

   (Naturally, since this example uses random numbers your own answer
will be slightly different from the one shown here!)

   If you typed `v .' and `t .' before, type them again to return to
full-sized display of vectors.

File: calc,  Node: List Answer 12,  Next: List Answer 13,  Prev: List Answer 11,  Up: Answers to Exercises

4.7.30 List Tutorial Exercise 12
--------------------------------

This problem can be made a lot easier by taking advantage of some
symmetries.  First of all, after some thought it's clear that the `y'
axis can be ignored altogether.  Just pick a random `x' component for
one end of the match, pick a random direction `theta', and see if `x'
and `x + cos(theta)' (which is the `x' coordinate of the other
endpoint) cross a line.  The lines are at integer coordinates, so this
happens when the two numbers surround an integer.

   Since the two endpoints are equivalent, we may as well choose the
leftmost of the two endpoints as `x'.  Then `theta' is an angle pointing
to the right, in the range -90 to 90 degrees.  (We could use radians,
but it would feel like cheating to refer to `pi/2' radians while trying
to estimate `pi'!)

   In fact, since the field of lines is infinite we can choose the
coordinates 0 and 1 for the lines on either side of the leftmost
endpoint.  The rightmost endpoint will be between 0 and 1 if the match
does not cross a line, or between 1 and 2 if it does.  So: Pick random
`x' and `theta', compute `x + cos(theta)', and count how many of the
results are greater than one.  Simple!

   We can make this go a bit faster by using the `v .' and `t .'
commands.

     1:  [0.52, 0.71, ..., 0.72]    2:  [0.52, 0.71, ..., 0.72]
         .                          1:  [78.4, 64.5, ..., -42.9]
                                        .

     v . t . 1. v b 100 <RET>  V M k r    180. v b 100 <RET>  V M k r  90 -

(The next step may be slow, depending on the speed of your computer.)

     2:  [0.52, 0.71, ..., 0.72]    1:  [0.72, 1.14, ..., 1.45]
     1:  [0.20, 0.43, ..., 0.73]        .
         .

         m d  V M C                     +

     1:  [0, 1, ..., 1]       1:  0.64            1:  3.125
         .                        .                   .

         1 V M a >                V R + 100 /         2 <TAB> /

   Let's try the third method, too.  We'll use random integers up to
one million.  The `k r' command with an integer argument picks a random
integer.

     2:  [1000000, 1000000, ..., 1000000]   2:  [78489, 527587, ..., 814975]
     1:  [1000000, 1000000, ..., 1000000]   1:  [324014, 358783, ..., 955450]
         .                                      .

         1000000 v b 100 <RET> <RET>                V M k r  <TAB>  V M k r

     1:  [1, 1, ..., 25]      1:  [1, 1, ..., 0]     1:  0.56
         .                        .                      .

         V M k g                  1 V M a =              V R + 100 /

     1:  10.714        1:  3.273
         .                 .

         6 <TAB> /           Q

   For a proof of this property of the GCD function, see section 4.5.2,
exercise 10, of Knuth's _Art of Computer Programming_, volume II.

   If you typed `v .' and `t .' before, type them again to return to
full-sized display of vectors.

File: calc,  Node: List Answer 13,  Next: List Answer 14,  Prev: List Answer 12,  Up: Answers to Exercises

4.7.31 List Tutorial Exercise 13
--------------------------------

First, we put the string on the stack as a vector of ASCII codes.

     1:  [84, 101, 115, ..., 51]
         .

         "Testing, 1, 2, 3 <RET>

Note that the `"' key, like `$', initiates algebraic entry so there was
no need to type an apostrophe.  Also, Calc didn't mind that we omitted
the closing `"'.  (The same goes for all closing delimiters like `)'
and `]' at the end of a formula.

   We'll show two different approaches here.  In the first, we note that
if the input vector is `[a, b, c, d]', then the hash code is `3 (3 (3a
+ b) + c) + d = 27a + 9b + 3c + d'.  In other words, it's a sum of
descending powers of three times the ASCII codes.

     2:  [84, 101, 115, ..., 51]    2:  [84, 101, 115, ..., 51]
     1:  16                         1:  [15, 14, 13, ..., 0]
         .                              .

         <RET> v l                        v x 16 <RET> -

     2:  [84, 101, 115, ..., 51]    1:  1960915098    1:  121
     1:  [14348907, ..., 1]             .                 .
         .

         3 <TAB> V M ^                    *                 511 %

Once again, `*' elegantly summarizes most of the computation.  But
there's an even more elegant approach:  Reduce the formula `3 $$ + $'
across the vector.  Recall that this represents a function of two
arguments that computes its first argument times three plus its second
argument.

     1:  [84, 101, 115, ..., 51]    1:  1960915098
         .                              .

         "Testing, 1, 2, 3 <RET>          V R ' 3$$+$ <RET>

If you did the decimal arithmetic exercise, this will be familiar.
Basically, we're turning a base-3 vector of digits into an integer,
except that our "digits" are much larger than real digits.

   Instead of typing `511 %' again to reduce the result, we can be
cleverer still and notice that rather than computing a huge integer and
taking the modulo at the end, we can take the modulo at each step
without affecting the result.  While this means there are more
arithmetic operations, the numbers we operate on remain small so the
operations are faster.

     1:  [84, 101, 115, ..., 51]    1:  121
         .                              .

         "Testing, 1, 2, 3 <RET>          V R ' (3$$+$)%511 <RET>

   Why does this work?  Think about a two-step computation:
`3 (3a + b) + c'.  Taking a result modulo 511 basically means
subtracting off enough 511's to put the result in the desired range.
So the result when we take the modulo after every step is,

     3 (3 a + b - 511 m) + c - 511 n

for some suitable integers `m' and `n'.  Expanding out by the
distributive law yields

     9 a + 3 b + c - 511*3 m - 511 n

The `m' term in the latter formula is redundant because any
contribution it makes could just as easily be made by the `n' term.  So
we can take it out to get an equivalent formula with `n' = 3m + n',

     9 a + 3 b + c - 511 n'

which is just the formula for taking the modulo only at the end of the
calculation.  Therefore the two methods are essentially the same.

   Later in the tutorial we will encounter "modulo forms", which
basically automate the idea of reducing every intermediate result
modulo some value M.

File: calc,  Node: List Answer 14,  Next: Types Answer 1,  Prev: List Answer 13,  Up: Answers to Exercises

4.7.32 List Tutorial Exercise 14
--------------------------------

We want to use `H V U' to nest a function which adds a random step to
an `(x,y)' coordinate.  The function is a bit long, but otherwise the
problem is quite straightforward.

     2:  [0, 0]     1:  [ [    0,       0    ]
     1:  50               [  0.4288, -0.1695 ]
         .                [ -0.4787, -0.9027 ]
                          ...

         [0,0] 50       H V U ' <# + [random(2.0)-1, random(2.0)-1]> <RET>

   Just as the text recommended, we used `< >' nameless function
notation to keep the two `random' calls from being evaluated before
nesting even begins.

   We now have a vector of `[x, y]' sub-vectors, which by Calc's rules
acts like a matrix.  We can transpose this matrix and unpack to get a
pair of vectors, `x' and `y', suitable for graphing.

     2:  [ 0, 0.4288, -0.4787, ... ]
     1:  [ 0, -0.1696, -0.9027, ... ]
         .

         v t  v u  g f

   Incidentally, because the `x' and `y' are completely independent in
this case, we could have done two separate commands to create our `x'
and `y' vectors of numbers directly.

   To make a random walk of unit steps, we note that `sincos' of a
random direction exactly gives us an `[x, y]' step of unit length; in
fact, the new nesting function is even briefer, though we might want to
lower the precision a bit for it.

     2:  [0, 0]     1:  [ [    0,      0    ]
     1:  50               [  0.1318, 0.9912 ]
         .                [ -0.5965, 0.3061 ]
                          ...

         [0,0] 50   m d  p 6 <RET>   H V U ' <# + sincos(random(360.0))> <RET>

   Another `v t v u g f' sequence will graph this new random walk.

   An interesting twist on these random walk functions would be to use
complex numbers instead of 2-vectors to represent points on the plane.
In the first example, we'd use something like `random + random*(0,1)',
and in the second we could use polar complex numbers with random phase
angles.  (This exercise was first suggested in this form by Randal
Schwartz.)

File: calc,  Node: Types Answer 1,  Next: Types Answer 2,  Prev: List Answer 14,  Up: Answers to Exercises

4.7.33 Types Tutorial Exercise 1
--------------------------------

If the number is the square root of `pi' times a rational number, then
its square, divided by `pi', should be a rational number.

     1:  1.26508260337    1:  0.509433962268   1:  2486645810:4881193627
         .                    .                    .

                              2 ^ P /              c F

Technically speaking this is a rational number, but not one that is
likely to have arisen in the original problem.  More likely, it just
happens to be the fraction which most closely represents some
irrational number to within 12 digits.

   But perhaps our result was not quite exact.  Let's reduce the
precision slightly and try again:

     1:  0.509433962268     1:  27:53
         .                      .

         U p 10 <RET>             c F

Aha!  It's unlikely that an irrational number would equal a fraction
this simple to within ten digits, so our original number was probably
`sqrt(27 pi / 53)'.

   Notice that we didn't need to re-round the number when we reduced the
precision.  Remember, arithmetic operations always round their inputs
to the current precision before they begin.

File: calc,  Node: Types Answer 2,  Next: Types Answer 3,  Prev: Types Answer 1,  Up: Answers to Exercises

4.7.34 Types Tutorial Exercise 2
--------------------------------

`inf / inf = nan'.  Perhaps `1' is the "obvious" answer.  But if
`17 inf = inf', then `17 inf / inf = inf / inf = 17', too.

   `exp(inf) = inf'.  It's tempting to say that the exponential of
infinity must be "bigger" than "regular" infinity, but as far as Calc
is concerned all infinities are as just as big.  In other words, as `x'
goes to infinity, `e^x' also goes to infinity, but the fact the `e^x'
grows much faster than `x' is not relevant here.

   `exp(-inf) = 0'.  Here we have a finite answer even though the input
is infinite.

   `sqrt(-inf) = (0, 1) inf'.  Remember that `(0, 1)' represents the
imaginary number `i'.  Here's a derivation: `sqrt(-inf) =
sqrt((-1) * inf) = sqrt(-1) * sqrt(inf)'.  The first part is, by
definition, `i'; the second is `inf' because, once again, all
infinities are the same size.

   `sqrt(uinf) = uinf'.  In fact, we do know something about the
direction because `sqrt' is defined to return a value in the right half
of the complex plane.  But Calc has no notation for this, so it settles
for the conservative answer `uinf'.

   `abs(uinf) = inf'.  No matter which direction `x' points, `abs(x)'
always points along the positive real axis.

   `ln(0) = -inf'.  Here we have an infinite answer to a finite input.
As in the `1 / 0' case, Calc will only use infinities here if you have
turned on Infinite mode.  Otherwise, it will treat `ln(0)' as an error.

File: calc,  Node: Types Answer 3,  Next: Types Answer 4,  Prev: Types Answer 2,  Up: Answers to Exercises

4.7.35 Types Tutorial Exercise 3
--------------------------------

We can make `inf - inf' be any real number we like, say, `a', just by
claiming that we added `a' to the first infinity but not to the second.
This is just as true for complex values of `a', so `nan' can stand for
a complex number.  (And, similarly, `uinf' can stand for an infinity
that points in any direction in the complex plane, such as `(0, 1)
inf').

   In fact, we can multiply the first `inf' by two.  Surely
`2 inf - inf = inf', but also `2 inf - inf = inf - inf = nan'.  So
`nan' can even stand for infinity.  Obviously it's just as easy to make
it stand for minus infinity as for plus infinity.

   The moral of this story is that "infinity" is a slippery fish
indeed, and Calc tries to handle it by having a very simple model for
infinities (only the direction counts, not the "size"); but Calc is
careful to write `nan' any time this simple model is unable to tell
what the true answer is.

File: calc,  Node: Types Answer 4,  Next: Types Answer 5,  Prev: Types Answer 3,  Up: Answers to Exercises

4.7.36 Types Tutorial Exercise 4
--------------------------------

     2:  0@ 47' 26"              1:  0@ 2' 47.411765"
     1:  17                          .
         .

         0@ 47' 26" <RET> 17           /

The average song length is two minutes and 47.4 seconds.

     2:  0@ 2' 47.411765"     1:  0@ 3' 7.411765"    1:  0@ 53' 6.000005"
     1:  0@ 0' 20"                .                      .
         .

         20"                      +                      17 *

The album would be 53 minutes and 6 seconds long.

File: calc,  Node: Types Answer 5,  Next: Types Answer 6,  Prev: Types Answer 4,  Up: Answers to Exercises

4.7.37 Types Tutorial Exercise 5
--------------------------------

Let's suppose it's January 14, 1991.  The easiest thing to do is to
keep trying 13ths of months until Calc reports a Friday.  We can do
this by manually entering dates, or by using `t I':

     1:  <Wed Feb 13, 1991>    1:  <Wed Mar 13, 1991>   1:  <Sat Apr 13, 1991>
         .                         .                        .

         ' <2/13> <RET>       <DEL>    ' <3/13> <RET>             t I

(Calc assumes the current year if you don't say otherwise.)

   This is getting tedious--we can keep advancing the date by typing `t
I' over and over again, but let's automate the job by using vector
mapping.  The `t I' command actually takes a second "how-many-months"
argument, which defaults to one.  This argument is exactly what we want
to map over:

     2:  <Sat Apr 13, 1991>     1:  [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
     1:  [1, 2, 3, 4, 5, 6]          <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
         .                           <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
                                    .

         v x 6 <RET>                  V M t I

Et voila`, September 13, 1991 is a Friday.

     1:  242
         .

     ' <sep 13> - <jan 14> <RET>

And the answer to our original question:  242 days to go.

File: calc,  Node: Types Answer 6,  Next: Types Answer 7,  Prev: Types Answer 5,  Up: Answers to Exercises

4.7.38 Types Tutorial Exercise 6
--------------------------------

The full rule for leap years is that they occur in every year divisible
by four, except that they don't occur in years divisible by 100, except
that they _do_ in years divisible by 400.  We could work out the answer
by carefully counting the years divisible by four and the exceptions,
but there is a much simpler way that works even if we don't know the
leap year rule.

   Let's assume the present year is 1991.  Years have 365 days, except
that leap years (whenever they occur) have 366 days.  So let's count
the number of days between now and then, and compare that to the number
of years times 365.  The number of extra days we find must be equal to
the number of leap years there were.

     1:  <Mon Jan 1, 10001>     2:  <Mon Jan 1, 10001>     1:  2925593
         .                      1:  <Tue Jan 1, 1991>          .
                                    .

       ' <jan 1 10001> <RET>         ' <jan 1 1991> <RET>          -

     3:  2925593       2:  2925593     2:  2925593     1:  1943
     2:  10001         1:  8010        1:  2923650         .
     1:  1991              .               .
         .

       10001 <RET> 1991      -               365 *           -

There will be 1943 leap years before the year 10001.  (Assuming, of
course, that the algorithm for computing leap years remains unchanged
for that long.  *Note Date Forms::, for some interesting background
information in that regard.)

File: calc,  Node: Types Answer 7,  Next: Types Answer 8,  Prev: Types Answer 6,  Up: Answers to Exercises

4.7.39 Types Tutorial Exercise 7
--------------------------------

The relative errors must be converted to absolute errors so that `+/-'
notation may be used.

     1:  1.              2:  1.
         .               1:  0.2
                             .

         20 <RET> .05 *        4 <RET> .05 *

   Now we simply chug through the formula.

     1:  19.7392088022    1:  394.78 +/- 19.739    1:  6316.5 +/- 706.21
         .                    .                        .

         2 P 2 ^ *            20 p 1 *                 4 p .2 <RET> 2 ^ *

   It turns out the `v u' command will unpack an error form as well as
a vector.  This saves us some retyping of numbers.

     3:  6316.5 +/- 706.21     2:  6316.5 +/- 706.21
     2:  6316.5                1:  0.1118
     1:  706.21                    .
         .

         <RET> v u                   <TAB> /

Thus the volume is 6316 cubic centimeters, within about 11 percent.

File: calc,  Node: Types Answer 8,  Next: Types Answer 9,  Prev: Types Answer 7,  Up: Answers to Exercises

4.7.40 Types Tutorial Exercise 8
--------------------------------

The first answer is pretty simple:  `1 / (0 .. 10) = (0.1 .. inf)'.
Since a number in the interval `(0 .. 10)' can get arbitrarily close to
zero, its reciprocal can get arbitrarily large, so the answer is an
interval that effectively means, "any number greater than 0.1" but with
no upper bound.

   The second answer, similarly, is `1 / (-10 .. 0) = (-inf .. -0.1)'.

   Calc normally treats division by zero as an error, so that the
formula `1 / 0' is left unsimplified.  Our third problem,
`1 / [0 .. 10]', also (potentially) divides by zero because zero is now
a member of the interval.  So Calc leaves this one unevaluated, too.

   If you turn on Infinite mode by pressing `m i', you will instead get
the answer `[0.1 .. inf]', which includes infinity as a possible value.

   The fourth calculation, `1 / (-10 .. 10)', has the same problem.
Zero is buried inside the interval, but it's still a possible value.
It's not hard to see that the actual result of `1 / (-10 .. 10)' will
be either greater than 0.1, or less than -0.1.  Thus the interval goes
from minus infinity to plus infinity, with a "hole" in it from -0.1 to
0.1.  Calc doesn't have any way to represent this, so it just reports
`[-inf .. inf]' as the answer.  It may be disappointing to hear "the
answer lies somewhere between minus infinity and plus infinity,
inclusive," but that's the best that interval arithmetic can do in this
case.

File: calc,  Node: Types Answer 9,  Next: Types Answer 10,  Prev: Types Answer 8,  Up: Answers to Exercises

4.7.41 Types Tutorial Exercise 9
--------------------------------

     1:  [-3 .. 3]       2:  [-3 .. 3]     2:  [0 .. 9]
         .               1:  [0 .. 9]      1:  [-9 .. 9]
                             .                 .

         [ 3 n .. 3 ]        <RET> 2 ^           <TAB> <RET> *

In the first case the result says, "if a number is between -3 and 3,
its square is between 0 and 9."  The second case says, "the product of
two numbers each between -3 and 3 is between -9 and 9."

   An interval form is not a number; it is a symbol that can stand for
many different numbers.  Two identical-looking interval forms can stand
for different numbers.

   The same issue arises when you try to square an error form.

File: calc,  Node: Types Answer 10,  Next: Types Answer 11,  Prev: Types Answer 9,  Up: Answers to Exercises

4.7.42 Types Tutorial Exercise 10
---------------------------------

Testing the first number, we might arbitrarily choose 17 for `x'.

     1:  17 mod 811749613   2:  17 mod 811749613   1:  533694123 mod 811749613
         .                      811749612              .
                                .

         17 M 811749613 <RET>     811749612              ^

Since 533694123 is (considerably) different from 1, the number 811749613
must not be prime.

   It's awkward to type the number in twice as we did above.  There are
various ways to avoid this, and algebraic entry is one.  In fact, using
a vector mapping operation we can perform several tests at once.  Let's
use this method to test the second number.

     2:  [17, 42, 100000]               1:  [1 mod 15485863, 1 mod ... ]
     1:  15485863                           .
         .

      [17 42 100000] 15485863 <RET>           V M ' ($$ mod $)^($-1) <RET>

The result is three ones (modulo `n'), so it's very probable that
15485863 is prime.  (In fact, this number is the millionth prime.)

   Note that the functions `($$^($-1)) mod $' or `$$^($-1) % $' would
have been hopelessly inefficient, since they would have calculated the
power using full integer arithmetic.

   Calc has a `k p' command that does primality testing.  For small
numbers it does an exact test; for large numbers it uses a variant of
the Fermat test we used here.  You can use `k p' repeatedly to prove
that a large integer is prime with any desired probability.

File: calc,  Node: Types Answer 11,  Next: Types Answer 12,  Prev: Types Answer 10,  Up: Answers to Exercises

4.7.43 Types Tutorial Exercise 11
---------------------------------

There are several ways to insert a calculated number into an HMS form.
One way to convert a number of seconds to an HMS form is simply to
multiply the number by an HMS form representing one second:

     1:  31415926.5359     2:  31415926.5359     1:  8726@ 38' 46.5359"
         .                 1:  0@ 0' 1"              .
                               .

         P 1e7 *               0@ 0' 1"              *

     2:  8726@ 38' 46.5359"             1:  6@ 6' 2.5359" mod 24@ 0' 0"
     1:  15@ 27' 16" mod 24@ 0' 0"          .
         .

         x time <RET>                         +

It will be just after six in the morning.

   The algebraic `hms' function can also be used to build an HMS form:

     1:  hms(0, 0, 10000000. pi)       1:  8726@ 38' 46.5359"
         .                                 .

       ' hms(0, 0, 1e7 pi) <RET>             =

The `=' key is necessary to evaluate the symbol `pi' to the actual
number 3.14159...

File: calc,  Node: Types Answer 12,  Next: Types Answer 13,  Prev: Types Answer 11,  Up: Answers to Exercises

4.7.44 Types Tutorial Exercise 12
---------------------------------

As we recall, there are 17 songs of about 2 minutes and 47 seconds each.

     2:  0@ 2' 47"                    1:  [0@ 3' 7" .. 0@ 3' 47"]
     1:  [0@ 0' 20" .. 0@ 1' 0"]          .
         .

         [ 0@ 20" .. 0@ 1' ]              +

     1:  [0@ 52' 59." .. 1@ 4' 19."]
         .

         17 *

No matter how long it is, the album will fit nicely on one CD.

File: calc,  Node: Types Answer 13,  Next: Types Answer 14,  Prev: Types Answer 12,  Up: Answers to Exercises

4.7.45 Types Tutorial Exercise 13
---------------------------------

Type `' 1 yr <RET> u c s <RET>'.  The answer is 31557600 seconds.

File: calc,  Node: Types Answer 14,  Next: Types Answer 15,  Prev: Types Answer 13,  Up: Answers to Exercises

4.7.46 Types Tutorial Exercise 14
---------------------------------

How long will it take for a signal to get from one end of the computer
to the other?

     1:  m / c         1:  3.3356 ns
         .                 .

      ' 1 m / c <RET>        u c ns <RET>

(Recall, `c' is a "unit" corresponding to the speed of light.)

     1:  3.3356 ns     1:  0.81356 ns / ns     1:  0.81356
     2:  4.1 ns            .                       .
         .

       ' 4.1 ns <RET>        /                       u s

Thus a signal could take up to 81 percent of a clock cycle just to go
from one place to another inside the computer, assuming the signal
could actually attain the full speed of light.  Pretty tight!

File: calc,  Node: Types Answer 15,  Next: Algebra Answer 1,  Prev: Types Answer 14,  Up: Answers to Exercises

4.7.47 Types Tutorial Exercise 15
---------------------------------

The speed limit is 55 miles per hour on most highways.  We want to find
the ratio of Sam's speed to the US speed limit.

     1:  55 mph         2:  55 mph           3:  11 hr mph / yd
         .              1:  5 yd / hr            .
                            .

       ' 55 mph <RET>       ' 5 yd/hr <RET>          /

   The `u s' command cancels out these units to get a plain number.
Now we take the logarithm base two to find the final answer, assuming
that each successive pill doubles his speed.

     1:  19360.       2:  19360.       1:  14.24
         .            1:  2                .
                          .

         u s              2                B

Thus Sam can take up to 14 pills without a worry.

File: calc,  Node: Algebra Answer 1,  Next: Algebra Answer 2,  Prev: Types Answer 15,  Up: Answers to Exercises

4.7.48 Algebra Tutorial Exercise 1
----------------------------------

The result `sqrt(x)^2' is simplified back to `x' by the Calculator, but
`sqrt(x^2)' is not.  (Consider what happens if `x = -4'.)  If `x' is
real, this formula could be simplified to `abs(x)', but for general
complex arguments even that is not safe.  (*Note Declarations::, for a
way to tell Calc that `x' is known to be real.)

File: calc,  Node: Algebra Answer 2,  Next: Algebra Answer 3,  Prev: Algebra Answer 1,  Up: Answers to Exercises

4.7.49 Algebra Tutorial Exercise 2
----------------------------------

Suppose our roots are `[a, b, c]'.  We want a polynomial which is zero
when `x' is any of these values.  The trivial polynomial `x-a' is zero
when `x=a', so the product `(x-a)(x-b)(x-c)' will do the job.  We can
use `a c x' to write this in a more familiar form.

     1:  34 x - 24 x^3          1:  [1.19023, -1.19023, 0]
         .                          .

         r 2                        a P x <RET>

     1:  [x - 1.19023, x + 1.19023, x]     1:  (x - 1.19023) (x + 1.19023) x
         .                                     .

         V M ' x-$ <RET>                         V R *

     1:  x^3 - 1.41666 x        1:  34 x - 24 x^3
         .                          .

         a c x <RET>                  24 n *  a x

Sure enough, our answer (multiplied by a suitable constant) is the same
as the original polynomial.

File: calc,  Node: Algebra Answer 3,  Next: Algebra Answer 4,  Prev: Algebra Answer 2,  Up: Answers to Exercises

4.7.50 Algebra Tutorial Exercise 3
----------------------------------

     1:  x sin(pi x)         1:  (sin(pi x) - pi x cos(pi x)) / pi^2
         .                       .

       ' x sin(pi x) <RET>   m r   a i x <RET>

     1:  [y, 1]
     2:  (sin(pi x) - pi x cos(pi x)) / pi^2
         .

       ' [y,1] <RET> <TAB>

     1:  [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
         .

         V M $ <RET>

     1:  (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
         .

         V R -

     1:  (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
         .

         =

     1:  [0., -0.95493, 0.63662, -1.5915, 1.2732]
         .

         v x 5 <RET>  <TAB>  V M $ <RET>

File: calc,  Node: Algebra Answer 4,  Next: Rewrites Answer 1,  Prev: Algebra Answer 3,  Up: Answers to Exercises

4.7.51 Algebra Tutorial Exercise 4
----------------------------------

The hard part is that `V R +' is no longer sufficient to add up all the
contributions from the slices, since the slices have varying
coefficients.  So first we must come up with a vector of these
coefficients.  Here's one way:

     2:  -1                 2:  3                    1:  [4, 2, ..., 4]
     1:  [1, 2, ..., 9]     1:  [-1, 1, ..., -1]         .
         .                      .

         1 n v x 9 <RET>          V M ^  3 <TAB>             -

     1:  [4, 2, ..., 4, 1]      1:  [1, 4, 2, ..., 4, 1]
         .                          .

         1 |                        1 <TAB> |

Now we compute the function values.  Note that for this method we need
eleven values, including both endpoints of the desired interval.

     2:  [1, 4, 2, ..., 4, 1]
     1:  [1, 1.1, 1.2,  ...  , 1.8, 1.9, 2.]
         .

      11 <RET> 1 <RET> .1 <RET>  C-u v x

     2:  [1, 4, 2, ..., 4, 1]
     1:  [0., 0.084941, 0.16993, ... ]
         .

         ' sin(x) ln(x) <RET>   m r  p 5 <RET>   V M $ <RET>

Once again this calls for `V M * V R +'; a simple `*' does the same
thing.

     1:  11.22      1:  1.122      1:  0.374
         .              .              .

         *              .1 *           3 /

Wow!  That's even better than the result from the Taylor series method.

File: calc,  Node: Rewrites Answer 1,  Next: Rewrites Answer 2,  Prev: Algebra Answer 4,  Up: Answers to Exercises

4.7.52 Rewrites Tutorial Exercise 1
-----------------------------------

We'll use Big mode to make the formulas more readable.

                                                    ___
                                               2 + V 2
     1:  (2 + sqrt(2)) / (1 + sqrt(2))     1:  --------
         .                                          ___
                                               1 + V 2

                                               .

       ' (2+sqrt(2)) / (1+sqrt(2)) <RET>         d B

Multiplying by the conjugate helps because `(a+b) (a-b) = a^2 - b^2'.

               ___    ___
     1:  (2 + V 2 ) (V 2  - 1)
         .

       a r a/(b+c) := a*(b-c) / (b^2-c^2) <RET>

              ___                         ___
     1:  2 + V 2  - 2                1:  V 2
         .                               .

       a r a*(b+c) := a*b + a*c          a s

(We could have used `a x' instead of a rewrite rule for the second
step.)

   The multiply-by-conjugate rule turns out to be useful in many
different circumstances, such as when the denominator involves sines
and cosines or the imaginary constant `i'.

File: calc,  Node: Rewrites Answer 2,  Next: Rewrites Answer 3,  Prev: Rewrites Answer 1,  Up: Answers to Exercises

4.7.53 Rewrites Tutorial Exercise 2
-----------------------------------

Here is the rule set:

     [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
       fib(1, x, y) := x,
       fib(n, x, y) := fib(n-1, y, x+y) ]

The first rule turns a one-argument `fib' that people like to write
into a three-argument `fib' that makes computation easier.  The second
rule converts back from three-argument form once the computation is
done.  The third rule does the computation itself.  It basically says
that if `x' and `y' are two consecutive Fibonacci numbers, then `y' and
`x+y' are the next (overlapping) pair of Fibonacci numbers.

   Notice that because the number `n' was "validated" by the conditions
on the first rule, there is no need to put conditions on the other
rules because the rule set would never get that far unless the input
were valid.  That further speeds computation, since no extra conditions
need to be checked at every step.

   Actually, a user with a nasty sense of humor could enter a bad
three-argument `fib' call directly, say, `fib(0, 1, 1)', which would
get the rules into an infinite loop.  One thing that would help keep
this from happening by accident would be to use something like `ZzFib'
instead of `fib' as the name of the three-argument function.

File: calc,  Node: Rewrites Answer 3,  Next: Rewrites Answer 4,  Prev: Rewrites Answer 2,  Up: Answers to Exercises

4.7.54 Rewrites Tutorial Exercise 3
-----------------------------------

He got an infinite loop.  First, Calc did as expected and rewrote
`2 + 3 x' to `f(2, 3, x)'.  Then it looked for ways to apply the rule
again, and found that `f(2, 3, x)' looks like `a + b x' with `a = 0'
and `b = 1', so it rewrote to `f(0, 1, f(2, 3, x))'.  It then wrapped
another `f(0, 1, ...)' around that, and so on, ad infinitum.  Joe
should have used `M-1 a r' to make sure the rule applied only once.

   (Actually, even the first step didn't work as he expected.  What Calc
really gives for `M-1 a r' in this situation is `f(3 x, 1, 2)',
treating 2 as the "variable," and `3 x' as a constant being added to
it.  While this may seem odd, it's just as valid a solution as the
"obvious" one.  One way to fix this would be to add the condition `::
variable(x)' to the rule, to make sure the thing that matches `x' is
indeed a variable, or to change `x' to `quote(x)' on the lefthand side,
so that the rule matches the actual variable `x' rather than letting
`x' stand for something else.)

File: calc,  Node: Rewrites Answer 4,  Next: Rewrites Answer 5,  Prev: Rewrites Answer 3,  Up: Answers to Exercises

4.7.55 Rewrites Tutorial Exercise 4
-----------------------------------

Here is a suitable set of rules to solve the first part of the problem:

     [ seq(n, c) := seq(n/2,  c+1) :: n%2 = 0,
       seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]

   Given the initial formula `seq(6, 0)', application of these rules
produces the following sequence of formulas:

     seq( 3, 1)
     seq(10, 2)
     seq( 5, 3)
     seq(16, 4)
     seq( 8, 5)
     seq( 4, 6)
     seq( 2, 7)
     seq( 1, 8)

whereupon neither of the rules match, and rewriting stops.

   We can pretty this up a bit with a couple more rules:

     [ seq(n) := seq(n, 0),
       seq(1, c) := c,
       ... ]

Now, given `seq(6)' as the starting configuration, we get 8 as the
result.

   The change to return a vector is quite simple:

     [ seq(n) := seq(n, []) :: integer(n) :: n > 0,
       seq(1, v) := v | 1,
       seq(n, v) := seq(n/2,  v | n) :: n%2 = 0,
       seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]

Given `seq(6)', the result is `[6, 3, 10, 5, 16, 8, 4, 2, 1]'.

   Notice that the `n > 1' guard is no longer necessary on the last
rule since the `n = 1' case is now detected by another rule.  But a
guard has been added to the initial rule to make sure the initial value
is suitable before the computation begins.

   While still a good idea, this guard is not as vitally important as it
was for the `fib' function, since calling, say, `seq(x, [])' will not
get into an infinite loop.  Calc will not be able to prove the symbol
`x' is either even or odd, so none of the rules will apply and the
rewrites will stop right away.

File: calc,  Node: Rewrites Answer 5,  Next: Rewrites Answer 6,  Prev: Rewrites Answer 4,  Up: Answers to Exercises

4.7.56 Rewrites Tutorial Exercise 5
-----------------------------------

If `x' is the sum `a + b', then `nterms(X)' must be `nterms(A)' plus
`nterms(B)'.  If `x' is not a sum, then `nterms(X)' = 1.

     [ nterms(a + b) := nterms(a) + nterms(b),
       nterms(x)     := 1 ]

Here we have taken advantage of the fact that earlier rules always
match before later rules; `nterms(x)' will only be tried if we already
know that `x' is not a sum.

File: calc,  Node: Rewrites Answer 6,  Next: Programming Answer 1,  Prev: Rewrites Answer 5,  Up: Answers to Exercises

4.7.57 Rewrites Tutorial Exercise 6
-----------------------------------

Here is a rule set that will do the job:

     [ a*(b + c) := a*b + a*c,
       opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
          :: constant(a) :: constant(b),
       opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
          :: constant(a) :: constant(b),
       a O(x^n) := O(x^n) :: constant(a),
       x^opt(m) O(x^n) := O(x^(n+m)),
       O(x^n) O(x^m) := O(x^(n+m)) ]

   If we really want the `+' and `*' keys to operate naturally on power
series, we should put these rules in `EvalRules'.  For testing
purposes, it is better to put them in a different variable, say, `O',
first.

   The first rule just expands products of sums so that the rest of the
rules can assume they have an expanded-out polynomial to work with.
Note that this rule does not mention `O' at all, so it will apply to
any product-of-sum it encounters--this rule may surprise you if you put
it into `EvalRules'!

   In the second rule, the sum of two O's is changed to the smaller O.
The optional constant coefficients are there mostly so that `O(x^2) -
O(x^3)' and `O(x^3) - O(x^2)' are handled as well as `O(x^2) + O(x^3)'.

   The third rule absorbs higher powers of `x' into O's.

   The fourth rule says that a constant times a negligible quantity is
still negligible.  (This rule will also match `O(x^3) / 4', with `a =
1/4'.)

   The fifth rule rewrites, for example, `x^2 O(x^3)' to `O(x^5)'.  (It
is easy to see that if one of these forms is negligible, the other is,
too.)  Notice the `x^opt(m)' to pick up terms like `x O(x^3)'.
Optional powers will match `x' as `x^1' but not 1 as `x^0'.  This turns
out to be exactly what we want here.

   The sixth rule is the corresponding rule for products of two O's.

   Another way to solve this problem would be to create a new "data
type" that represents truncated power series.  We might represent these
as function calls `series(COEFS, X)' where COEFS is a vector of
coefficients for `x^0', `x^1', `x^2', and so on.  Rules would exist for
sums and products of such `series' objects, and as an optional
convenience could also know how to combine a `series' object with a
normal polynomial.  (With this, and with a rule that rewrites `O(x^n)'
to the equivalent `series' form, you could still enter power series in
exactly the same notation as before.)  Operations on such objects would
probably be more efficient, although the objects would be a bit harder
to read.

   Some other symbolic math programs provide a power series data type
similar to this.  Mathematica, for example, has an object that looks
like `PowerSeries[X, X0, COEFS, NMIN, NMAX, DEN]', where X0 is the
point about which the power series is taken (we've been assuming this
was always zero), and NMIN, NMAX, and DEN allow pseudo-power-series
with fractional or negative powers.  Also, the `PowerSeries' objects
have a special display format that makes them look like `2 x^2 +
O(x^4)' when they are printed out.  (*Note Compositions::, for a way to
do this in Calc, although for something as involved as this it would
probably be better to write the formatting routine in Lisp.)

File: calc,  Node: Programming Answer 1,  Next: Programming Answer 2,  Prev: Rewrites Answer 6,  Up: Answers to Exercises

4.7.58 Programming Tutorial Exercise 1
--------------------------------------

Just enter the formula `ninteg(sin(t)/t, t, 0, x)', type `Z F', and
answer the questions.  Since this formula contains two variables, the
default argument list will be `(t x)'.  We want to change this to `(x)'
since `t' is really a dummy variable to be used within `ninteg'.

   The exact keystrokes are `Z F s Si <RET> <RET> C-b C-b <DEL> <DEL>
<RET> y'.  (The `C-b C-b <DEL> <DEL>' are what fix the argument list.)

File: calc,  Node: Programming Answer 2,  Next: Programming Answer 3,  Prev: Programming Answer 1,  Up: Answers to Exercises

4.7.59 Programming Tutorial Exercise 2
--------------------------------------

One way is to move the number to the top of the stack, operate on it,
then move it back:  `C-x ( M-<TAB> n M-<TAB> M-<TAB> C-x )'.

   Another way is to negate the top three stack entries, then negate
again the top two stack entries:  `C-x ( M-3 n M-2 n C-x )'.

   Finally, it turns out that a negative prefix argument causes a
command like `n' to operate on the specified stack entry only, which is
just what we want:  `C-x ( M-- 3 n C-x )'.

   Just for kicks, let's also do it algebraically:
`C-x ( ' -$$$, $$, $ <RET> C-x )'.

File: calc,  Node: Programming Answer 3,  Next: Programming Answer 4,  Prev: Programming Answer 2,  Up: Answers to Exercises

4.7.60 Programming Tutorial Exercise 3
--------------------------------------

Each of these functions can be computed using the stack, or using
algebraic entry, whichever way you prefer:

Computing `sin(x) / x':

   Using the stack:  `C-x (  <RET> S <TAB> /  C-x )'.

   Using algebraic entry:  `C-x (  ' sin($)/$ <RET>  C-x )'.

Computing the logarithm:

   Using the stack:  `C-x (  <TAB> B  C-x )'

   Using algebraic entry:  `C-x (  ' log($,$$) <RET>  C-x )'.

Computing the vector of integers:

   Using the stack:  `C-x (  1 <RET> 1  C-u v x  C-x )'.  (Recall that
`C-u v x' takes the vector size, starting value, and increment from the
stack.)

   Alternatively:  `C-x (  ~ v x  C-x )'.  (The `~' key pops a number
from the stack and uses it as the prefix argument for the next command.)

   Using algebraic entry:  `C-x (  ' index($) <RET>  C-x )'.

File: calc,  Node: Programming Answer 4,  Next: Programming Answer 5,  Prev: Programming Answer 3,  Up: Answers to Exercises

4.7.61 Programming Tutorial Exercise 4
--------------------------------------

Here's one way:  `C-x ( <RET> V R + <TAB> v l / C-x )'.

File: calc,  Node: Programming Answer 5,  Next: Programming Answer 6,  Prev: Programming Answer 4,  Up: Answers to Exercises

4.7.62 Programming Tutorial Exercise 5
--------------------------------------

     2:  1              1:  1.61803398502         2:  1.61803398502
     1:  20                 .                     1:  1.61803398875
         .                                            .

        1 <RET> 20         Z < & 1 + Z >                I H P

This answer is quite accurate.

File: calc,  Node: Programming Answer 6,  Next: Programming Answer 7,  Prev: Programming Answer 5,  Up: Answers to Exercises

4.7.63 Programming Tutorial Exercise 6
--------------------------------------

Here is the matrix:

     [ [ 0, 1 ]   * [a, b] = [b, a + b]
       [ 1, 1 ] ]

Thus `[0, 1; 1, 1]^n * [1, 1]' computes Fibonacci numbers `n+1' and
`n+2'.  Here's one program that does the job:

     C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] <RET> v u <DEL> C-x )

This program is quite efficient because Calc knows how to raise a
matrix (or other value) to the power `n' in only `log(n,2)' steps.  For
example, this program can compute the 1000th Fibonacci number (a
209-digit integer!) in about 10 steps; even though the `Z < ... Z >'
solution had much simpler steps, it would have required so many steps
that it would not have been practical.

File: calc,  Node: Programming Answer 7,  Next: Programming Answer 8,  Prev: Programming Answer 6,  Up: Answers to Exercises

4.7.64 Programming Tutorial Exercise 7
--------------------------------------

The trick here is to compute the harmonic numbers differently, so that
the loop counter itself accumulates the sum of reciprocals.  We use a
separate variable to hold the integer counter.

     1:  1          2:  1       1:  .
         .          1:  4
                        .

         1 t 1       1 <RET> 4      Z ( t 2 r 1 1 + s 1 & Z )

The body of the loop goes as follows:  First save the harmonic sum so
far in variable 2.  Then delete it from the stack; the for loop itself
will take care of remembering it for us.  Next, recall the count from
variable 1, add one to it, and feed its reciprocal to the for loop to
use as the step value.  The for loop will increase the "loop counter"
by that amount and keep going until the loop counter exceeds 4.

     2:  31                  3:  31
     1:  3.99498713092       2:  3.99498713092
         .                   1:  4.02724519544
                                 .

         r 1 r 2                 <RET> 31 & +

   Thus we find that the 30th harmonic number is 3.99, and the 31st
harmonic number is 4.02.

File: calc,  Node: Programming Answer 8,  Next: Programming Answer 9,  Prev: Programming Answer 7,  Up: Answers to Exercises

4.7.65 Programming Tutorial Exercise 8
--------------------------------------

The first step is to compute the derivative `f'(x)' and thus the formula
`x - f(x)/f'(x)'.

   (Because this definition is long, it will be repeated in concise form
below.  You can use `C-x * m' to load it from there.  While you are
entering a `Z ` Z '' body in a macro, Calc simply collects keystrokes
without executing them.  In the following diagrams we'll pretend Calc
actually executed the keystrokes as you typed them, just for purposes
of illustration.)

     2:  sin(cos(x)) - 0.5            3:  4.5
     1:  4.5                          2:  sin(cos(x)) - 0.5
         .                            1:  -(sin(x) cos(cos(x)))
                                          .

     ' sin(cos(x))-0.5 <RET> 4.5  m r  C-x ( Z `  <TAB> <RET> a d x <RET>

     2:  4.5
     1:  x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
         .

         /  ' x <RET> <TAB> -   t 1

   Now, we enter the loop.  We'll use a repeat loop with a 20-repetition
limit just in case the method fails to converge for some reason.
(Normally, the `Z /' command will stop the loop before all 20
repetitions are done.)

     1:  4.5         3:  4.5                     2:  4.5
         .           2:  x + (sin(cos(x)) ...    1:  5.24196456928
                     1:  4.5                         .
                         .

       20 Z <          <RET> r 1 <TAB>                 s l x <RET>

   This is the new guess for `x'.  Now we compare it with the old one
to see if we've converged.

     3:  5.24196     2:  5.24196     1:  5.24196     1:  5.26345856348
     2:  5.24196     1:  0               .               .
     1:  4.5             .
         .

       <RET> M-<TAB>         a =             Z /             Z > Z ' C-x )

   The loop converges in just a few steps to this value.  To check the
result, we can simply substitute it back into the equation.

     2:  5.26345856348
     1:  0.499999999997
         .

      <RET> ' sin(cos($)) <RET>

   Let's test the new definition again:

     2:  x^2 - 9           1:  3.
     1:  1                     .
         .

       ' x^2-9 <RET> 1           X

   Once again, here's the full Newton's Method definition:

     C-x ( Z `  <TAB> <RET> a d x <RET>  /  ' x <RET> <TAB> -  t 1
                20 Z <  <RET> r 1 <TAB>  s l x <RET>
                        <RET> M-<TAB>  a =  Z /
                   Z >
           Z '
     C-x )

   It turns out that Calc has a built-in command for applying a formula
repeatedly until it converges to a number.  *Note Nesting and Fixed
Points::, to see how to use it.

   Also, of course, `a R' is a built-in command that uses Newton's
method (among others) to look for numerical solutions to any equation.
*Note Root Finding::.

File: calc,  Node: Programming Answer 9,  Next: Programming Answer 10,  Prev: Programming Answer 8,  Up: Answers to Exercises

4.7.66 Programming Tutorial Exercise 9
--------------------------------------

The first step is to adjust `z' to be greater than 5.  A simple "for"
loop will do the job here.  If `z' is less than 5, we reduce the
problem using `psi(z) = psi(z+1) - 1/z'.  We go on to compute
`psi(z+1)', and remember to add back a factor of `-1/z' when we're
done.  This step is repeated until `z > 5'.

   (Because this definition is long, it will be repeated in concise form
below.  You can use `C-x * m' to load it from there.  While you are
entering a `Z ` Z '' body in a macro, Calc simply collects keystrokes
without executing them.  In the following diagrams we'll pretend Calc
actually executed the keystrokes as you typed them, just for purposes
of illustration.)

     1:  1.             1:  1.
         .                  .

      1.0 <RET>       C-x ( Z `  s 1  0 t 2

   Here, variable 1 holds `z' and variable 2 holds the adjustment
factor.  If `z < 5', we use a loop to increase it.

   (By the way, we started with `1.0' instead of the integer 1 because
otherwise the calculation below will try to do exact fractional
arithmetic, and will never converge because fractions compare equal
only if they are exactly equal, not just equal to within the current
precision.)

     3:  1.      2:  1.       1:  6.
     2:  1.      1:  1            .
     1:  5           .
         .

       <RET> 5        a <    Z [  5 Z (  & s + 2  1 s + 1  1 Z ) r 1  Z ]

   Now we compute the initial part of the sum: `ln(z) - 1/2z' minus the
adjustment factor.

     2:  1.79175946923      2:  1.7084261359      1:  -0.57490719743
     1:  0.0833333333333    1:  2.28333333333         .
         .                      .

         L  r 1 2 * &           -  r 2                -

   Now we evaluate the series.  We'll use another "for" loop counting
up the value of `2 n'.  (Calc does have a summation command, `a +', but
we'll use loops just to get more practice with them.)

     3:  -0.5749       3:  -0.5749        4:  -0.5749      2:  -0.5749
     2:  2             2:  1:6            3:  1:6          1:  2.3148e-3
     1:  40            1:  2              2:  2                .
         .                 .              1:  36.
                                              .

        2 <RET> 40        Z ( <RET> k b <TAB>     <RET> r 1 <TAB> ^      * /

     3:  -0.5749       3:  -0.5772      2:  -0.5772     1:  -0.577215664892
     2:  -0.5749       2:  -0.5772      1:  0               .
     1:  2.3148e-3     1:  -0.5749          .
         .                 .

       <TAB> <RET> M-<TAB>       - <RET> M-<TAB>      a =     Z /    2  Z )  Z ' C-x )

   This is the value of `- gamma', with a slight bit of roundoff error.
To get a full 12 digits, let's use a higher precision:

     2:  -0.577215664892      2:  -0.577215664892
     1:  1.                   1:  -0.577215664901532

         1. <RET>                   p 16 <RET> X

   Here's the complete sequence of keystrokes:

     C-x ( Z `  s 1  0 t 2
                <RET> 5 a <  Z [  5 Z (  & s + 2  1 s + 1  1 Z ) r 1  Z ]
                L r 1 2 * & - r 2 -
                2 <RET> 40  Z (  <RET> k b <TAB> <RET> r 1 <TAB> ^ * /
                               <TAB> <RET> M-<TAB> - <RET> M-<TAB> a = Z /
                       2  Z )
           Z '
     C-x )

File: calc,  Node: Programming Answer 10,  Next: Programming Answer 11,  Prev: Programming Answer 9,  Up: Answers to Exercises

4.7.67 Programming Tutorial Exercise 10
---------------------------------------

Taking the derivative of a term of the form `x^n' will produce a term
like `n x^(n-1)'.  Taking the derivative of a constant produces zero.
From this it is easy to see that the `n'th derivative of a polynomial,
evaluated at `x = 0', will equal the coefficient on the `x^n' term
times `n!'.

   (Because this definition is long, it will be repeated in concise form
below.  You can use `C-x * m' to load it from there.  While you are
entering a `Z ` Z '' body in a macro, Calc simply collects keystrokes
without executing them.  In the following diagrams we'll pretend Calc
actually executed the keystrokes as you typed them, just for purposes
of illustration.)

     2:  5 x^4 + (x + 1)^2          3:  5 x^4 + (x + 1)^2
     1:  6                          2:  0
         .                          1:  6
                                        .

       ' 5 x^4 + (x+1)^2 <RET> 6        C-x ( Z `  [ ] t 1  0 <TAB>

Variable 1 will accumulate the vector of coefficients.

     2:  0              3:  0                  2:  5 x^4 + ...
     1:  5 x^4 + ...    2:  5 x^4 + ...        1:  1
         .              1:  1                      .
                            .

        Z ( <TAB>         <RET> 0 s l x <RET>            M-<TAB> ! /  s | 1

Note that `s | 1' appends the top-of-stack value to the vector in a
variable; it is completely analogous to `s + 1'.  We could have written
instead, `r 1 <TAB> | t 1'.

     1:  20 x^3 + 2 x + 2      1:  0         1:  [1, 2, 1, 0, 5, 0, 0]
         .                         .             .

         a d x <RET>                 1 Z )         <DEL> r 1  Z ' C-x )

   To convert back, a simple method is just to map the coefficients
against a table of powers of `x'.

     2:  [1, 2, 1, 0, 5, 0, 0]    2:  [1, 2, 1, 0, 5, 0, 0]
     1:  6                        1:  [0, 1, 2, 3, 4, 5, 6]
         .                            .

         6 <RET>                        1 + 0 <RET> 1 C-u v x

     2:  [1, 2, 1, 0, 5, 0, 0]    2:  1 + 2 x + x^2 + 5 x^4
     1:  [1, x, x^2, x^3, ... ]       .
         .

         ' x <RET> <TAB> V M ^            *

   Once again, here are the whole polynomial to/from vector programs:

     C-x ( Z `  [ ] t 1  0 <TAB>
                Z (  <TAB> <RET> 0 s l x <RET> M-<TAB> ! /  s | 1
                     a d x <RET>
              1 Z ) r 1
           Z '
     C-x )

     C-x (  1 + 0 <RET> 1 C-u v x ' x <RET> <TAB> V M ^ *  C-x )

File: calc,  Node: Programming Answer 11,  Next: Programming Answer 12,  Prev: Programming Answer 10,  Up: Answers to Exercises

4.7.68 Programming Tutorial Exercise 11
---------------------------------------

First we define a dummy program to go on the `z s' key.  The true `z s'
key is supposed to take two numbers from the stack and return one
number, so <DEL> as a dummy definition will make sure the stack comes
out right.

     2:  4          1:  4                         2:  4
     1:  2              .                         1:  2
         .                                            .

       4 <RET> 2       C-x ( <DEL> C-x )  Z K s <RET>       2

   The last step replaces the 2 that was eaten during the creation of
the dummy `z s' command.  Now we move on to the real definition.  The
recurrence needs to be rewritten slightly, to the form `s(n,m) =
s(n-1,m-1) - (n-1) s(n-1,m)'.

   (Because this definition is long, it will be repeated in concise form
below.  You can use `C-x * m' to load it from there.)

     2:  4        4:  4       3:  4       2:  4
     1:  2        3:  2       2:  2       1:  2
         .        2:  4       1:  0           .
                  1:  2           .
                      .

       C-x (       M-2 <RET>        a =         Z [  <DEL> <DEL> 1  Z :

     4:  4       2:  4                     2:  3      4:  3    4:  3    3:  3
     3:  2       1:  2                     1:  2      3:  2    3:  2    2:  2
     2:  2           .                         .      2:  3    2:  3    1:  3
     1:  0                                            1:  2    1:  1        .
         .                                                .        .

       <RET> 0   a = Z [  <DEL> <DEL> 0  Z :  <TAB> 1 - <TAB>   M-2 <RET>     1 -      z s

(Note that the value 3 that our dummy `z s' produces is not correct; it
is merely a placeholder that will do just as well for now.)

     3:  3               4:  3           3:  3       2:  3      1:  -6
     2:  3               3:  3           2:  3       1:  9          .
     1:  2               2:  3           1:  3           .
         .               1:  2               .
                             .

      M-<TAB> M-<TAB>     <TAB> <RET> M-<TAB>         z s          *          -

     1:  -6                          2:  4          1:  11      2:  11
         .                           1:  2              .       1:  11
                                         .                          .

       Z ] Z ] C-x )   Z K s <RET>      <DEL> 4 <RET> 2       z s      M-<RET> k s

   Even though the result that we got during the definition was highly
bogus, once the definition is complete the `z s' command gets the right
answers.

   Here's the full program once again:

     C-x (  M-2 <RET> a =
            Z [  <DEL> <DEL> 1
            Z :  <RET> 0 a =
                 Z [  <DEL> <DEL> 0
                 Z :  <TAB> 1 - <TAB> M-2 <RET> 1 - z s
                      M-<TAB> M-<TAB> <TAB> <RET> M-<TAB> z s * -
                 Z ]
            Z ]
     C-x )

   You can read this definition using `C-x * m' (`read-kbd-macro')
followed by `Z K s', without having to make a dummy definition first,
because `read-kbd-macro' doesn't need to execute the definition as it
reads it in.  For this reason, `C-x * m' is often the easiest way to
create recursive programs in Calc.

File: calc,  Node: Programming Answer 12,  Prev: Programming Answer 11,  Up: Answers to Exercises

4.7.69 Programming Tutorial Exercise 12
---------------------------------------

This turns out to be a much easier way to solve the problem.  Let's
denote Stirling numbers as calls of the function `s'.

   First, we store the rewrite rules corresponding to the definition of
Stirling numbers in a convenient variable:

     s e StirlingRules <RET>
     [ s(n,n) := 1  :: n >= 0,
       s(n,0) := 0  :: n > 0,
       s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
     C-c C-c

   Now, it's just a matter of applying the rules:

     2:  4          1:  s(4, 2)              1:  11
     1:  2              .                        .
         .

       4 <RET> 2       C-x (  ' s($$,$) <RET>     a r StirlingRules <RET>  C-x )

   As in the case of the `fib' rules, it would be useful to put these
rules in `EvalRules' and to add a `:: remember' condition to the last
rule.

File: calc,  Node: Introduction,  Next: Data Types,  Prev: Tutorial,  Up: Top

5 Introduction
**************

This chapter is the beginning of the Calc reference manual.  It covers
basic concepts such as the stack, algebraic and numeric entry, undo,
numeric prefix arguments, etc.

* Menu:

* Basic Commands::
* Help Commands::
* Stack Basics::
* Numeric Entry::
* Algebraic Entry::
* Quick Calculator::
* Prefix Arguments::
* Undo::
* Error Messages::
* Multiple Calculators::
* Troubleshooting Commands::

File: calc,  Node: Basic Commands,  Next: Help Commands,  Prev: Introduction,  Up: Introduction

5.1 Basic Commands
==================

To start the Calculator in its standard interface, type `M-x calc'.  By
default this creates a pair of small windows, `*Calculator*' and `*Calc
Trail*'.  The former displays the contents of the Calculator stack and
is manipulated exclusively through Calc commands.  It is possible
(though not usually necessary) to create several Calc mode buffers each
of which has an independent stack, undo list, and mode settings.  There
is exactly one Calc Trail buffer; it records a list of the results of
all calculations that have been done.  The Calc Trail buffer uses a
variant of Calc mode, so Calculator commands still work when the trail
buffer's window is selected.  It is possible to turn the trail window
off, but the `*Calc Trail*' buffer itself still exists and is updated
silently.  *Note Trail Commands::.

   In most installations, the `C-x * c' key sequence is a more
convenient way to start the Calculator.  Also, `C-x * *' is a synonym
for `C-x * c' unless you last used Calc in its Keypad mode.

   Most Calc commands use one or two keystrokes.  Lower- and upper-case
letters are distinct.  Commands may also be entered in full `M-x' form;
for some commands this is the only form.  As a convenience, the `x' key
(`calc-execute-extended-command') is like `M-x' except that it enters
the initial string `calc-' for you.  For example, the following key
sequences are equivalent: `S', `M-x calc-sin <RET>', `x sin <RET>'.

   Although Calc is designed to be used from the keyboard, some of
Calc's more common commands are available from a menu.  In the menu, the
arguments to the functions are given by referring to their stack level
numbers.

   The Calculator exists in many parts.  When you type `C-x * c', the
Emacs "auto-load" mechanism will bring in only the first part, which
contains the basic arithmetic functions.  The other parts will be
auto-loaded the first time you use the more advanced commands like trig
functions or matrix operations.  This is done to improve the response
time of the Calculator in the common case when all you need to do is a
little arithmetic.  If for some reason the Calculator fails to load an
extension module automatically, you can force it to load all the
extensions by using the `C-x * L' (`calc-load-everything') command.
*Note Mode Settings::.

   If you type `M-x calc' or `C-x * c' with any numeric prefix argument,
the Calculator is loaded if necessary, but it is not actually started.
If the argument is positive, the `calc-ext' extensions are also loaded
if necessary.  User-written Lisp code that wishes to make use of Calc's
arithmetic routines can use `(calc 0)' or `(calc 1)' to auto-load the
Calculator.

   If you type `C-x * b', then next time you use `C-x * c' you will get
a Calculator that uses the full height of the Emacs screen.  When
full-screen mode is on, `C-x * c' runs the `full-calc' command instead
of `calc'.  From the Unix shell you can type `emacs -f full-calc' to
start a new Emacs specifically for use as a calculator.  When Calc is
started from the Emacs command line like this, Calc's normal "quit"
commands actually quit Emacs itself.

   The `C-x * o' command is like `C-x * c' except that the Calc window
is not actually selected.  If you are already in the Calc window, `C-x
* o' switches you out of it.  (The regular Emacs `C-x o' command would
also work for this, but it has a tendency to drop you into the Calc
Trail window instead, which `C-x * o' takes care not to do.)

   For one quick calculation, you can type `C-x * q' (`quick-calc')
which prompts you for a formula (like `2+3/4').  The result is
displayed at the bottom of the Emacs screen without ever creating any
special Calculator windows.  *Note Quick Calculator::.

   Finally, if you are using the X window system you may want to try
`C-x * k' (`calc-keypad') which runs Calc with a "calculator keypad"
picture as well as a stack display.  Click on the keys with the mouse
to operate the calculator.  *Note Keypad Mode::.

   The `q' key (`calc-quit') exits Calc mode and closes the
Calculator's window(s).  It does not delete the Calculator buffers.  If
you type `M-x calc' again, the Calculator will reappear with the
contents of the stack intact.  Typing `C-x * c' or `C-x * *' again from
inside the Calculator buffer is equivalent to executing `calc-quit';
you can think of `C-x * *' as toggling the Calculator on and off.

   The `C-x * x' command also turns the Calculator off, no matter which
user interface (standard, Keypad, or Embedded) is currently active.  It
also cancels `calc-edit' mode if used from there.

   The `d <SPC>' key sequence (`calc-refresh') redraws the contents of
the Calculator buffer from memory.  Use this if the contents of the
buffer have been damaged somehow.

   The `o' key (`calc-realign') moves the cursor back to its "home"
position at the bottom of the Calculator buffer.

   The `<' and `>' keys are bound to `calc-scroll-left' and
`calc-scroll-right'.  These are just like the normal horizontal
scrolling commands except that they scroll one half-screen at a time by
default.  (Calc formats its output to fit within the bounds of the
window whenever it can.)

   The `{' and `}' keys are bound to `calc-scroll-down' and
`calc-scroll-up'.  They scroll up or down by one-half the height of the
Calc window.

   The `C-x * 0' command (`calc-reset'; that's `C-x *' followed by a
zero) resets the Calculator to its initial state.  This clears the
stack, resets all the modes to their initial values (the values that
were saved with `m m' (`calc-save-modes')), clears the caches (*note
Caches::), and so on.  (It does _not_ erase the values of any
variables.) With an argument of 0, Calc will be reset to its default
state; namely, the modes will be given their default values.  With a
positive prefix argument, `C-x * 0' preserves the contents of the stack
but resets everything else to its initial state; with a negative prefix
argument, `C-x * 0' preserves the contents of the stack but resets
everything else to its default state.

File: calc,  Node: Help Commands,  Next: Stack Basics,  Prev: Basic Commands,  Up: Introduction

5.2 Help Commands
=================

The `?' key (`calc-help') displays a series of brief help messages.
Some keys (such as `b' and `d') are prefix keys, like Emacs' <ESC> and
`C-x' prefixes.  You can type `?' after a prefix to see a list of
commands beginning with that prefix.  (If the message includes
`[MORE]', press `?' again to see additional commands for that prefix.)

   The `h h' (`calc-full-help') command displays all the `?' responses
at once.  When printed, this makes a nice, compact (three pages)
summary of Calc keystrokes.

   In general, the `h' key prefix introduces various commands that
provide help within Calc.  Many of the `h' key functions are
Calc-specific analogues to the `C-h' functions for Emacs help.

   The `h i' (`calc-info') command runs the Emacs Info system to read
this manual on-line.  This is basically the same as typing `C-h i' (the
regular way to run the Info system), then, if Info is not already in
the Calc manual, selecting the beginning of the manual.  The `C-x * i'
command is another way to read the Calc manual; it is different from `h
i' in that it works any time, not just inside Calc.  The plain `i' key
is also equivalent to `h i', though this key is obsolete and may be
replaced with a different command in a future version of Calc.

   The `h t' (`calc-tutorial') command runs the Info system on the
Tutorial section of the Calc manual.  It is like `h i', except that it
selects the starting node of the tutorial rather than the beginning of
the whole manual.  (It actually selects the node "Interactive Tutorial"
which tells a few things about using the Info system before going on to
the actual tutorial.)  The `C-x * t' key is equivalent to `h t' (but it
works at all times).

   The `h s' (`calc-info-summary') command runs the Info system on the
Summary node of the Calc manual.  *Note Summary::.  The `C-x * s' key
is equivalent to `h s'.

   The `h k' (`calc-describe-key') command looks up a key sequence in
the Calc manual.  For example, `h k H a S' looks up the documentation
on the `H a S' (`calc-solve-for') command.  This works by looking up
the textual description of the key(s) in the Key Index of the manual,
then jumping to the node indicated by the index.

   Most Calc commands do not have traditional Emacs documentation
strings, since the `h k' command is both more convenient and more
instructive.  This means the regular Emacs `C-h k' (`describe-key')
command will not be useful for Calc keystrokes.

   The `h c' (`calc-describe-key-briefly') command reads a key sequence
and displays a brief one-line description of it at the bottom of the
screen.  It looks for the key sequence in the Summary node of the Calc
manual; if it doesn't find the sequence there, it acts just like its
regular Emacs counterpart `C-h c' (`describe-key-briefly').  For
example, `h c H a S' gives the description:

     H a S runs calc-solve-for:  a `H a S' v  => fsolve(a,v)  (?=notes)

which means the command `H a S' or `H M-x calc-solve-for' takes a value
`a' from the stack, prompts for a value `v', then applies the algebraic
function `fsolve' to these values.  The `?=notes' message means you can
now type `?' to see additional notes from the summary that apply to
this command.

   The `h f' (`calc-describe-function') command looks up an algebraic
function or a command name in the Calc manual.  Enter an algebraic
function name to look up that function in the Function Index or enter a
command name beginning with `calc-' to look it up in the Command Index.
This command will also look up operator symbols that can appear in
algebraic formulas, like `%' and `=>'.

   The `h v' (`calc-describe-variable') command looks up a variable in
the Calc manual.  Enter a variable name like `pi' or `PlotRejects'.

   The `h b' (`calc-describe-bindings') command is just like `C-h b',
except that only local (Calc-related) key bindings are listed.

   The `h n' or `h C-n' (`calc-view-news') command displays the "news"
or change history of Calc.  This is kept in the file `README', which
Calc looks for in the same directory as the Calc source files.

   The `h C-c', `h C-d', and `h C-w' keys display copying,
distribution, and warranty information about Calc.  These work by
pulling up the appropriate parts of the "Copying" or "Reporting Bugs"
sections of the manual.

File: calc,  Node: Stack Basics,  Next: Numeric Entry,  Prev: Help Commands,  Up: Introduction

5.3 Stack Basics
================

Calc uses RPN notation.  If you are not familiar with RPN, *note RPN
Tutorial::.

   To add the numbers 1 and 2 in Calc you would type the keys: `1 <RET>
2 +'.  (<RET> corresponds to the <ENTER> key on most calculators.)  The
first three keystrokes "push" the numbers 1 and 2 onto the stack.  The
`+' key always "pops" the top two numbers from the stack, adds them,
and pushes the result (3) back onto the stack.  This number is ready for
further calculations:  `5 -' pushes 5 onto the stack, then pops the 3
and 5, subtracts them, and pushes the result (-2).

   Note that the "top" of the stack actually appears at the _bottom_ of
the buffer.  A line containing a single `.' character signifies the end
of the buffer; Calculator commands operate on the number(s) directly
above this line.  The `d t' (`calc-truncate-stack') command allows you
to move the `.' marker up and down in the stack; *note Truncating the
Stack::.

   Stack elements are numbered consecutively, with number 1 being the
top of the stack.  These line numbers are ordinarily displayed on the
lefthand side of the window.  The `d l' (`calc-line-numbering') command
controls whether these numbers appear.  (Line numbers may be turned off
since they slow the Calculator down a bit and also clutter the display.)

   The unshifted letter `o' (`calc-realign') command repositions the
cursor to its top-of-stack "home" position.  It also undoes any
horizontal scrolling in the window.  If you give it a numeric prefix
argument, it instead moves the cursor to the specified stack element.

   The <RET> (or equivalent <SPC>) key is only required to separate two
consecutive numbers.  (After all, if you typed `1 2' by themselves the
Calculator would enter the number 12.)  If you press <RET> or <SPC>
_not_ right after typing a number, the key duplicates the number on the
top of the stack.  `<RET> *' is thus a handy way to square a number.

   The <DEL> key pops and throws away the top number on the stack.  The
<TAB> key swaps the top two objects on the stack.  *Note Stack and
Trail::, for descriptions of these and other stack-related commands.

File: calc,  Node: Numeric Entry,  Next: Algebraic Entry,  Prev: Stack Basics,  Up: Introduction

5.4 Numeric Entry
=================

Pressing a digit or other numeric key begins numeric entry using the
minibuffer.  The number is pushed on the stack when you press the <RET>
or <SPC> keys.  If you press any other non-numeric key, the number is
pushed onto the stack and the appropriate operation is performed.  If
you press a numeric key which is not valid, the key is ignored.

   There are three different concepts corresponding to the word "minus,"
typified by `a-b' (subtraction), `-x' (change-sign), and `-5' (negative
number).  Calc uses three different keys for these operations,
respectively: `-', `n', and `_' (the underscore).  The `-' key subtracts
the two numbers on the top of the stack.  The `n' key changes the sign
of the number on the top of the stack or the number currently being
entered.  The `_' key begins entry of a negative number or changes the
sign of the number currently being entered.  The following sequences
all enter the number -5 onto the stack:  `0 <RET> 5 -', `5 n <RET>', `5
<RET> n', `_ 5 <RET>', `5 _ <RET>'.

   Some other keys are active during numeric entry, such as `#' for
non-decimal numbers, `:' for fractions, and `@' for HMS forms.  These
notations are described later in this manual with the corresponding
data types.  *Note Data Types::.

   During numeric entry, the only editing key available is <DEL>.

File: calc,  Node: Algebraic Entry,  Next: Quick Calculator,  Prev: Numeric Entry,  Up: Introduction

5.5 Algebraic Entry
===================

The `'' (`calc-algebraic-entry') command can be used to enter
calculations in algebraic form.  This is accomplished by typing the
apostrophe key, ', followed by the expression in standard format:

     ' 2+3*4 <RET>.

This will compute `2+(3*4) = 14' and push it on the stack.  If you wish
you can ignore the RPN aspect of Calc altogether and simply enter
algebraic expressions in this way.  You may want to use <DEL> every so
often to clear previous results off the stack.

   You can press the apostrophe key during normal numeric entry to
switch the half-entered number into Algebraic entry mode.  One reason
to do this would be to fix a typo, as the full Emacs cursor motion and
editing keys are available during algebraic entry but not during
numeric entry.

   In the same vein, during either numeric or algebraic entry you can
press ``' (backquote) to switch to `calc-edit' mode, where you complete
your half-finished entry in a separate buffer.  *Note Editing Stack
Entries::.

   If you prefer algebraic entry, you can use the command `m a'
(`calc-algebraic-mode') to set Algebraic mode.  In this mode, digits
and other keys that would normally start numeric entry instead start
full algebraic entry; as long as your formula begins with a digit you
can omit the apostrophe.  Open parentheses and square brackets also
begin algebraic entry.  You can still do RPN calculations in this mode,
but you will have to press <RET> to terminate every number: `2 <RET> 3
<RET> * 4 <RET> +' would accomplish the same thing as `2*3+4 <RET>'.

   If you give a numeric prefix argument like `C-u' to the `m a'
command, it enables Incomplete Algebraic mode; this is like regular
Algebraic mode except that it applies to the `(' and `[' keys only.
Numeric keys still begin a numeric entry in this mode.

   The `m t' (`calc-total-algebraic-mode') gives you an even stronger
algebraic-entry mode, in which _all_ regular letter and punctuation
keys begin algebraic entry.  Use this if you prefer typing `sqrt( )'
instead of `Q', `factor( )' instead of `a f', and so on.  To type
regular Calc commands when you are in Total Algebraic mode, hold down
the <META> key.  Thus `M-q' is the command to quit Calc, `M-p' sets the
precision, and `M-m t' (or `M-m M-t', if you prefer) turns Total
Algebraic mode back off again.  Meta keys also terminate algebraic
entry, so that `2+3 M-S' is equivalent to `2+3 <RET> M-S'.  The symbol
`Alg*' will appear in the mode line whenever you are in this mode.

   Pressing `'' (the apostrophe) a second time re-enters the previous
algebraic formula.  You can then use the normal Emacs editing keys to
modify this formula to your liking before pressing <RET>.

   Within a formula entered from the keyboard, the symbol `$'
represents the number on the top of the stack.  If an entered formula
contains any `$' characters, the Calculator replaces the top of stack
with that formula rather than simply pushing the formula onto the
stack.  Thus, `' 1+2 <RET>' pushes 3 on the stack, and `$*2 <RET>'
replaces it with 6.  Note that the `$' key always initiates algebraic
entry; the `'' is unnecessary if `$' is the first character in the new
formula.

   Higher stack elements can be accessed from an entered formula with
the symbols `$$', `$$$', and so on.  The number of stack elements
removed (to be replaced by the entered values) equals the number of
dollar signs in the longest such symbol in the formula.  For example,
`$$+$$$' adds the second and third stack elements, replacing the top
three elements with the answer.  (All information about the top stack
element is thus lost since no single `$' appears in this formula.)

   A slightly different way to refer to stack elements is with a dollar
sign followed by a number:  `$1', `$2', and so on are much like `$',
`$$', etc., except that stack entries referred to numerically are not
replaced by the algebraic entry.  That is, while `$+1' replaces 5 on
the stack with 6, `$1+1' leaves the 5 on the stack and pushes an
additional 6.

   If a sequence of formulas are entered separated by commas, each
formula is pushed onto the stack in turn.  For example, `1,2,3' pushes
those three numbers onto the stack (leaving the 3 at the top), and
`$+1,$-1' replaces a 5 on the stack with 4 followed by 6.  Also, `$,$$'
exchanges the top two elements of the stack, just like the <TAB> key.

   You can finish an algebraic entry with `M-=' or `M-<RET>' instead of
<RET>.  This uses `=' to evaluate the variables in each formula that
goes onto the stack.  (Thus `' pi <RET>' pushes the variable `pi', but
`' pi M-<RET>' pushes 3.1415.)

   If you finish your algebraic entry by pressing <LFD> (or `C-j')
instead of <RET>, Calc disables the default simplifications (as if by
`m O'; *note Simplification Modes::) while the entry is being pushed on
the stack.  Thus `' 1+2 <RET>' pushes 3 on the stack, but `' 1+2 <LFD>'
pushes the formula `1+2'; you might then press `=' when it is time to
evaluate this formula.

File: calc,  Node: Quick Calculator,  Next: Prefix Arguments,  Prev: Algebraic Entry,  Up: Introduction

5.6 "Quick Calculator" Mode
===========================

There is another way to invoke the Calculator if all you need to do is
make one or two quick calculations.  Type `C-x * q' (or `M-x
quick-calc'), then type any formula as an algebraic entry.  The
Calculator will compute the result and display it in the echo area,
without ever actually putting up a Calc window.

   You can use the `$' character in a Quick Calculator formula to refer
to the previous Quick Calculator result.  Older results are not
retained; the Quick Calculator has no effect on the full Calculator's
stack or trail.  If you compute a result and then forget what it was,
just run `C-x * q' again and enter `$' as the formula.

   If this is the first time you have used the Calculator in this Emacs
session, the `C-x * q' command will create the `*Calculator*' buffer
and perform all the usual initializations; it simply will refrain from
putting that buffer up in a new window.  The Quick Calculator refers to
the `*Calculator*' buffer for all mode settings.  Thus, for example, to
set the precision that the Quick Calculator uses, simply run the full
Calculator momentarily and use the regular `p' command.

   If you use `C-x * q' from inside the Calculator buffer, the effect
is the same as pressing the apostrophe key (algebraic entry).

   The result of a Quick calculation is placed in the Emacs "kill ring"
as well as being displayed.  A subsequent `C-y' command will yank the
result into the editing buffer.  You can also use this to yank the
result into the next `C-x * q' input line as a more explicit
alternative to `$' notation, or to yank the result into the Calculator
stack after typing `C-x * c'.

   If you finish your formula by typing <LFD> (or `C-j') instead of
<RET>, the result is inserted immediately into the current buffer
rather than going into the kill ring.

   Quick Calculator results are actually evaluated as if by the `=' key
(which replaces variable names by their stored values, if any).  If the
formula you enter is an assignment to a variable using the `:='
operator, say, `foo := 2 + 3' or `foo := foo + 1', then the result of
the evaluation is stored in that Calc variable.  *Note Store and
Recall::.

   If the result is an integer and the current display radix is decimal,
the number will also be displayed in hex, octal and binary formats.  If
the integer is in the range from 1 to 126, it will also be displayed as
an ASCII character.

   For example, the quoted character `"x"' produces the vector result
`[120]' (because 120 is the ASCII code of the lower-case `x'; *note
Strings::).  Since this is a vector, not an integer, it is displayed
only according to the current mode settings.  But running Quick Calc
again and entering `120' will produce the result `120 (16#78, 8#170,
x)' which shows the number in its decimal, hexadecimal, octal, and
ASCII forms.

   Please note that the Quick Calculator is not any faster at loading
or computing the answer than the full Calculator; the name "quick"
merely refers to the fact that it's much less hassle to use for small
calculations.

File: calc,  Node: Prefix Arguments,  Next: Undo,  Prev: Quick Calculator,  Up: Introduction

5.7 Numeric Prefix Arguments
============================

Many Calculator commands use numeric prefix arguments.  Some, such as
`d s' (`calc-sci-notation'), set a parameter to the value of the prefix
argument or use a default if you don't use a prefix.  Others (like `d
f' (`calc-fix-notation')) require an argument and prompt for a number
if you don't give one as a prefix.

   As a rule, stack-manipulation commands accept a numeric prefix
argument which is interpreted as an index into the stack.  A positive
argument operates on the top N stack entries; a negative argument
operates on the Nth stack entry in isolation; and a zero argument
operates on the entire stack.

   Most commands that perform computations (such as the arithmetic and
scientific functions) accept a numeric prefix argument that allows the
operation to be applied across many stack elements.  For unary
operations (that is, functions of one argument like absolute value or
complex conjugate), a positive prefix argument applies that function to
the top N stack entries simultaneously, and a negative argument applies
it to the Nth stack entry only.  For binary operations (functions of
two arguments like addition, GCD, and vector concatenation), a positive
prefix argument "reduces" the function across the top N stack elements
(for example, `C-u 5 +' sums the top 5 stack entries; *note Reducing
and Mapping::), and a negative argument maps the next-to-top N stack
elements with the top stack element as a second argument (for example,
`7 c-u -5 +' adds 7 to the top 5 stack elements).  This feature is not
available for operations which use the numeric prefix argument for some
other purpose.

   Numeric prefixes are specified the same way as always in Emacs:
Press a sequence of <META>-digits, or press <ESC> followed by digits,
or press `C-u' followed by digits.  Some commands treat plain `C-u'
(without any actual digits) specially.

   You can type `~' (`calc-num-prefix') to pop an integer from the top
of the stack and enter it as the numeric prefix for the next command.
For example, `C-u 16 p' sets the precision to 16 digits; an alternate
(silly) way to do this would be `2 <RET> 4 ^ ~ p', i.e., compute 2 to
the fourth power and set the precision to that value.

   Conversely, if you have typed a numeric prefix argument the `~' key
pushes it onto the stack in the form of an integer.

File: calc,  Node: Undo,  Next: Error Messages,  Prev: Prefix Arguments,  Up: Introduction

5.8 Undoing Mistakes
====================

The shift-`U' key (`calc-undo') undoes the most recent operation.  If
that operation added or dropped objects from the stack, those objects
are removed or restored.  If it was a "store" operation, you are
queried whether or not to restore the variable to its original value.
The `U' key may be pressed any number of times to undo successively
farther back in time; with a numeric prefix argument it undoes a
specified number of operations.  The undo history is cleared only by the
`q' (`calc-quit') command.  (Recall that `C-x * c' is synonymous with
`calc-quit' while inside the Calculator; this also clears the undo
history.)

   Currently the mode-setting commands (like `calc-precision') are not
undoable.  You can undo past a point where you changed a mode, but you
will need to reset the mode yourself.

   The shift-`D' key (`calc-redo') redoes an operation that was
mistakenly undone.  Pressing `U' with a negative prefix argument is
equivalent to executing `calc-redo'.  You can redo any number of times,
up to the number of recent consecutive undo commands.  Redo information
is cleared whenever you give any command that adds new undo
information, i.e., if you undo, then enter a number on the stack or make
any other change, then it will be too late to redo.

   The `M-<RET>' key (`calc-last-args') is like undo in that it
restores the arguments of the most recent command onto the stack;
however, it does not remove the result of that command.  Given a numeric
prefix argument, this command applies to the `n'th most recent command
which removed items from the stack; it pushes those items back onto the
stack.

   The `K' (`calc-keep-args') command provides a related function to
`M-<RET>'.  *Note Stack and Trail::.

   It is also possible to recall previous results or inputs using the
trail.  *Note Trail Commands::.

   The standard Emacs `C-_' undo key is recognized as a synonym for `U'.

File: calc,  Node: Error Messages,  Next: Multiple Calculators,  Prev: Undo,  Up: Introduction

5.9 Error Messages
==================

Many situations that would produce an error message in other calculators
simply create unsimplified formulas in the Emacs Calculator.  For
example, `1 <RET> 0 /' pushes the formula `1 / 0'; `0 L' pushes the
formula `ln(0)'.  Floating-point overflow and underflow are also
reasons for this to happen.

   When a function call must be left in symbolic form, Calc usually
produces a message explaining why.  Messages that are probably
surprising or indicative of user errors are displayed automatically.
Other messages are simply kept in Calc's memory and are displayed only
if you type `w' (`calc-why').  You can also press `w' if the same
computation results in several messages.  (The first message will end
with `[w=more]' in this case.)

   The `d w' (`calc-auto-why') command controls when error messages are
displayed automatically.  (Calc effectively presses `w' for you after
your computation finishes.)  By default, this occurs only for
"important" messages.  The other possible modes are to report _all_
messages automatically, or to report none automatically (so that you
must always press `w' yourself to see the messages).

File: calc,  Node: Multiple Calculators,  Next: Troubleshooting Commands,  Prev: Error Messages,  Up: Introduction

5.10 Multiple Calculators
=========================

It is possible to have any number of Calc mode buffers at once.
Usually this is done by executing `M-x another-calc', which is similar
to `C-x * c' except that if a `*Calculator*' buffer already exists, a
new, independent one with a name of the form `*Calculator*<N>' is
created.  You can also use the command `calc-mode' to put any buffer
into Calculator mode, but this would ordinarily never be done.

   The `q' (`calc-quit') command does not destroy a Calculator buffer;
it only closes its window.  Use `M-x kill-buffer' to destroy a
Calculator buffer.

   Each Calculator buffer keeps its own stack, undo list, and mode
settings such as precision, angular mode, and display formats.  In
Emacs terms, variables such as `calc-stack' are buffer-local variables.
The global default values of these variables are used only when a new
Calculator buffer is created.  The `calc-quit' command saves the stack
and mode settings of the buffer being quit as the new defaults.

   There is only one trail buffer, `*Calc Trail*', used by all
Calculator buffers.

File: calc,  Node: Troubleshooting Commands,  Prev: Multiple Calculators,  Up: Introduction

5.11 Troubleshooting Commands
=============================

This section describes commands you can use in case a computation
incorrectly fails or gives the wrong answer.

   *Note Reporting Bugs::, if you find a problem that appears to be due
to a bug or deficiency in Calc.

* Menu:

* Autoloading Problems::
* Recursion Depth::
* Caches::
* Debugging Calc::

File: calc,  Node: Autoloading Problems,  Next: Recursion Depth,  Prev: Troubleshooting Commands,  Up: Troubleshooting Commands

5.11.1 Autoloading Problems
---------------------------

The Calc program is split into many component files; components are
loaded automatically as you use various commands that require them.
Occasionally Calc may lose track of when a certain component is
necessary; typically this means you will type a command and it won't
work because some function you've never heard of was undefined.

   If this happens, the easiest workaround is to type `C-x * L'
(`calc-load-everything') to force all the parts of Calc to be loaded
right away.  This will cause Emacs to take up a lot more memory than it
would otherwise, but it's guaranteed to fix the problem.

File: calc,  Node: Recursion Depth,  Next: Caches,  Prev: Autoloading Problems,  Up: Troubleshooting Commands

5.11.2 Recursion Depth
----------------------

Calc uses recursion in many of its calculations.  Emacs Lisp keeps a
variable `max-lisp-eval-depth' which limits the amount of recursion
possible in an attempt to recover from program bugs.  If a calculation
ever halts incorrectly with the message "Computation got stuck or ran
too long," use the `M' command (`calc-more-recursion-depth') to
increase this limit.  (Of course, this will not help if the calculation
really did get stuck due to some problem inside Calc.)

   The limit is always increased (multiplied) by a factor of two.  There
is also an `I M' (`calc-less-recursion-depth') command which decreases
this limit by a factor of two, down to a minimum value of 200.  The
default value is 1000.

   These commands also double or halve `max-specpdl-size', another
internal Lisp recursion limit.  The minimum value for this limit is 600.

File: calc,  Node: Caches,  Next: Debugging Calc,  Prev: Recursion Depth,  Up: Troubleshooting Commands

5.11.3 Caches
-------------

Calc saves certain values after they have been computed once.  For
example, the `P' (`calc-pi') command initially "knows" the constant
`pi' to about 20 decimal places; if the current precision is greater
than this, it will recompute `pi' using a series approximation.  This
value will not need to be recomputed ever again unless you raise the
precision still further.  Many operations such as logarithms and sines
make use of similarly cached values such as `pi/4' and `ln(2)'.  The
visible effect of caching is that high-precision computations may seem
to do extra work the first time.  Other things cached include powers of
two (for the binary arithmetic functions), matrix inverses and
determinants, symbolic integrals, and data points computed by the
graphing commands.

   If you suspect a Calculator cache has become corrupt, you can use the
`calc-flush-caches' command to reset all caches to the empty state.
(This should only be necessary in the event of bugs in the Calculator.)
The `C-x * 0' (with the zero key) command also resets caches along with
all other aspects of the Calculator's state.

File: calc,  Node: Debugging Calc,  Prev: Caches,  Up: Troubleshooting Commands

5.11.4 Debugging Calc
---------------------

A few commands exist to help in the debugging of Calc commands.  *Note
Programming::, to see the various ways that you can write your own Calc
commands.

   The `Z T' (`calc-timing') command turns on and off a mode in which
the timing of slow commands is reported in the Trail.  Any Calc command
that takes two seconds or longer writes a line to the Trail showing how
many seconds it took.  This value is accurate only to within one second.

   All steps of executing a command are included; in particular, time
taken to format the result for display in the stack and trail is
counted.  Some prompts also count time taken waiting for them to be
answered, while others do not; this depends on the exact implementation
of the command.  For best results, if you are timing a sequence that
includes prompts or multiple commands, define a keyboard macro to run
the whole sequence at once.  Calc's `X' command (*note Keyboard
Macros::) will then report the time taken to execute the whole macro.

   Another advantage of the `X' command is that while it is executing,
the stack and trail are not updated from step to step.  So if you
expect the output of your test sequence to leave a result that may take
a long time to format and you don't wish to count this formatting time,
end your sequence with a <DEL> keystroke to clear the result from the
stack.  When you run the sequence with `X', Calc will never bother to
format the large result.

   Another thing `Z T' does is to increase the Emacs variable
`gc-cons-threshold' to a much higher value (two million; the usual
default in Calc is 250,000) for the duration of each command.  This
generally prevents garbage collection during the timing of the command,
though it may cause your Emacs process to grow abnormally large.
(Garbage collection time is a major unpredictable factor in the timing
of Emacs operations.)

   Another command that is useful when debugging your own Lisp
extensions to Calc is `M-x calc-pass-errors', which disables the error
handler that changes the "`max-lisp-eval-depth' exceeded" message to
the much more friendly "Computation got stuck or ran too long."  This
handler interferes with the Emacs Lisp debugger's `debug-on-error'
mode.  Errors are reported in the handler itself rather than at the
true location of the error.  After you have executed
`calc-pass-errors', Lisp errors will be reported correctly but the
user-friendly message will be lost.

File: calc,  Node: Data Types,  Next: Stack and Trail,  Prev: Introduction,  Up: Top

6 Data Types
************

This chapter discusses the various types of objects that can be placed
on the Calculator stack, how they are displayed, and how they are
entered.  (*Note Data Type Formats::, for information on how these data
types are represented as underlying Lisp objects.)

   Integers, fractions, and floats are various ways of describing real
numbers.  HMS forms also for many purposes act as real numbers.  These
types can be combined to form complex numbers, modulo forms, error
forms, or interval forms.  (But these last four types cannot be combined
arbitrarily: error forms may not contain modulo forms, for example.)
Finally, all these types of numbers may be combined into vectors,
matrices, or algebraic formulas.

* Menu:

* Integers::                The most basic data type.
* Fractions::               This and above are called "rationals".
* Floats::                  This and above are called "reals".
* Complex Numbers::         This and above are called "numbers".
* Infinities::
* Vectors and Matrices::
* Strings::
* HMS Forms::
* Date Forms::
* Modulo Forms::
* Error Forms::
* Interval Forms::
* Incomplete Objects::
* Variables::
* Formulas::

File: calc,  Node: Integers,  Next: Fractions,  Prev: Data Types,  Up: Data Types

6.1 Integers
============

The Calculator stores integers to arbitrary precision.  Addition,
subtraction, and multiplication of integers always yields an exact
integer result.  (If the result of a division or exponentiation of
integers is not an integer, it is expressed in fractional or
floating-point form according to the current Fraction mode.  *Note
Fraction Mode::.)

   A decimal integer is represented as an optional sign followed by a
sequence of digits.  Grouping (*note Grouping Digits::) can be used to
insert a comma at every third digit for display purposes, but you must
not type commas during the entry of numbers.

   A non-decimal integer is represented as an optional sign, a radix
between 2 and 36, a `#' symbol, and one or more digits.  For radix 11
and above, the letters A through Z (upper- or lower-case) count as
digits and do not terminate numeric entry mode.  *Note Radix Modes::,
for how to set the default radix for display of integers.  Numbers of
any radix may be entered at any time.  If you press `#' at the
beginning of a number, the current display radix is used.

File: calc,  Node: Fractions,  Next: Floats,  Prev: Integers,  Up: Data Types

6.2 Fractions
=============

A "fraction" is a ratio of two integers.  Fractions are traditionally
written "2/3" but Calc uses the notation `2:3'.  (The `/' key performs
RPN division; the following two sequences push the number `2:3' on the
stack:  `2 : 3 <RET>', or `2 <RET> 3 /' assuming Fraction mode has been
enabled.)  When the Calculator produces a fractional result it always
reduces it to simplest form, which may in fact be an integer.

   Fractions may also be entered in a three-part form, where `2:3:4'
represents two-and-three-quarters.  *Note Fraction Formats::, for
fraction display formats.

   Non-decimal fractions are entered and displayed as `RADIX#NUM:DENOM'
(or in the analogous three-part form).  The numerator and denominator
always use the same radix.

File: calc,  Node: Floats,  Next: Complex Numbers,  Prev: Fractions,  Up: Data Types

6.3 Floats
==========

A floating-point number or "float" is a number stored in scientific
notation.  The number of significant digits in the fractional part is
governed by the current floating precision (*note Precision::).  The
range of acceptable values is from `10^-3999999' (inclusive) to
`10^4000000' (exclusive), plus the corresponding negative values and
zero.

   Calculations that would exceed the allowable range of values (such
as `exp(exp(20))') are left in symbolic form by Calc.  The messages
"floating-point overflow" or "floating-point underflow" indicate that
during the calculation a number would have been produced that was too
large or too close to zero, respectively, to be represented by Calc.
This does not necessarily mean the final result would have overflowed,
just that an overflow occurred while computing the result.  (In fact,
it could report an underflow even though the final result would have
overflowed!)

   If a rational number and a float are mixed in a calculation, the
result will in general be expressed as a float.  Commands that require
an integer value (such as `k g' [`gcd']) will also accept integer-valued
floats, i.e., floating-point numbers with nothing after the decimal
point.

   Floats are identified by the presence of a decimal point and/or an
exponent.  In general a float consists of an optional sign, digits
including an optional decimal point, and an optional exponent consisting
of an `e', an optional sign, and up to seven exponent digits.  For
example, `23.5e-2' is 23.5 times ten to the minus-second power, or
0.235.

   Floating-point numbers are normally displayed in decimal notation
with all significant figures shown.  Exceedingly large or small numbers
are displayed in scientific notation.  Various other display options are
available.  *Note Float Formats::.

   Floating-point numbers are stored in decimal, not binary.  The result
of each operation is rounded to the nearest value representable in the
number of significant digits specified by the current precision,
rounding away from zero in the case of a tie.  Thus (in the default
display mode) what you see is exactly what you get.  Some operations
such as square roots and transcendental functions are performed with
several digits of extra precision and then rounded down, in an effort
to make the final result accurate to the full requested precision.
However, accuracy is not rigorously guaranteed.  If you suspect the
validity of a result, try doing the same calculation in a higher
precision.  The Calculator's arithmetic is not intended to be
IEEE-conformant in any way.

   While floats are always _stored_ in decimal, they can be entered and
displayed in any radix just like integers and fractions.  Since a float
that is entered in a radix other that 10 will be converted to decimal,
the number that Calc stores may not be exactly the number that was
entered, it will be the closest decimal approximation given the current
precison.  The notation `RADIX#DDD.DDD' is a floating-point number
whose digits are in the specified radix.  Note that the `.'  is more
aptly referred to as a "radix point" than as a decimal point in this
case.  The number `8#123.4567' is defined as `8#1234567 * 8^-4'.  If
the radix is 14 or less, you can use `e' notation to write a
non-decimal number in scientific notation.  The exponent is written in
decimal, and is considered to be a power of the radix: `8#1234567e-4'.
If the radix is 15 or above, the letter `e' is a digit, so scientific
notation must be written out, e.g., `16#123.4567*16^2'.  The first two
exercises of the Modes Tutorial explore some of the properties of
non-decimal floats.

File: calc,  Node: Complex Numbers,  Next: Infinities,  Prev: Floats,  Up: Data Types

6.4 Complex Numbers
===================

There are two supported formats for complex numbers: rectangular and
polar.  The default format is rectangular, displayed in the form
`(REAL,IMAG)' where REAL is the real part and IMAG is the imaginary
part, each of which may be any real number.  Rectangular complex
numbers can also be displayed in `A+Bi' notation; *note Complex
Formats::.

   Polar complex numbers are displayed in the form `(R;THETA)' where R
is the nonnegative magnitude and THETA is the argument or phase angle.
The range of THETA depends on the current angular mode (*note Angular
Modes::); it is generally between -180 and +180 degrees or the
equivalent range in radians.

   Complex numbers are entered in stages using incomplete objects.
*Note Incomplete Objects::.

   Operations on rectangular complex numbers yield rectangular complex
results, and similarly for polar complex numbers.  Where the two types
are mixed, or where new complex numbers arise (as for the square root of
a negative real), the current "Polar mode" is used to determine the
type.  *Note Polar Mode::.

   A complex result in which the imaginary part is zero (or the phase
angle is 0 or 180 degrees or `pi' radians) is automatically converted
to a real number.

File: calc,  Node: Infinities,  Next: Vectors and Matrices,  Prev: Complex Numbers,  Up: Data Types

6.5 Infinities
==============

The word `inf' represents the mathematical concept of "infinity".  Calc
actually has three slightly different infinity-like values: `inf',
`uinf', and `nan'.  These are just regular variable names (*note
Variables::); you should avoid using these names for your own variables
because Calc gives them special treatment.  Infinities, like all
variable names, are normally entered using algebraic entry.

   Mathematically speaking, it is not rigorously correct to treat
"infinity" as if it were a number, but mathematicians often do so
informally.  When they say that `1 / inf = 0', what they really mean is
that `1 / x', as `x' becomes larger and larger, becomes arbitrarily
close to zero.  So you can imagine that if `x' got "all the way to
infinity," then `1 / x' would go all the way to zero.  Similarly, when
they say that `exp(inf) = inf', they mean that `exp(x)' grows without
bound as `x' grows.  The symbol `-inf' likewise stands for an
infinitely negative real value; for example, we say that `exp(-inf) =
0'.  You can have an infinity pointing in any direction on the complex
plane:  `sqrt(-inf) = i inf'.

   The same concept of limits can be used to define `1 / 0'.  We really
want the value that `1 / x' approaches as `x' approaches zero.  But if
all we have is `1 / 0', we can't tell which direction `x' was coming
from.  If `x' was positive and decreasing toward zero, then we should
say that `1 / 0 = inf'.  But if `x' was negative and increasing toward
zero, the answer is `1 / 0 = -inf'.  In fact, `x' could be an imaginary
number, giving the answer `i inf' or `-i inf'.  Calc uses the special
symbol `uinf' to mean "undirected infinity", i.e., a value which is
infinitely large but with an unknown sign (or direction on the complex
plane).

   Calc actually has three modes that say how infinities are handled.
Normally, infinities never arise from calculations that didn't already
have them.  Thus, `1 / 0' is treated simply as an error and left
unevaluated.  The `m i' (`calc-infinite-mode') command (*note Infinite
Mode::) enables a mode in which `1 / 0' evaluates to `uinf' instead.
There is also an alternative type of infinite mode which says to treat
zeros as if they were positive, so that `1 / 0 = inf'.  While this is
less mathematically correct, it may be the answer you want in some
cases.

   Since all infinities are "as large" as all others, Calc simplifies,
e.g., `5 inf' to `inf'.  Another example is `5 - inf = -inf', where the
`-inf' is so large that adding a finite number like five to it does not
affect it.  Note that `a - inf' also results in `-inf'; Calc assumes
that variables like `a' always stand for finite quantities.  Just to
show that infinities really are all the same size, note that `sqrt(inf)
= inf^2 = exp(inf) = inf' in Calc's notation.

   It's not so easy to define certain formulas like `0 * inf' and `inf
/ inf'.  Depending on where these zeros and infinities came from, the
answer could be literally anything.  The latter formula could be the
limit of `x / x' (giving a result of one), or `2 x / x' (giving two),
or `x^2 / x' (giving `inf'), or `x / x^2' (giving zero).  Calc uses the
symbol `nan' to represent such an "indeterminate" value.  (The name
"nan" comes from analogy with the "NAN" concept of IEEE standard
arithmetic; it stands for "Not A Number."  This is somewhat of a
misnomer, since `nan' _does_ stand for some number or infinity, it's
just that _which_ number it stands for cannot be determined.)  In
Calc's notation, `0 * inf = nan' and `inf / inf = nan'.  A few other
common indeterminate expressions are `inf - inf' and `inf ^ 0'.  Also,
`0 / 0 = nan' if you have turned on Infinite mode (as described above).

   Infinities are especially useful as parts of "intervals".  *Note
Interval Forms::.

File: calc,  Node: Vectors and Matrices,  Next: Strings,  Prev: Infinities,  Up: Data Types

6.6 Vectors and Matrices
========================

The "vector" data type is flexible and general.  A vector is simply a
list of zero or more data objects.  When these objects are numbers, the
whole is a vector in the mathematical sense.  When these objects are
themselves vectors of equal (nonzero) length, the whole is a "matrix".
A vector which is not a matrix is referred to here as a "plain vector".

   A vector is displayed as a list of values separated by commas and
enclosed in square brackets:  `[1, 2, 3]'.  Thus the following is a 2
row by 3 column matrix:  `[[1, 2, 3], [4, 5, 6]]'.  Vectors, like
complex numbers, are entered as incomplete objects.  *Note Incomplete
Objects::.  During algebraic entry, vectors are entered all at once in
the usual brackets-and-commas form.  Matrices may be entered
algebraically as nested vectors, or using the shortcut notation
`[1, 2, 3; 4, 5, 6]', with rows separated by semicolons.  The commas
may usually be omitted when entering vectors:  `[1 2 3]'.  Curly braces
may be used in place of brackets: `{1, 2, 3}', but the commas are
required in this case.

   Traditional vector and matrix arithmetic is also supported; *note
Basic Arithmetic:: and *note Matrix Functions::.  Many other operations
are applied to vectors element-wise.  For example, the complex
conjugate of a vector is a vector of the complex conjugates of its
elements.

   Algebraic functions for building vectors include `vec(a, b, c)' to
build `[a, b, c]', `cvec(a, n, m)' to build an NxM matrix of `a's, and
`index(n)' to build a vector of integers from 1 to `n'.

File: calc,  Node: Strings,  Next: HMS Forms,  Prev: Vectors and Matrices,  Up: Data Types

6.7 Strings
===========

Character strings are not a special data type in the Calculator.
Rather, a string is represented simply as a vector all of whose
elements are integers in the range 0 to 255 (ASCII codes).  You can
enter a string at any time by pressing the `"' key.  Quotation marks
and backslashes are written `\"' and `\\', respectively, inside
strings.  Other notations introduced by backslashes are:

     \a     7          \^@    0
     \b     8          \^a-z  1-26
     \e     27         \^[    27
     \f     12         \^\\   28
     \n     10         \^]    29
     \r     13         \^^    30
     \t     9          \^_    31
                       \^?    127

Finally, a backslash followed by three octal digits produces any
character from its ASCII code.

   Strings are normally displayed in vector-of-integers form.  The
`d "' (`calc-display-strings') command toggles a mode in which any
vectors of small integers are displayed as quoted strings instead.

   The backslash notations shown above are also used for displaying
strings.  Characters 128 and above are not translated by Calc; unless
you have an Emacs modified for 8-bit fonts, these will show up in
backslash-octal-digits notation.  For characters below 32, and for
character 127, Calc uses the backslash-letter combination if there is
one, or otherwise uses a `\^' sequence.

   The only Calc feature that uses strings is "compositions"; *note
Compositions::.  Strings also provide a convenient way to do
conversions between ASCII characters and integers.

   There is a `string' function which provides a different display
format for strings.  Basically, `string(S)', where S is a vector of
integers in the proper range, is displayed as the corresponding string
of characters with no surrounding quotation marks or other
modifications.  Thus `string("ABC")' (or `string([65 66 67])') will
look like `ABC' on the stack.  This happens regardless of whether `d "'
has been used.  The only way to turn it off is to use `d U'
(unformatted language mode) which will display `string("ABC")' instead.

   Control characters are displayed somewhat differently by `string'.
Characters below 32, and character 127, are shown using `^' notation
(same as shown above, but without the backslash).  The quote and
backslash characters are left alone, as are characters 128 and above.

   The `bstring' function is just like `string' except that the
resulting string is breakable across multiple lines if it doesn't fit
all on one line.  Potential break points occur at every space character
in the string.

File: calc,  Node: HMS Forms,  Next: Date Forms,  Prev: Strings,  Up: Data Types

6.8 HMS Forms
=============

"HMS" stands for Hours-Minutes-Seconds; when used as an angular
argument, the interpretation is Degrees-Minutes-Seconds.  All functions
that operate on angles accept HMS forms.  These are interpreted as
degrees regardless of the current angular mode.  It is also possible to
use HMS as the angular mode so that calculated angles are expressed in
degrees, minutes, and seconds.

   The default format for HMS values is `HOURS@ MINS' SECS"'.  During
entry, the letters `h' (for "hours") or `o' (approximating the
"degrees" symbol) are accepted as well as `@', `m' is accepted in place
of `'', and `s' is accepted in place of `"'.  The HOURS value is an
integer (or integer-valued float).  The MINS value is an integer or
integer-valued float between 0 and 59.  The SECS value is a real number
between 0 (inclusive) and 60 (exclusive).  A positive HMS form is
interpreted as HOURS + MINS/60 + SECS/3600.  A negative HMS form is
interpreted as - HOURS - MINS/60 - SECS/3600.  Display format for HMS
forms is quite flexible.  *Note HMS Formats::.

   HMS forms can be added and subtracted.  When they are added to
numbers, the numbers are interpreted according to the current angular
mode.  HMS forms can also be multiplied and divided by real numbers.
Dividing two HMS forms produces a real-valued ratio of the two angles.

   Just for kicks, `M-x calc-time' pushes the current time of day on
the stack as an HMS form.

File: calc,  Node: Date Forms,  Next: Modulo Forms,  Prev: HMS Forms,  Up: Data Types

6.9 Date Forms
==============

A "date form" represents a date and possibly an associated time.
Simple date arithmetic is supported:  Adding a number to a date
produces a new date shifted by that many days; adding an HMS form to a
date shifts it by that many hours.  Subtracting two date forms computes
the number of days between them (represented as a simple number).  Many
other operations, such as multiplying two date forms, are nonsensical
and are not allowed by Calc.

   Date forms are entered and displayed enclosed in `< >' brackets.
The default format is, e.g., `<Wed Jan 9, 1991>' for dates, or
`<3:32:20pm Wed Jan 9, 1991>' for dates with times.  Input is flexible;
date forms can be entered in any of the usual notations for dates and
times.  *Note Date Formats::.

   Date forms are stored internally as numbers, specifically the number
of days since midnight on the morning of January 1 of the year 1 AD.
If the internal number is an integer, the form represents a date only;
if the internal number is a fraction or float, the form represents a
date and time.  For example, `<6:00am Wed Jan 9, 1991>' is represented
by the number 726842.25.  The standard precision of 12 decimal digits
is enough to ensure that a (reasonable) date and time can be stored
without roundoff error.

   If the current precision is greater than 12, date forms will keep
additional digits in the seconds position.  For example, if the
precision is 15, the seconds will keep three digits after the decimal
point.  Decreasing the precision below 12 may cause the time part of a
date form to become inaccurate.  This can also happen if astronomically
high years are used, though this will not be an issue in everyday (or
even everymillennium) use.  Note that date forms without times are
stored as exact integers, so roundoff is never an issue for them.

   You can use the `v p' (`calc-pack') and `v u' (`calc-unpack')
commands to get at the numerical representation of a date form.  *Note
Packing and Unpacking::.

   Date forms can go arbitrarily far into the future or past.  Negative
year numbers represent years BC.  Calc uses a combination of the
Gregorian and Julian calendars, following the history of Great Britain
and the British colonies.  This is the same calendar that is used by
the `cal' program in most Unix implementations.

   Some historical background:  The Julian calendar was created by
Julius Caesar in the year 46 BC as an attempt to fix the gradual drift
caused by the lack of leap years in the calendar used until that time.
The Julian calendar introduced an extra day in all years divisible by
four.  After some initial confusion, the calendar was adopted around
the year we call 8 AD.  Some centuries later it became apparent that
the Julian year of 365.25 days was itself not quite right.  In 1582
Pope Gregory XIII introduced the Gregorian calendar, which added the
new rule that years divisible by 100, but not by 400, were not to be
considered leap years despite being divisible by four.  Many countries
delayed adoption of the Gregorian calendar because of religious
differences; in Britain it was put off until the year 1752, by which
time the Julian calendar had fallen eleven days behind the true
seasons.  So the switch to the Gregorian calendar in early September
1752 introduced a discontinuity:  The day after Sep 2, 1752 is Sep 14,
1752.  Calc follows this convention.  To take another example, Russia
waited until 1918 before adopting the new calendar, and thus needed to
remove thirteen days (between Feb 1, 1918 and Feb 14, 1918).  This
means that Calc's reckoning will be inconsistent with Russian history
between 1752 and 1918, and similarly for various other countries.

   Today's timekeepers introduce an occasional "leap second" as well,
but Calc does not take these minor effects into account.  (If it did,
it would have to report a non-integer number of days between, say,
`<12:00am Mon Jan 1, 1900>' and `<12:00am Sat Jan 1, 2000>'.)

   Calc uses the Julian calendar for all dates before the year 1752,
including dates BC when the Julian calendar technically had not yet
been invented.  Thus the claim that day number -10000 is called "August
16, 28 BC" should be taken with a grain of salt.

   Please note that there is no "year 0"; the day before `<Sat Jan 1,
+1>' is `<Fri Dec 31, -1>'.  These are days 0 and -1 respectively in
Calc's internal numbering scheme.

   Another day counting system in common use is, confusingly, also
called "Julian."  The Julian day number is the numbers of days since
12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT) is
-1721423.5 (recall that Calc starts at midnight instead of noon).  Thus
to convert a Calc date code obtained by unpacking a date form into a
Julian day number, simply add 1721423.5 after compensating for the time
zone difference.  The built-in `t J' command performs this conversion
for you.

   The Julian day number is based on the Julian cycle, which was
invented in 1583 by Joseph Justus Scaliger.  Scaliger named it the
Julian cycle since it is involves the Julian calendar, but some have
suggested that Scaliger named it in honor of his father, Julius Caesar
Scaliger.  The Julian cycle is based it on three other cycles: the
indiction cycle, the Metonic cycle, and the solar cycle.  The indiction
cycle is a 15 year cycle originally used by the Romans for tax purposes
but later used to date medieval documents.  The Metonic cycle is a 19
year cycle; 19 years is close to being a common multiple of a solar year
and a lunar month, and so every 19 years the phases of the moon will
occur on the same days of the year.  The solar cycle is a 28 year
cycle; the Julian calendar repeats itself every 28 years.  The smallest
time period which contains multiples of all three cycles is the least
common multiple of 15 years, 19 years and 28 years, which (since
they're pairwise relatively prime) is 15*19*28 = 7980 years.  This is
the length of a Julian cycle.  Working backwards, the previous year in
which all three cycles began was 4713 BC, and so Scalinger chose that
year as the beginning of a Julian cycle.  Since at the time there were
no historical records from before 4713 BC, using this year as a
starting point had the advantage of avoiding negative year numbers.  In
1849, the astronomer John Herschel (son of William Herschel) suggested
using the number of days since the beginning of the Julian cycle as an
astronomical dating system; this idea was taken up by other
astronomers.  (At the time, noon was the start of the astronomical day.
Herschel originally suggested counting the days since Jan 1, 4713 BC at
noon Alexandria time; this was later amended to noon GMT.)  Julian day
numbering is largely used in astronomy.

   The Unix operating system measures time as an integer number of
seconds since midnight, Jan 1, 1970.  To convert a Calc date value into
a Unix time stamp, first subtract 719164 (the code for `<Jan 1,
1970>'), then multiply by 86400 (the number of seconds in a day) and
press `R' to round to the nearest integer.  If you have a date form,
you can simply subtract the day `<Jan 1, 1970>' instead of unpacking
and subtracting 719164.  Likewise, divide by 86400 and add `<Jan 1,
1970>' to convert from Unix time to a Calc date form.  (Note that Unix
normally maintains the time in the GMT time zone; you may need to
subtract five hours to get New York time, or eight hours for California
time.  The same is usually true of Julian day counts.)  The built-in `t
U' command performs these conversions.

File: calc,  Node: Modulo Forms,  Next: Error Forms,  Prev: Date Forms,  Up: Data Types

6.10 Modulo Forms
=================

A "modulo form" is a real number which is taken modulo (i.e., within an
integer multiple of) some value M.  Arithmetic modulo M often arises in
number theory.  Modulo forms are written `A mod M', where A and M are
real numbers or HMS forms, and `0 <= a < M'.  In many applications `a'
and `M' will be integers but this is not required.

   To create a modulo form during numeric entry, press the shift-`M'
key to enter the word `mod'.  As a special convenience, pressing
shift-`M' a second time automatically enters the value of `M' that was
most recently used before.  During algebraic entry, either type `mod'
by hand or press `M-m' (that's `<META>-m').  Once again, pressing this
a second time enters the current modulo.

   Modulo forms are not to be confused with the modulo operator `%'.
The expression `27 % 10' means to compute 27 modulo 10 to produce the
result 7.  Further computations treat this 7 as just a regular integer.
The expression `27 mod 10' produces the result `7 mod 10'; further
computations with this value are again reduced modulo 10 so that the
result always lies in the desired range.

   When two modulo forms with identical `M''s are added or multiplied,
the Calculator simply adds or multiplies the values, then reduces modulo
`M'.  If one argument is a modulo form and the other a plain number,
the plain number is treated like a compatible modulo form.  It is also
possible to raise modulo forms to powers; the result is the value raised
to the power, then reduced modulo `M'.  (When all values involved are
integers, this calculation is done much more efficiently than actually
computing the power and then reducing.)

   Two modulo forms `A mod M' and `B mod M' can be divided if `a', `b',
and `M' are all integers.  The result is the modulo form which, when
multiplied by `B mod M', produces `A mod M'.  If there is no solution
to this equation (which can happen only when `M' is non-prime), or if
any of the arguments are non-integers, the division is left in symbolic
form.  Other operations, such as square roots, are not yet supported
for modulo forms.  (Note that, although `(A mod M)^.5' will compute a
"modulo square root" in the sense of reducing `sqrt(a)' modulo `M',
this is not a useful definition from the number-theoretical point of
view.)

   It is possible to mix HMS forms and modulo forms.  For example, an
HMS form modulo 24 could be used to manipulate clock times; an HMS form
modulo 360 would be suitable for angles.  Making the modulo `M' also be
an HMS form eliminates troubles that would arise if the angular mode
were inadvertently set to Radians, in which case `2@ 0' 0" mod 24'
would be interpreted as two degrees modulo 24 radians!

   Modulo forms cannot have variables or formulas for components.  If
you enter the formula `(x + 2) mod 5', Calc propagates the modulus to
each of the coefficients:  `(1 mod 5) x + (2 mod 5)'.

   You can use `v p' and `%' to modify modulo forms.  *Note Packing and
Unpacking::.  *Note Basic Arithmetic::.

   The algebraic function `makemod(a, m)' builds the modulo form
`a mod m'.

File: calc,  Node: Error Forms,  Next: Interval Forms,  Prev: Modulo Forms,  Up: Data Types

6.11 Error Forms
================

An "error form" is a number with an associated standard deviation, as
in `2.3 +/- 0.12'.  The notation `X +/- sigma' stands for an uncertain
value which follows a normal or Gaussian distribution of mean `x' and
standard deviation or "error" `sigma'.  Both the mean and the error can
be either numbers or formulas.  Generally these are real numbers but
the mean may also be complex.  If the error is negative or complex, it
is changed to its absolute value.  An error form with zero error is
converted to a regular number by the Calculator.

   All arithmetic and transcendental functions accept error forms as
input.  Operations on the mean-value part work just like operations on
regular numbers.  The error part for any function `f(x)' (such as
`sin(x)') is defined by the error of `x' times the derivative of `f'
evaluated at the mean value of `x'.  For a two-argument function
`f(x,y)' (such as addition) the error is the square root of the sum of
the squares of the errors due to `x' and `y'.  Note that this
definition assumes the errors in `x' and `y' are uncorrelated.  A side
effect of this definition is that `(2 +/- 1) * (2 +/- 1)' is not the
same as `(2 +/- 1)^2'; the former represents the product of two
independent values which happen to have the same probability
distributions, and the latter is the product of one random value with
itself.  The former will produce an answer with less error, since on
the average the two independent errors can be expected to cancel out.

   Consult a good text on error analysis for a discussion of the proper
use of standard deviations.  Actual errors often are neither
Gaussian-distributed nor uncorrelated, and the above formulas are valid
only when errors are small.  As an example, the error arising from
`sin(X +/- SIGMA)' is `SIGMA abs(cos(X))'.  When `x' is close to zero,
`cos(x)' is close to one so the error in the sine is close to `sigma';
this makes sense, since `sin(x)' is approximately `x' near zero, so a
given error in `x' will produce about the same error in the sine.
Likewise, near 90 degrees `cos(x)' is nearly zero and so the computed
error is small:  The sine curve is nearly flat in that region, so an
error in `x' has relatively little effect on the value of `sin(x)'.
However, consider `sin(90 +/- 1000)'.  The cosine of 90 is zero, so
Calc will report zero error!  We get an obviously wrong result because
we have violated the small-error approximation underlying the error
analysis.  If the error in `x' had been small, the error in `sin(x)'
would indeed have been negligible.

   To enter an error form during regular numeric entry, use the `p'
("plus-or-minus") key to type the `+/-' symbol.  (If you try actually
typing `+/-' the `+' key will be interpreted as the Calculator's `+'
command!)  Within an algebraic formula, you can press `M-+' to type the
`+/-' symbol, or type it out by hand.

   Error forms and complex numbers can be mixed; the formulas shown
above are used for complex numbers, too; note that if the error part
evaluates to a complex number its absolute value (or the square root of
the sum of the squares of the absolute values of the two error
contributions) is used.  Mathematically, this corresponds to a radially
symmetric Gaussian distribution of numbers on the complex plane.
However, note that Calc considers an error form with real components to
represent a real number, not a complex distribution around a real mean.

   Error forms may also be composed of HMS forms.  For best results,
both the mean and the error should be HMS forms if either one is.

   The algebraic function `sdev(a, b)' builds the error form `a +/- b'.

File: calc,  Node: Interval Forms,  Next: Incomplete Objects,  Prev: Error Forms,  Up: Data Types

6.12 Interval Forms
===================

An "interval" is a subset of consecutive real numbers.  For example,
the interval `[2 .. 4]' represents all the numbers from 2 to 4,
inclusive.  If you multiply it by the interval `[0.5 .. 2]' you obtain
`[1 .. 8]'.  This calculation represents the fact that if you multiply
some number in the range `[2 .. 4]' by some other number in the range
`[0.5 .. 2]', your result will lie in the range from 1 to 8.  Interval
arithmetic is used to get a worst-case estimate of the possible range
of values a computation will produce, given the set of possible values
of the input.

   Calc supports several varieties of intervals, including "closed"
intervals of the type shown above, "open" intervals such as `(2 .. 4)',
which represents the range of numbers from 2 to 4 _exclusive_, and
"semi-open" intervals in which one end uses a round parenthesis and the
other a square bracket.  In mathematical terms, `[2 .. 4]' means `2 <=
x <= 4', whereas `[2 .. 4)' represents `2 <= x < 4', `(2 .. 4]'
represents `2 < x <= 4', and `(2 .. 4)' represents `2 < x < 4'.

   The lower and upper limits of an interval must be either real numbers
(or HMS or date forms), or symbolic expressions which are assumed to be
real-valued, or `-inf' and `inf'.  In general the lower limit must be
less than the upper limit.  A closed interval containing only one
value, `[3 .. 3]', is converted to a plain number (3) automatically.
An interval containing no values at all (such as `[3 .. 2]' or `[2 ..
2)') can be represented but is not guaranteed to behave well when used
in arithmetic.  Note that the interval `[3 .. inf)' represents all real
numbers greater than or equal to 3, and `(-inf .. inf)' represents all
real numbers.  In fact, `[-inf .. inf]' represents all real numbers
including the real infinities.

   Intervals are entered in the notation shown here, either as algebraic
formulas, or using incomplete forms.  (*Note Incomplete Objects::.)  In
algebraic formulas, multiple periods in a row are collected from left
to right, so that `1...1e2' is interpreted as `1.0 .. 1e2' rather than
`1 .. 0.1e2'.  Add spaces or zeros if you want to get the other
interpretation.  If you omit the lower or upper limit, a default of
`-inf' or `inf' (respectively) is furnished.

   Infinite mode also affects operations on intervals (*note
Infinities::).  Calc will always introduce an open infinity, as in `1 /
(0 .. 2] = [0.5 .. inf)'.  But closed infinities,
`1 / [0 .. 2] = [0.5 .. inf]', arise only in Infinite mode; otherwise
they are left unevaluated.  Note that the "direction" of a zero is not
an issue in this case since the zero is always assumed to be continuous
with the rest of the interval.  For intervals that contain zero inside
them Calc is forced to give the result, `1 / (-2 .. 2) = [-inf .. inf]'.

   While it may seem that intervals and error forms are similar, they
are based on entirely different concepts of inexact quantities.  An
error form `X +/- SIGMA' means a variable is random, and its value could
be anything but is "probably" within one SIGMA of the mean value `x'.
An interval `[A .. B]' means a variable's value is unknown, but
guaranteed to lie in the specified range.  Error forms are statistical
or "average case" approximations; interval arithmetic tends to produce
"worst case" bounds on an answer.

   Intervals may not contain complex numbers, but they may contain HMS
forms or date forms.

   *Note Set Operations::, for commands that interpret interval forms
as subsets of the set of real numbers.

   The algebraic function `intv(n, a, b)' builds an interval form from
`a' to `b'; `n' is an integer code which must be 0 for `(..)', 1 for
`(..]', 2 for `[..)', or 3 for `[..]'.

   Please note that in fully rigorous interval arithmetic, care would be
taken to make sure that the computation of the lower bound rounds toward
minus infinity, while upper bound computations round toward plus
infinity.  Calc's arithmetic always uses a round-to-nearest mode, which
means that roundoff errors could creep into an interval calculation to
produce intervals slightly smaller than they ought to be.  For example,
entering `[1..2]' and pressing `Q 2 ^' should yield the interval
`[1..2]' again, but in fact it yields the (slightly too small) interval
`[1..1.9999999]' due to roundoff error.

File: calc,  Node: Incomplete Objects,  Next: Variables,  Prev: Interval Forms,  Up: Data Types

6.13 Incomplete Objects
=======================

When `(' or `[' is typed to begin entering a complex number or vector,
respectively, the effect is to push an "incomplete" complex number or
vector onto the stack.  The `,' key adds the value(s) at the top of the
stack onto the current incomplete object.  The `)' and `]' keys "close"
the incomplete object after adding any values on the top of the stack
in front of the incomplete object.

   As a result, the sequence of keystrokes `[ 2 , 3 <RET> 2 * , 9 ]'
pushes the vector `[2, 6, 9]' onto the stack.  Likewise, `( 1 , 2 Q )'
pushes the complex number `(1, 1.414)' (approximately).

   If several values lie on the stack in front of the incomplete object,
all are collected and appended to the object.  Thus the `,' key is
redundant:  `[ 2 <RET> 3 <RET> 2 * 9 ]'.  Some people prefer the
equivalent <SPC> key to <RET>.

   As a special case, typing `,' immediately after `(', `[', or `,'
adds a zero or duplicates the preceding value in the list being formed.
Typing <DEL> during incomplete entry removes the last item from the
list.

   The `;' key is used in the same way as `,' to create polar complex
numbers:  `( 1 ; 2 )'.  When entering a vector, `;' is useful for
creating a matrix.  In particular, `[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]' is
equivalent to `[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]'.

   Incomplete entry is also used to enter intervals.  For example, `[ 2
.. 4 )' enters a semi-open interval.  Note that when you type the first
period, it will be interpreted as a decimal point, but when you type a
second period immediately afterward, it is re-interpreted as part of
the interval symbol.  Typing `..' corresponds to executing the
`calc-dots' command.

   If you find incomplete entry distracting, you may wish to enter
vectors and complex numbers as algebraic formulas by pressing the
apostrophe key.

File: calc,  Node: Variables,  Next: Formulas,  Prev: Incomplete Objects,  Up: Data Types

6.14 Variables
==============

A "variable" is somewhere between a storage register on a conventional
calculator, and a variable in a programming language.  (In fact, a Calc
variable is really just an Emacs Lisp variable that contains a Calc
number or formula.)  A variable's name is normally composed of letters
and digits.  Calc also allows apostrophes and `#' signs in variable
names.  (The Calc variable `foo' corresponds to the Emacs Lisp variable
`var-foo', but unless you access the variable from within Emacs Lisp,
you don't need to worry about it.  Variable names in algebraic formulas
implicitly have `var-' prefixed to their names.  The `#' character in
variable names used in algebraic formulas corresponds to a dash `-' in
the Lisp variable name.  If the name contains any dashes, the prefix
`var-' is _not_ automatically added.  Thus the two formulas `foo + 1'
and `var#foo + 1' both refer to the same variable.)

   In a command that takes a variable name, you can either type the full
name of a variable, or type a single digit to use one of the special
convenience variables `q0' through `q9'.  For example, `3 s s 2' stores
the number 3 in variable `q2', and `3 s s foo <RET>' stores that number
in variable `foo'.

   To push a variable itself (as opposed to the variable's value) on the
stack, enter its name as an algebraic expression using the apostrophe
(<'>) key.

   The `=' (`calc-evaluate') key "evaluates" a formula by replacing all
variables in the formula which have been given values by a `calc-store'
or `calc-let' command by their stored values.  Other variables are left
alone.  Thus a variable that has not been stored acts like an abstract
variable in algebra; a variable that has been stored acts more like a
register in a traditional calculator.  With a positive numeric prefix
argument, `=' evaluates the top N stack entries; with a negative
argument, `=' evaluates the Nth stack entry.

   A few variables are called "special constants".  Their names are
`e', `pi', `i', `phi', and `gamma'.  (*Note Scientific Functions::.)
When they are evaluated with `=', their values are calculated if
necessary according to the current precision or complex polar mode.  If
you wish to use these symbols for other purposes, simply undefine or
redefine them using `calc-store'.

   The variables `inf', `uinf', and `nan' stand for infinite or
indeterminate values.  It's best not to use them as regular variables,
since Calc uses special algebraic rules when it manipulates them.  Calc
displays a warning message if you store a value into any of these
special variables.

   *Note Store and Recall::, for a discussion of commands dealing with
variables.

File: calc,  Node: Formulas,  Prev: Variables,  Up: Data Types

6.15 Formulas
=============

When you press the apostrophe key you may enter any expression or
formula in algebraic form.  (Calc uses the terms "expression" and
"formula" interchangeably.)  An expression is built up of numbers,
variable names, and function calls, combined with various arithmetic
operators.  Parentheses may be used to indicate grouping.  Spaces are
ignored within formulas, except that spaces are not permitted within
variable names or numbers.  Arithmetic operators, in order from highest
to lowest precedence, and with their equivalent function names, are:

   `_' [`subscr'] (subscripts);

   postfix `%' [`percent'] (as in `25% = 0.25');

   prefix `!' [`lnot'] (logical "not," as in `!x');

   `+/-' [`sdev'] (the standard deviation symbol) and `mod' [`makemod']
(the symbol for modulo forms);

   postfix `!' [`fact'] (factorial, as in `n!') and postfix `!!'
[`dfact'] (double factorial);

   `^' [`pow'] (raised-to-the-power-of);

   prefix `+' and `-' [`neg'] (as in `-x');

   `*' [`mul'];

   `/' [`div'], `%' [`mod'] (modulo), and `\' [`idiv'] (integer
division);

   infix `+' [`add'] and `-' [`sub'] (as in `x-y');

   `|' [`vconcat'] (vector concatenation);

   relations `=' [`eq'], `!=' [`neq'], `<' [`lt'], `>' [`gt'], `<='
[`leq'], and `>=' [`geq'];

   `&&' [`land'] (logical "and");

   `||' [`lor'] (logical "or");

   the C-style "if" operator `a?b:c' [`if'];

   `!!!' [`pnot'] (rewrite pattern "not");

   `&&&' [`pand'] (rewrite pattern "and");

   `|||' [`por'] (rewrite pattern "or");

   `:=' [`assign'] (for assignments and rewrite rules);

   `::' [`condition'] (rewrite pattern condition);

   `=>' [`evalto'].

   Note that, unlike in usual computer notation, multiplication binds
more strongly than division:  `a*b/c*d' is equivalent to `(a*b)/(c*d)'.

   The multiplication sign `*' may be omitted in many cases.  In
particular, if the righthand side is a number, variable name, or
parenthesized expression, the `*' may be omitted.  Implicit
multiplication has the same precedence as the explicit `*' operator.
The one exception to the rule is that a variable name followed by a
parenthesized expression, as in `f(x)', is interpreted as a function
call, not an implicit `*'.  In many cases you must use a space if you
omit the `*':  `2a' is the same as `2*a', and `a b' is the same as
`a*b', but `ab' is a variable called `ab', _not_ the product of `a' and
`b'!  Also note that `f (x)' is still a function call.

   The rules are slightly different for vectors written with square
brackets.  In vectors, the space character is interpreted (like the
comma) as a separator of elements of the vector.  Thus `[ 2a b+c d ]' is
equivalent to `[2*a, b+c, d]', whereas `2a b+c d' is equivalent to
`2*a*b + c*d'.  Note that spaces around the brackets, and around
explicit commas, are ignored.  To force spaces to be interpreted as
multiplication you can enclose a formula in parentheses as in `[(a b)
2(c d)]', which is interpreted as `[a*b, 2*c*d]'.  An implicit comma is
also inserted between `][', as in the matrix `[[1 2][3 4]]'.

   Vectors that contain commas (not embedded within nested parentheses
or brackets) do not treat spaces specially:  `[a b, 2 c d]' is a vector
of two elements.  Also, if it would be an error to treat spaces as
separators, but not otherwise, then Calc will ignore spaces: `[a - b]'
is a vector of one element, but `[a -b]' is a vector of two elements.
Finally, vectors entered with curly braces instead of square brackets
do not give spaces any special treatment.  When Calc displays a vector
that does not contain any commas, it will insert parentheses if
necessary to make the meaning clear: `[(a b)]'.

   The expression `5%-2' is ambiguous; is this five-percent minus two,
or five modulo minus-two?  Calc always interprets the leftmost symbol as
an infix operator preferentially (modulo, in this case), so you would
need to write `(5%)-2' to get the former interpretation.

   A function call is, e.g., `sin(1+x)'.  (The Calc algebraic function
`foo' corresponds to the Emacs Lisp function `calcFunc-foo', but unless
you access the function from within Emacs Lisp, you don't need to worry
about it.)  Most mathematical Calculator commands like `calc-sin' have
function equivalents like `sin'.  If no Lisp function is defined for a
function called by a formula, the call is left as it is during
algebraic manipulation: `f(x+y)' is left alone.  Beware that many
innocent-looking short names like `in' and `re' have predefined
meanings which could surprise you; however, single letters or single
letters followed by digits are always safe to use for your own function
names.  *Note Function Index::.

   In the documentation for particular commands, the notation `H S'
(`calc-sinh') [`sinh'] means that the key sequence `H S', the command
`M-x calc-sinh', and the algebraic function `sinh(x)' all represent the
same operation.

   Commands that interpret ("parse") text as algebraic formulas include
algebraic entry (`''), editing commands like ``' which parse the
contents of the editing buffer when you finish, the `C-x * g' and
`C-x * r' commands, the `C-y' command, the X window system "paste"
mouse operation, and Embedded mode.  All of these operations use the
same rules for parsing formulas; in particular, language modes (*note
Language Modes::) affect them all in the same way.

   When you read a large amount of text into the Calculator (say a
vector which represents a big set of rewrite rules; *note Rewrite
Rules::), you may wish to include comments in the text.  Calc's formula
parser ignores the symbol `%%' and anything following it on a line:

     [ a + b,   %% the sum of "a" and "b"
       c + d,
       %% last line is coming up:
       e + f ]

This is parsed exactly the same as `[ a + b, c + d, e + f ]'.

   *Note Syntax Tables::, for a way to create your own operators and
other input notations.  *Note Compositions::, for a way to create new
display formats.

   *Note Algebra::, for commands for manipulating formulas symbolically.

File: calc,  Node: Stack and Trail,  Next: Mode Settings,  Prev: Data Types,  Up: Top

7 Stack and Trail Commands
**************************

This chapter describes the Calc commands for manipulating objects on the
stack and in the trail buffer.  (These commands operate on objects of
any type, such as numbers, vectors, formulas, and incomplete objects.)

* Menu:

* Stack Manipulation::
* Editing Stack Entries::
* Trail Commands::
* Keep Arguments::

File: calc,  Node: Stack Manipulation,  Next: Editing Stack Entries,  Prev: Stack and Trail,  Up: Stack and Trail

7.1 Stack Manipulation Commands
===============================

To duplicate the top object on the stack, press <RET> or <SPC> (two
equivalent keys for the `calc-enter' command).  Given a positive
numeric prefix argument, these commands duplicate several elements at
the top of the stack.  Given a negative argument, these commands
duplicate the specified element of the stack.  Given an argument of
zero, they duplicate the entire stack.  For example, with `10 20 30' on
the stack, <RET> creates `10 20 30 30', `C-u 2 <RET>' creates `10 20 30
20 30', `C-u - 2 <RET>' creates `10 20 30 20', and `C-u 0 <RET>'
creates `10 20 30 10 20 30'.

   The <LFD> (`calc-over') command (on a key marked Line-Feed if you
have it, else on `C-j') is like `calc-enter' except that the sign of
the numeric prefix argument is interpreted oppositely.  Also, with no
prefix argument the default argument is 2.  Thus with `10 20 30' on the
stack, <LFD> and `C-u 2 <LFD>' are both equivalent to `C-u - 2 <RET>',
producing `10 20 30 20'.

   To remove the top element from the stack, press <DEL> (`calc-pop').
The `C-d' key is a synonym for <DEL>.  (If the top element is an
incomplete object with at least one element, the last element is
removed from it.)  Given a positive numeric prefix argument, several
elements are removed.  Given a negative argument, the specified element
of the stack is deleted.  Given an argument of zero, the entire stack
is emptied.  For example, with `10 20 30' on the stack, <DEL> leaves
`10 20', `C-u 2 <DEL>' leaves `10', `C-u - 2 <DEL>' leaves `10 30', and
`C-u 0 <DEL>' leaves an empty stack.

   The `M-<DEL>' (`calc-pop-above') command is to <DEL> what <LFD> is
to <RET>:  It interprets the sign of the numeric prefix argument in the
opposite way, and the default argument is 2.  Thus `M-<DEL>' by itself
removes the second-from-top stack element, leaving the first, third,
fourth, and so on; `M-3 M-<DEL>' deletes the third stack element.

   To exchange the top two elements of the stack, press <TAB>
(`calc-roll-down').  Given a positive numeric prefix argument, the
specified number of elements at the top of the stack are rotated
downward.  Given a negative argument, the entire stack is rotated
downward the specified number of times.  Given an argument of zero, the
entire stack is reversed top-for-bottom.  For example, with `10 20 30
40 50' on the stack, <TAB> creates `10 20 30 50 40', `C-u 3 <TAB>'
creates `10 20 50 30 40', `C-u - 2 <TAB>' creates `40 50 10 20 30', and
`C-u 0 <TAB>' creates `50 40 30 20 10'.

   The command `M-<TAB>' (`calc-roll-up') is analogous to <TAB> except
that it rotates upward instead of downward.  Also, the default with no
prefix argument is to rotate the top 3 elements.  For example, with `10
20 30 40 50' on the stack, `M-<TAB>' creates `10 20 40 50 30', `C-u 4
M-<TAB>' creates `10 30 40 50 20', `C-u - 2 M-<TAB>' creates `30 40 50
10 20', and `C-u 0 M-<TAB>' creates `50 40 30 20 10'.

   A good way to view the operation of <TAB> and `M-<TAB>' is in terms
of moving a particular element to a new position in the stack.  With a
positive argument N, <TAB> moves the top stack element down to level N,
making room for it by pulling all the intervening stack elements toward
the top.  `M-<TAB>' moves the element at level N up to the top.
(Compare with <LFD>, which copies instead of moving the element in
level N.)

   With a negative argument -N, <TAB> rotates the stack to move the
object in level N to the deepest place in the stack, and the object in
level N+1 to the top.  `M-<TAB>' rotates the deepest stack element to
be in level n, also putting the top stack element in level N+1.

   *Note Selecting Subformulas::, for a way to apply these commands to
any portion of a vector or formula on the stack.

File: calc,  Node: Editing Stack Entries,  Next: Trail Commands,  Prev: Stack Manipulation,  Up: Stack and Trail

7.2 Editing Stack Entries
=========================

The ``' (`calc-edit') command creates a temporary buffer (`*Calc
Edit*') for editing the top-of-stack value using regular Emacs
commands.  Note that ``' is a backquote, not a quote. With a numeric
prefix argument, it edits the specified number of stack entries at
once.  (An argument of zero edits the entire stack; a negative argument
edits one specific stack entry.)

   When you are done editing, press `C-c C-c' to finish and return to
Calc.  The <RET> and <LFD> keys also work to finish most sorts of
editing, though in some cases Calc leaves <RET> with its usual meaning
("insert a newline") if it's a situation where you might want to insert
new lines into the editing buffer.

   When you finish editing, the Calculator parses the lines of text in
the `*Calc Edit*' buffer as numbers or formulas, replaces the original
stack elements in the original buffer with these new values, then kills
the `*Calc Edit*' buffer.  The original Calculator buffer continues to
exist during editing, but for best results you should be careful not to
change it until you have finished the edit.  You can also cancel the
edit by killing the buffer with `C-x k'.

   The formula is normally reevaluated as it is put onto the stack.
For example, editing `a + 2' to `3 + 2' and pressing `C-c C-c' will
push 5 on the stack.  If you use <LFD> to finish, Calc will put the
result on the stack without evaluating it.

   If you give a prefix argument to `C-c C-c', Calc will not kill the
`*Calc Edit*' buffer.  You can switch back to that buffer and continue
editing if you wish.  However, you should understand that if you
initiated the edit with ``', the `C-c C-c' operation will be programmed
to replace the top of the stack with the new edited value, and it will
do this even if you have rearranged the stack in the meanwhile.  This
is not so much of a problem with other editing commands, though, such
as `s e' (`calc-edit-variable'; *note Operations on Variables::).

   If the `calc-edit' command involves more than one stack entry, each
line of the `*Calc Edit*' buffer is interpreted as a separate formula.
Otherwise, the entire buffer is interpreted as one formula, with line
breaks ignored.  (You can use `C-o' or `C-q C-j' to insert a newline in
the buffer without pressing <RET>.)

   The ``' key also works during numeric or algebraic entry.  The text
entered so far is moved to the `*Calc Edit*' buffer for more extensive
editing than is convenient in the minibuffer.

File: calc,  Node: Trail Commands,  Next: Keep Arguments,  Prev: Editing Stack Entries,  Up: Stack and Trail

7.3 Trail Commands
==================

The commands for manipulating the Calc Trail buffer are two-key
sequences beginning with the `t' prefix.

   The `t d' (`calc-trail-display') command turns display of the trail
on and off.  Normally the trail display is toggled on if it was off,
off if it was on.  With a numeric prefix of zero, this command always
turns the trail off; with a prefix of one, it always turns the trail on.
The other trail-manipulation commands described here automatically turn
the trail on.  Note that when the trail is off values are still recorded
there; they are simply not displayed.  To set Emacs to turn the trail
off by default, type `t d' and then save the mode settings with `m m'
(`calc-save-modes').

   The `t i' (`calc-trail-in') and `t o' (`calc-trail-out') commands
switch the cursor into and out of the Calc Trail window.  In practice
they are rarely used, since the commands shown below are a more
convenient way to move around in the trail, and they work "by remote
control" when the cursor is still in the Calculator window.

   There is a "trail pointer" which selects some entry of the trail at
any given time.  The trail pointer looks like a `>' symbol right before
the selected number.  The following commands operate on the trail
pointer in various ways.

   The `t y' (`calc-trail-yank') command reads the selected value in
the trail and pushes it onto the Calculator stack.  It allows you to
re-use any previously computed value without retyping.  With a numeric
prefix argument N, it yanks the value N lines above the current trail
pointer.

   The `t <' (`calc-trail-scroll-left') and `t >'
(`calc-trail-scroll-right') commands horizontally scroll the trail
window left or right by one half of its width.

   The `t n' (`calc-trail-next') and `t p' (`calc-trail-previous)'
commands move the trail pointer down or up one line.  The `t f'
(`calc-trail-forward') and `t b' (`calc-trail-backward') commands move
the trail pointer down or up one screenful at a time.  All of these
commands accept numeric prefix arguments to move several lines or
screenfuls at a time.

   The `t [' (`calc-trail-first') and `t ]' (`calc-trail-last')
commands move the trail pointer to the first or last line of the trail.
The `t h' (`calc-trail-here') command moves the trail pointer to the
cursor position; unlike the other trail commands, `t h' works only when
Calc Trail is the selected window.

   The `t s' (`calc-trail-isearch-forward') and `t r'
(`calc-trail-isearch-backward') commands perform an incremental search
forward or backward through the trail.  You can press <RET> to
terminate the search; the trail pointer moves to the current line.  If
you cancel the search with `C-g', the trail pointer stays where it was
when the search began.

   The `t m' (`calc-trail-marker') command allows you to enter a line
of text of your own choosing into the trail.  The text is inserted
after the line containing the trail pointer; this usually means it is
added to the end of the trail.  Trail markers are useful mainly as the
targets for later incremental searches in the trail.

   The `t k' (`calc-trail-kill') command removes the selected line from
the trail.  The line is saved in the Emacs kill ring suitable for
yanking into another buffer, but it is not easy to yank the text back
into the trail buffer.  With a numeric prefix argument, this command
kills the N lines below or above the selected one.

   The `t .' (`calc-full-trail-vectors') command is described
elsewhere; *note Vector and Matrix Formats::.

File: calc,  Node: Keep Arguments,  Prev: Trail Commands,  Up: Stack and Trail

7.4 Keep Arguments
==================

The `K' (`calc-keep-args') command acts like a prefix for the following
command.  It prevents that command from removing its arguments from the
stack.  For example, after `2 <RET> 3 +', the stack contains the sole
number 5, but after `2 <RET> 3 K +', the stack contains the arguments
and the result: `2 3 5'.

   With the exception of keyboard macros, this works for all commands
that take arguments off the stack. (To avoid potentially unpleasant
behavior, a `K' prefix before a keyboard macro will be ignored.  A `K'
prefix called _within_ the keyboard macro will still take effect.)  As
another example, `K a s' simplifies a formula, pushing the simplified
version of the formula onto the stack after the original formula
(rather than replacing the original formula).  Note that you could get
the same effect by typing `<RET> a s', copying the formula and then
simplifying the copy. One difference is that for a very large formula
the time taken to format the intermediate copy in `<RET> a s' could be
noticeable; `K a s' would avoid this extra work.

   Even stack manipulation commands are affected.  <TAB> works by
popping two values and pushing them back in the opposite order, so `2
<RET> 3 K <TAB>' produces `2 3 3 2'.

   A few Calc commands provide other ways of doing the same thing.  For
example, `' sin($)' replaces the number on the stack with its sine
using algebraic entry; to push the sine and keep the original argument
you could use either `' sin($1)' or `K ' sin($)'.  *Note Algebraic
Entry::.  Also, the `s s' command is effectively the same as `K s t'.
*Note Storing Variables::.

   If you execute a command and then decide you really wanted to keep
the argument, you can press `M-<RET>' (`calc-last-args').  This command
pushes the last arguments that were popped by any command onto the
stack.  Note that the order of things on the stack will be different
than with `K':  `2 <RET> 3 + M-<RET>' leaves `5 2 3' on the stack
instead of `2 3 5'.  *Note Undo::.

File: calc,  Node: Mode Settings,  Next: Arithmetic,  Prev: Stack and Trail,  Up: Top

8 Mode Settings
***************

This chapter describes commands that set modes in the Calculator.  They
do not affect the contents of the stack, although they may change the
_appearance_ or _interpretation_ of the stack's contents.

* Menu:

* General Mode Commands::
* Precision::
* Inverse and Hyperbolic::
* Calculation Modes::
* Simplification Modes::
* Declarations::
* Display Modes::
* Language Modes::
* Modes Variable::
* Calc Mode Line::

File: calc,  Node: General Mode Commands,  Next: Precision,  Prev: Mode Settings,  Up: Mode Settings

8.1 General Mode Commands
=========================

You can save all of the current mode settings in your Calc init file
(the file given by the variable `calc-settings-file', typically
`~/.calc.el') with the `m m' (`calc-save-modes') command.  This will
cause Emacs to reestablish these modes each time it starts up.  The
modes saved in the file include everything controlled by the `m' and
`d' prefix keys, the current precision and binary word size, whether or
not the trail is displayed, the current height of the Calc window, and
more.  The current interface (used when you type `C-x * *') is also
saved.  If there were already saved mode settings in the file, they are
replaced.  Otherwise, the new mode information is appended to the end
of the file.

   The `m R' (`calc-mode-record-mode') command tells Calc to record all
the mode settings (as if by pressing `m m') every time a mode setting
changes.  If the modes are saved this way, then this "automatic mode
recording" mode is also saved.  Type `m R' again to disable this method
of recording the mode settings.  To turn it off permanently, the `m m'
command will also be necessary.   (If Embedded mode is enabled, other
options for recording the modes are available; *note Mode Settings in
Embedded Mode::.)

   The `m F' (`calc-settings-file-name') command allows you to choose a
different file than the current value of `calc-settings-file' for `m
m', `Z P', and similar commands to save permanent information.  You are
prompted for a file name.  All Calc modes are then reset to their
default values, then settings from the file you named are loaded if
this file exists, and this file becomes the one that Calc will use in
the future for commands like `m m'.  The default settings file name is
`~/.calc.el'.  You can see the current file name by giving a blank
response to the `m F' prompt.  See also the discussion of the
`calc-settings-file' variable; *note Customizing Calc::.

   If the file name you give is your user init file (typically
`~/.emacs'), `m F' will not automatically load the new file.  This is
because your user init file may contain other things you don't want to
reread.  You can give a numeric prefix argument of 1 to `m F' to force
it to read the file no matter what.  Conversely, an argument of -1 tells
`m F' _not_ to read the new file.  An argument of 2 or -2 tells `m F'
not to reset the modes to their defaults beforehand, which is useful if
you intend your new file to have a variant of the modes present in the
file you were using before.

   The `m x' (`calc-always-load-extensions') command enables a mode in
which the first use of Calc loads the entire program, including all
extensions modules.  Otherwise, the extensions modules will not be
loaded until the various advanced Calc features are used.  Since this
mode only has effect when Calc is first loaded, `m x' is usually
followed by `m m' to make the mode-setting permanent.  To load all of
Calc just once, rather than always in the future, you can press `C-x *
L'.

   The `m S' (`calc-shift-prefix') command enables a mode in which all
of Calc's letter prefix keys may be typed shifted as well as unshifted.
If you are typing, say, `a S' (`calc-solve-for') quite often you might
find it easier to turn this mode on so that you can type `A S' instead.
When this mode is enabled, the commands that used to be on those single
shifted letters (e.g., `A' (`calc-abs')) can now be invoked by pressing
the shifted letter twice: `A A'.  Note that the `v' prefix key always
works both shifted and unshifted, and the `z' and `Z' prefix keys are
always distinct.  Also, the `h' prefix is not affected by this mode.
Press `m S' again to disable shifted-prefix mode.

File: calc,  Node: Precision,  Next: Inverse and Hyperbolic,  Prev: General Mode Commands,  Up: Mode Settings

8.2 Precision
=============

The `p' (`calc-precision') command controls the precision to which
floating-point calculations are carried.  The precision must be at
least 3 digits and may be arbitrarily high, within the limits of memory
and time.  This affects only floats:  Integer and rational calculations
are always carried out with as many digits as necessary.

   The `p' key prompts for the current precision.  If you wish you can
instead give the precision as a numeric prefix argument.

   Many internal calculations are carried to one or two digits higher
precision than normal.  Results are rounded down afterward to the
current precision.  Unless a special display mode has been selected,
floats are always displayed with their full stored precision, i.e.,
what you see is what you get.  Reducing the current precision does not
round values already on the stack, but those values will be rounded
down before being used in any calculation.  The `c 0' through `c 9'
commands (*note Conversions::) can be used to round an existing value
to a new precision.

   It is important to distinguish the concepts of "precision" and
"accuracy".  In the normal usage of these words, the number 123.4567
has a precision of 7 digits but an accuracy of 4 digits.  The precision
is the total number of digits not counting leading or trailing zeros
(regardless of the position of the decimal point).  The accuracy is
simply the number of digits after the decimal point (again not counting
trailing zeros).  In Calc you control the precision, not the accuracy
of computations.  If you were to set the accuracy instead, then
calculations like `exp(100)' would generate many more digits than you
would typically need, while `exp(-100)' would probably round to zero!
In Calc, both these computations give you exactly 12 (or the requested
number of) significant digits.

   The only Calc features that deal with accuracy instead of precision
are fixed-point display mode for floats (`d f'; *note Float Formats::),
and the rounding functions like `floor' and `round' (*note Integer
Truncation::).  Also, `c 0' through `c 9' deal with both precision and
accuracy depending on the magnitudes of the numbers involved.

   If you need to work with a particular fixed accuracy (say, dollars
and cents with two digits after the decimal point), one solution is to
work with integers and an "implied" decimal point.  For example, $8.99
divided by 6 would be entered `899 <RET> 6 /', yielding 149.833
(actually $1.49833 with our implied decimal point); pressing `R' would
round this to 150 cents, i.e., $1.50.

   *Note Floats::, for still more on floating-point precision and
related issues.

File: calc,  Node: Inverse and Hyperbolic,  Next: Calculation Modes,  Prev: Precision,  Up: Mode Settings

8.3 Inverse and Hyperbolic Flags
================================

There is no single-key equivalent to the `calc-arcsin' function.
Instead, you must first press `I' (`calc-inverse') to set the "Inverse
Flag", then press `S' (`calc-sin').  The `I' key actually toggles the
Inverse Flag.  When this flag is set, the word `Inv' appears in the
mode line.

   Likewise, the `H' key (`calc-hyperbolic') sets or clears the
Hyperbolic Flag, which transforms `calc-sin' into `calc-sinh'.  If both
of these flags are set at once, the effect will be `calc-arcsinh'.
(The Hyperbolic flag is also used by some non-trigonometric commands;
for example `H L' computes a base-10, instead of base-e, logarithm.)

   Command names like `calc-arcsin' are provided for completeness, and
may be executed with `x' or `M-x'.  Their effect is simply to toggle
the Inverse and/or Hyperbolic flags and then execute the corresponding
base command (`calc-sin' in this case).

   The Inverse and Hyperbolic flags apply only to the next Calculator
command, after which they are automatically cleared.  (They are also
cleared if the next keystroke is not a Calc command.)  Digits you type
after `I' or `H' (or `K') are treated as prefix arguments for the next
command, not as numeric entries.  The same is true of `C-u', but not of
the minus sign (`K -' means to subtract and keep arguments).

   The third Calc prefix flag, `K' (keep-arguments), is discussed
elsewhere.  *Note Keep Arguments::.

File: calc,  Node: Calculation Modes,  Next: Simplification Modes,  Prev: Inverse and Hyperbolic,  Up: Mode Settings

8.4 Calculation Modes
=====================

The commands in this section are two-key sequences beginning with the
`m' prefix.  (That's the letter `m', not the <META> key.)  The `m a'
(`calc-algebraic-mode') command is described elsewhere (*note Algebraic
Entry::).

* Menu:

* Angular Modes::
* Polar Mode::
* Fraction Mode::
* Infinite Mode::
* Symbolic Mode::
* Matrix Mode::
* Automatic Recomputation::
* Working Message::

File: calc,  Node: Angular Modes,  Next: Polar Mode,  Prev: Calculation Modes,  Up: Calculation Modes

8.4.1 Angular Modes
-------------------

The Calculator supports three notations for angles: radians, degrees,
and degrees-minutes-seconds.  When a number is presented to a function
like `sin' that requires an angle, the current angular mode is used to
interpret the number as either radians or degrees.  If an HMS form is
presented to `sin', it is always interpreted as degrees-minutes-seconds.

   Functions that compute angles produce a number in radians, a number
in degrees, or an HMS form depending on the current angular mode.  If
the result is a complex number and the current mode is HMS, the number
is instead expressed in degrees.  (Complex-number calculations would
normally be done in Radians mode, though.  Complex numbers are converted
to degrees by calculating the complex result in radians and then
multiplying by 180 over `pi'.)

   The `m r' (`calc-radians-mode'), `m d' (`calc-degrees-mode'), and `m
h' (`calc-hms-mode') commands control the angular mode.  The current
angular mode is displayed on the Emacs mode line.  The default angular
mode is Degrees.

File: calc,  Node: Polar Mode,  Next: Fraction Mode,  Prev: Angular Modes,  Up: Calculation Modes

8.4.2 Polar Mode
----------------

The Calculator normally "prefers" rectangular complex numbers in the
sense that rectangular form is used when the proper form can not be
decided from the input.  This might happen by multiplying a rectangular
number by a polar one, by taking the square root of a negative real
number, or by entering `( 2 <SPC> 3 )'.

   The `m p' (`calc-polar-mode') command toggles complex-number
preference between rectangular and polar forms.  In Polar mode, all of
the above example situations would produce polar complex numbers.

File: calc,  Node: Fraction Mode,  Next: Infinite Mode,  Prev: Polar Mode,  Up: Calculation Modes

8.4.3 Fraction Mode
-------------------

Division of two integers normally yields a floating-point number if the
result cannot be expressed as an integer.  In some cases you would
rather get an exact fractional answer.  One way to accomplish this is
to use the `:' (`calc-fdiv') [`fdiv'] command, which divides the two
integers on the top of the stack to produce a fraction: `6 <RET> 4 :'
produces `3:2' even though `6 <RET> 4 /' produces `1.5'.

   To set the Calculator to produce fractional results for normal
integer divisions, use the `m f' (`calc-frac-mode') command.  For
example, `8/4' produces `2' in either mode, but `6/4' produces `3:2' in
Fraction mode, `1.5' in Float mode.

   At any time you can use `c f' (`calc-float') to convert a fraction
to a float, or `c F' (`calc-fraction') to convert a float to a
fraction.  *Note Conversions::.

File: calc,  Node: Infinite Mode,  Next: Symbolic Mode,  Prev: Fraction Mode,  Up: Calculation Modes

8.4.4 Infinite Mode
-------------------

The Calculator normally treats results like `1 / 0' as errors; formulas
like this are left in unsimplified form.  But Calc can be put into a
mode where such calculations instead produce "infinite" results.

   The `m i' (`calc-infinite-mode') command turns this mode on and off.
When the mode is off, infinities do not arise except in calculations
that already had infinities as inputs.  (One exception is that infinite
open intervals like `[0 .. inf)' can be generated; however, intervals
closed at infinity (`[0 .. inf]') will not be generated when Infinite
mode is off.)

   With Infinite mode turned on, `1 / 0' will generate `uinf', an
undirected infinity.  *Note Infinities::, for a discussion of the
difference between `inf' and `uinf'.  Also, `0 / 0' evaluates to `nan',
the "indeterminate" symbol.  Various other functions can also return
infinities in this mode; for example, `ln(0) = -inf', and `gamma(-7) =
uinf'.  Once again, note that `exp(inf) = inf' regardless of Infinite
mode because this calculation has infinity as an input.

   The `m i' command with a numeric prefix argument of zero, i.e., `C-u
0 m i', turns on a Positive Infinite mode in which zero is treated as
positive instead of being directionless.  Thus, `1 / 0 = inf' and `-1 /
0 = -inf' in this mode.  Note that zero never actually has a sign in
Calc; there are no separate representations for +0 and -0.  Positive
Infinite mode merely changes the interpretation given to the single
symbol, `0'.  One consequence of this is that, while you might expect
`1 / -0 = -inf', actually `1 / -0' is equivalent to `1 / 0', which is
equal to positive `inf'.

File: calc,  Node: Symbolic Mode,  Next: Matrix Mode,  Prev: Infinite Mode,  Up: Calculation Modes

8.4.5 Symbolic Mode
-------------------

Calculations are normally performed numerically wherever possible.  For
example, the `calc-sqrt' command, or `sqrt' function in an algebraic
expression, produces a numeric answer if the argument is a number or a
symbolic expression if the argument is an expression: `2 Q' pushes
1.4142 but `<'> x+1 <RET> Q' pushes `sqrt(x+1)'.

   In "Symbolic mode", controlled by the `m s' (`calc-symbolic-mode')
command, functions which would produce inexact, irrational results are
left in symbolic form.  Thus `16 Q' pushes 4, but `2 Q' pushes
`sqrt(2)'.

   The shift-`N' (`calc-eval-num') command evaluates numerically the
expression at the top of the stack, by temporarily disabling
`calc-symbolic-mode' and executing `=' (`calc-evaluate').  Given a
numeric prefix argument, it also sets the floating-point precision to
the specified value for the duration of the command.

   To evaluate a formula numerically without expanding the variables it
contains, you can use the key sequence `m s a v m s' (this uses
`calc-alg-evaluate', which resimplifies but doesn't evaluate variables.)

File: calc,  Node: Matrix Mode,  Next: Automatic Recomputation,  Prev: Symbolic Mode,  Up: Calculation Modes

8.4.6 Matrix and Scalar Modes
-----------------------------

Calc sometimes makes assumptions during algebraic manipulation that are
awkward or incorrect when vectors and matrices are involved.  Calc has
two modes, "Matrix mode" and "Scalar mode", which modify its behavior
around vectors in useful ways.

   Press `m v' (`calc-matrix-mode') once to enter Matrix mode.  In this
mode, all objects are assumed to be matrices unless provably otherwise.
One major effect is that Calc will no longer consider multiplication to
be commutative.  (Recall that in matrix arithmetic, `A*B' is not the
same as `B*A'.)  This assumption affects rewrite rules and algebraic
simplification.  Another effect of this mode is that calculations that
would normally produce constants like 0 and 1 (e.g., `a - a' and `a /
a', respectively) will now produce function calls that represent
"generic" zero or identity matrices: `idn(0)', `idn(1)'.  The `idn'
function `idn(A,N)' returns A times an NxN identity matrix; if N is
omitted, it doesn't know what dimension to use and so the `idn' call
remains in symbolic form.  However, if this generic identity matrix is
later combined with a matrix whose size is known, it will be converted
into a true identity matrix of the appropriate size.  On the other hand,
if it is combined with a scalar (as in `idn(1) + 2'), Calc will assume
it really was a scalar after all and produce, e.g., 3.

   Press `m v' a second time to get Scalar mode.  Here, objects are
assumed _not_ to be vectors or matrices unless provably so.  For
example, normally adding a variable to a vector, as in `[x, y, z] + a',
will leave the sum in symbolic form because as far as Calc knows, `a'
could represent either a number or another 3-vector.  In Scalar mode,
`a' is assumed to be a non-vector, and the addition is evaluated to
`[x+a, y+a, z+a]'.

   Press `m v' a third time to return to the normal mode of operation.

   If you press `m v' with a numeric prefix argument N, you get a
special "dimensioned" Matrix mode in which matrices of unknown size are
assumed to be NxN square matrices.  Then, the function call `idn(1)'
will expand into an actual matrix rather than representing a "generic"
matrix.  Simply typing `C-u m v' will get you a square Matrix mode, in
which matrices of unknown size are assumed to be square matrices of
unspecified size.

   Of course these modes are approximations to the true state of
affairs, which is probably that some quantities will be matrices and
others will be scalars.  One solution is to "declare" certain variables
or functions to be scalar-valued.  *Note Declarations::, to see how to
make declarations in Calc.

   There is nothing stopping you from declaring a variable to be scalar
and then storing a matrix in it; however, if you do, the results you
get from Calc may not be valid.  Suppose you let Calc get the result
`[x+a, y+a, z+a]' shown above, and then stored `[1, 2, 3]' in `a'.  The
result would not be the same as for `[x, y, z] + [1, 2, 3]', but that's
because you have broken your earlier promise to Calc that `a' would be
scalar.

   Another way to mix scalars and matrices is to use selections (*note
Selecting Subformulas::).  Use Matrix mode when operating on your
formula normally; then, to apply Scalar mode to a certain part of the
formula without affecting the rest just select that part, change into
Scalar mode and press `=' to resimplify the part under this mode, then
change back to Matrix mode before deselecting.

File: calc,  Node: Automatic Recomputation,  Next: Working Message,  Prev: Matrix Mode,  Up: Calculation Modes

8.4.7 Automatic Recomputation
-----------------------------

The "evaluates-to" operator, `=>', has the special property that any
`=>' formulas on the stack are recomputed whenever variable values or
mode settings that might affect them are changed.  *Note Evaluates-To
Operator::.

   The `m C' (`calc-auto-recompute') command turns this automatic
recomputation on and off.  If you turn it off, Calc will not update
`=>' operators on the stack (nor those in the attached Embedded mode
buffer, if there is one).  They will not be updated unless you
explicitly do so by pressing `=' or until you press `m C' to turn
recomputation back on.  (While automatic recomputation is off, you can
think of `m C m C' as a command to update all `=>' operators while
leaving recomputation off.)

   To update `=>' operators in an Embedded buffer while automatic
recomputation is off, use `C-x * u'.  *Note Embedded Mode::.

File: calc,  Node: Working Message,  Prev: Automatic Recomputation,  Up: Calculation Modes

8.4.8 Working Messages
----------------------

Since the Calculator is written entirely in Emacs Lisp, which is not
designed for heavy numerical work, many operations are quite slow.  The
Calculator normally displays the message `Working...' in the echo area
during any command that may be slow.  In addition, iterative operations
such as square roots and trigonometric functions display the
intermediate result at each step.  Both of these types of messages can
be disabled if you find them distracting.

   Type `m w' (`calc-working') with a numeric prefix of 0 to disable
all "working" messages.  Use a numeric prefix of 1 to enable only the
plain `Working...' message.  Use a numeric prefix of 2 to see
intermediate results as well.  With no numeric prefix this displays the
current mode.

   While it may seem that the "working" messages will slow Calc down
considerably, experiments have shown that their impact is actually
quite small.  But if your terminal is slow you may find that it helps
to turn the messages off.

File: calc,  Node: Simplification Modes,  Next: Declarations,  Prev: Calculation Modes,  Up: Mode Settings

8.5 Simplification Modes
========================

The current "simplification mode" controls how numbers and formulas are
"normalized" when being taken from or pushed onto the stack.  Some
normalizations are unavoidable, such as rounding floating-point results
to the current precision, and reducing fractions to simplest form.
Others, such as simplifying a formula like `a+a' (or `2+3'), are done
by default but can be turned off when necessary.

   When you press a key like `+' when `2' and `3' are on the stack,
Calc pops these numbers, normalizes them, creates the formula `2+3',
normalizes it, and pushes the result.  Of course the standard rules for
normalizing `2+3' will produce the result `5'.

   Simplification mode commands consist of the lower-case `m' prefix key
followed by a shifted letter.

   The `m O' (`calc-no-simplify-mode') command turns off all optional
simplifications.  These would leave a formula like `2+3' alone.  In
fact, nothing except simple numbers are ever affected by normalization
in this mode.

   The `m N' (`calc-num-simplify-mode') command turns off simplification
of any formulas except those for which all arguments are constants.  For
example, `1+2' is simplified to `3', and `a+(2-2)' is simplified to
`a+0' but no further, since one argument of the sum is not a constant.
Unfortunately, `(a+2)-2' is _not_ simplified because the top-level `-'
operator's arguments are not both constant numbers (one of them is the
formula `a+2').  A constant is a number or other numeric object (such
as a constant error form or modulo form), or a vector all of whose
elements are constant.

   The `m D' (`calc-default-simplify-mode') command restores the
default simplifications for all formulas.  This includes many easy and
fast algebraic simplifications such as `a+0' to `a', and `a + 2 a' to
`3 a', as well as evaluating functions like `deriv(x^2, x)' to `2 x'.

   The `m B' (`calc-bin-simplify-mode') mode applies the default
simplifications to a result and then, if the result is an integer, uses
the `b c' (`calc-clip') command to clip the integer according to the
current binary word size.  *Note Binary Functions::.  Real numbers are
rounded to the nearest integer and then clipped; other kinds of results
(after the default simplifications) are left alone.

   The `m A' (`calc-alg-simplify-mode') mode does algebraic
simplification; it applies all the default simplifications, and also
the more powerful (and slower) simplifications made by `a s'
(`calc-simplify').  *Note Algebraic Simplifications::.

   The `m E' (`calc-ext-simplify-mode') mode does "extended" algebraic
simplification, as by the `a e' (`calc-simplify-extended') command.
*Note Unsafe Simplifications::.

   The `m U' (`calc-units-simplify-mode') mode does units
simplification; it applies the command `u s' (`calc-simplify-units'),
which in turn is a superset of `a s'.  In this mode, variable names
which are identifiable as unit names (like `mm' for "millimeters") are
simplified with their unit definitions in mind.

   A common technique is to set the simplification mode down to the
lowest amount of simplification you will allow to be applied
automatically, then use manual commands like `a s' and `c c'
(`calc-clean') to perform higher types of simplifications on demand.
*Note Algebraic Definitions::, for another sample use of
No-Simplification mode.

File: calc,  Node: Declarations,  Next: Display Modes,  Prev: Simplification Modes,  Up: Mode Settings

8.6 Declarations
================

A "declaration" is a statement you make that promises you will use a
certain variable or function in a restricted way.  This may give Calc
the freedom to do things that it couldn't do if it had to take the
fully general situation into account.

* Menu:

* Declaration Basics::
* Kinds of Declarations::
* Functions for Declarations::

File: calc,  Node: Declaration Basics,  Next: Kinds of Declarations,  Prev: Declarations,  Up: Declarations

8.6.1 Declaration Basics
------------------------

The `s d' (`calc-declare-variable') command is the easiest way to make
a declaration for a variable.  This command prompts for the variable
name, then prompts for the declaration.  The default at the declaration
prompt is the previous declaration, if any.  You can edit this
declaration, or press `C-k' to erase it and type a new declaration.
(Or, erase it and press <RET> to clear the declaration, effectively
"undeclaring" the variable.)

   A declaration is in general a vector of "type symbols" and "range"
values.  If there is only one type symbol or range value, you can write
it directly rather than enclosing it in a vector.  For example, `s d
foo <RET> real <RET>' declares `foo' to be a real number, and `s d bar
<RET> [int, const, [1..6]] <RET>' declares `bar' to be a constant
integer between 1 and 6.  (Actually, you can omit the outermost
brackets and Calc will provide them for you: `s d bar <RET> int, const,
[1..6] <RET>'.)

   Declarations in Calc are kept in a special variable called `Decls'.
This variable encodes the set of all outstanding declarations in the
form of a matrix.  Each row has two elements:  A variable or vector of
variables declared by that row, and the declaration specifier as
described above.  You can use the `s D' command to edit this variable
if you wish to see all the declarations at once.  *Note Operations on
Variables::, for a description of this command and the `s p' command
that allows you to save your declarations permanently if you wish.

   Items being declared can also be function calls.  The arguments in
the call are ignored; the effect is to say that this function returns
values of the declared type for any valid arguments.  The `s d' command
declares only variables, so if you wish to make a function declaration
you will have to edit the `Decls' matrix yourself.

   For example, the declaration matrix

     [ [ foo,       real       ]
       [ [j, k, n], int        ]
       [ f(1,2,3),  [0 .. inf) ] ]

declares that `foo' represents a real number, `j', `k' and `n'
represent integers, and the function `f' always returns a real number
in the interval shown.

   If there is a declaration for the variable `All', then that
declaration applies to all variables that are not otherwise declared.
It does not apply to function names.  For example, using the row `[All,
real]' says that all your variables are real unless they are explicitly
declared without `real' in some other row.  The `s d' command declares
`All' if you give a blank response to the variable-name prompt.

File: calc,  Node: Kinds of Declarations,  Next: Functions for Declarations,  Prev: Declaration Basics,  Up: Declarations

8.6.2 Kinds of Declarations
---------------------------

The type-specifier part of a declaration (that is, the second prompt in
the `s d' command) can be a type symbol, an interval, or a vector
consisting of zero or more type symbols followed by zero or more
intervals or numbers that represent the set of possible values for the
variable.

     [ [ a, [1, 2, 3, 4, 5] ]
       [ b, [1 .. 5]        ]
       [ c, [int, 1 .. 5]   ] ]

   Here `a' is declared to contain one of the five integers shown; `b'
is any number in the interval from 1 to 5 (any real number since we
haven't specified), and `c' is any integer in that interval.  Thus the
declarations for `a' and `c' are nearly equivalent (see below).

   The type-specifier can be the empty vector `[]' to say that nothing
is known about a given variable's value.  This is the same as not
declaring the variable at all except that it overrides any `All'
declaration which would otherwise apply.

   The initial value of `Decls' is the empty vector `[]'.  If `Decls'
has no stored value or if the value stored in it is not valid, it is
ignored and there are no declarations as far as Calc is concerned.
(The `s d' command will replace such a malformed value with a fresh
empty matrix, `[]', before recording the new declaration.)
Unrecognized type symbols are ignored.

   The following type symbols describe what sorts of numbers will be
stored in a variable:

`int'
     Integers.

`numint'
     Numerical integers.  (Integers or integer-valued floats.)

`frac'
     Fractions.  (Rational numbers which are not integers.)

`rat'
     Rational numbers.  (Either integers or fractions.)

`float'
     Floating-point numbers.

`real'
     Real numbers.  (Integers, fractions, or floats.  Actually,
     intervals and error forms with real components also count as reals
     here.)

`pos'
     Positive real numbers.  (Strictly greater than zero.)

`nonneg'
     Nonnegative real numbers.  (Greater than or equal to zero.)

`number'
     Numbers.  (Real or complex.)

   Calc uses this information to determine when certain simplifications
of formulas are safe.  For example, `(x^y)^z' cannot be simplified to
`x^(y z)' in general; for example, `((-3)^2)^1:2' is 3, but
`(-3)^(2*1:2) = (-3)^1' is -3.  However, this simplification _is_ safe
if `z' is known to be an integer, or if `x' is known to be a nonnegative
real number.  If you have given declarations that allow Calc to deduce
either of these facts, Calc will perform this simplification of the
formula.

   Calc can apply a certain amount of logic when using declarations.
For example, `(x^y)^(2n+1)' will be simplified if `n' has been declared
`int'; Calc knows that an integer times an integer, plus an integer,
must always be an integer.  (In fact, Calc would simplify `(-x)^(2n+1)'
to `-(x^(2n+1))' since it is able to determine that `2n+1' must be an
odd integer.)

   Similarly, `(abs(x)^y)^z' will be simplified to `abs(x)^(y z)'
because Calc knows that the `abs' function always returns a nonnegative
real.  If you had a `myabs' function that also had this property, you
could get Calc to recognize it by adding the row `[myabs(), nonneg]' to
the `Decls' matrix.

   One instance of this simplification is `sqrt(x^2)' (since the `sqrt'
function is effectively a one-half power).  Normally Calc leaves this
formula alone.  After the command `s d x <RET> real <RET>', however, it
can simplify the formula to `abs(x)'.  And after `s d x <RET> nonneg
<RET>', Calc can simplify this formula all the way to `x'.

   If there are any intervals or real numbers in the type specifier,
they comprise the set of possible values that the variable or function
being declared can have.  In particular, the type symbol `real' is
effectively the same as the range `[-inf .. inf]' (note that infinity
is included in the range of possible values); `pos' is the same as `(0
.. inf]', and `nonneg' is the same as `[0 .. inf]'.  Saying `[real, [-5
.. 5]]' is redundant because the fact that the variable is real can be
deduced just from the interval, but `[int, [-5 .. 5]]' and `[rat, [-5
.. 5]]' are useful combinations.

   Note that the vector of intervals or numbers is in the same format
used by Calc's set-manipulation commands.  *Note Set Operations::.

   The type specifier `[1, 2, 3]' is equivalent to `[numint, 1, 2, 3]',
_not_ to `[int, 1, 2, 3]'.  In other words, the range of possible
values means only that the variable's value must be numerically equal
to a number in that range, but not that it must be equal in type as
well.  Calc's set operations act the same way; `in(2, [1., 2., 3.])'
and `in(1.5, [1:2, 3:2, 5:2])' both report "true."

   If you use a conflicting combination of type specifiers, the results
are unpredictable.  An example is `[pos, [0 .. 5]]', where the interval
does not lie in the range described by the type symbol.

   "Real" declarations mostly affect simplifications involving powers
like the one described above.  Another case where they are used is in
the `a P' command which returns a list of all roots of a polynomial; if
the variable has been declared real, only the real roots (if any) will
be included in the list.

   "Integer" declarations are used for simplifications which are valid
only when certain values are integers (such as `(x^y)^z' shown above).

   Another command that makes use of declarations is `a s', when
simplifying equations and inequalities.  It will cancel `x' from both
sides of `a x = b x' only if it is sure `x' is non-zero, say, because
it has a `pos' declaration.  To declare specifically that `x' is real
and non-zero, use `[[-inf .. 0), (0 .. inf]]'.  (There is no way in the
current notation to say that `x' is nonzero but not necessarily real.)
The `a e' command does "unsafe" simplifications, including cancelling
`x' from the equation when `x' is not known to be nonzero.

   Another set of type symbols distinguish between scalars and vectors.

`scalar'
     The value is not a vector.

`vector'
     The value is a vector.

`matrix'
     The value is a matrix (a rectangular vector of vectors).

`sqmatrix'
     The value is a square matrix.

   These type symbols can be combined with the other type symbols
described above; `[int, matrix]' describes an object which is a matrix
of integers.

   Scalar/vector declarations are used to determine whether certain
algebraic operations are safe.  For example, `[a, b, c] + x' is
normally not simplified to `[a + x, b + x, c + x]', but it will be if
`x' has been declared `scalar'.  On the other hand, multiplication is
usually assumed to be commutative, but the terms in `x y' will never be
exchanged if both `x' and `y' are known to be vectors or matrices.
(Calc currently never distinguishes between `vector' and `matrix'
declarations.)

   *Note Matrix Mode::, for a discussion of Matrix mode and Scalar
mode, which are similar to declaring `[All, matrix]' or `[All, scalar]'
but much more convenient.

   One more type symbol that is recognized is used with the `H a d'
command for taking total derivatives of a formula.  *Note Calculus::.

`const'
     The value is a constant with respect to other variables.

   Calc does not check the declarations for a variable when you store a
value in it.  However, storing -3.5 in a variable that has been
declared `pos', `int', or `matrix' may have unexpected effects; Calc
may evaluate `sqrt(x^2)' to `3.5' if it substitutes the value first, or
to `-3.5' if `x' was declared `pos' and the formula `sqrt(x^2)' is
simplified to `x' before the value is substituted.  Before using a
variable for a new purpose, it is best to use `s d' or `s D' to check
to make sure you don't still have an old declaration for the variable
that will conflict with its new meaning.

File: calc,  Node: Functions for Declarations,  Prev: Kinds of Declarations,  Up: Declarations

8.6.3 Functions for Declarations
--------------------------------

Calc has a set of functions for accessing the current declarations in a
convenient manner.  These functions return 1 if the argument can be
shown to have the specified property, or 0 if the argument can be shown
_not_ to have that property; otherwise they are left unevaluated.
These functions are suitable for use with rewrite rules (*note
Conditional Rewrite Rules::) or programming constructs (*note
Conditionals in Macros::).  They can be entered only using algebraic
notation.  *Note Logical Operations::, for functions that perform other
tests not related to declarations.

   For example, `dint(17)' returns 1 because 17 is an integer, as do
`dint(n)' and `dint(2 n - 3)' if `n' has been declared `int', but
`dint(2.5)' and `dint(n + 0.5)' return 0.  Calc consults knowledge of
its own built-in functions as well as your own declarations:
`dint(floor(x))' returns 1.

   The `dint' function checks if its argument is an integer.  The
`dnatnum' function checks if its argument is a natural number, i.e., a
nonnegative integer.  The `dnumint' function checks if its argument is
numerically an integer, i.e., either an integer or an integer-valued
float.  Note that these and the other data type functions also accept
vectors or matrices composed of suitable elements, and that real
infinities `inf' and `-inf' are considered to be integers for the
purposes of these functions.

   The `drat' function checks if its argument is rational, i.e., an
integer or fraction.  Infinities count as rational, but intervals and
error forms do not.

   The `dreal' function checks if its argument is real.  This includes
integers, fractions, floats, real error forms, and intervals.

   The `dimag' function checks if its argument is imaginary, i.e., is
mathematically equal to a real number times `i'.

   The `dpos' function checks for positive (but nonzero) reals.  The
`dneg' function checks for negative reals.  The `dnonneg' function
checks for nonnegative reals, i.e., reals greater than or equal to
zero.  Note that the `a s' command can simplify an expression like `x >
0' to 1 or 0 using `dpos', and that `a s' is effectively applied to all
conditions in rewrite rules, so the actual functions `dpos', `dneg',
and `dnonneg' are rarely necessary.

   The `dnonzero' function checks that its argument is nonzero.  This
includes all nonzero real or complex numbers, all intervals that do not
include zero, all nonzero modulo forms, vectors all of whose elements
are nonzero, and variables or formulas whose values can be deduced to
be nonzero.  It does not include error forms, since they represent
values which could be anything including zero.  (This is also the set
of objects considered "true" in conditional contexts.)

   The `deven' function returns 1 if its argument is known to be an
even integer (or integer-valued float); it returns 0 if its argument is
known not to be even (because it is known to be odd or a non-integer).
The `a s' command uses this to simplify a test of the form `x % 2 = 0'.
There is also an analogous `dodd' function.

   The `drange' function returns a set (an interval or a vector of
intervals and/or numbers; *note Set Operations::) that describes the
set of possible values of its argument.  If the argument is a variable
or a function with a declaration, the range is copied from the
declaration.  Otherwise, the possible signs of the expression are
determined using a method similar to `dpos', etc., and a suitable set
like `[0 .. inf]' is returned.  If the expression is not provably real,
the `drange' function remains unevaluated.

   The `dscalar' function returns 1 if its argument is provably scalar,
or 0 if its argument is provably non-scalar.  It is left unevaluated if
this cannot be determined.  (If Matrix mode or Scalar mode is in
effect, this function returns 1 or 0, respectively, if it has no other
information.)  When Calc interprets a condition (say, in a rewrite
rule) it considers an unevaluated formula to be "false."  Thus,
`dscalar(a)' is "true" only if `a' is provably scalar, and
`!dscalar(a)' is "true" only if `a' is provably non-scalar; both are
"false" if there is insufficient information to tell.

File: calc,  Node: Display Modes,  Next: Language Modes,  Prev: Declarations,  Up: Mode Settings

8.7 Display Modes
=================

The commands in this section are two-key sequences beginning with the
`d' prefix.  The `d l' (`calc-line-numbering') and `d b'
(`calc-line-breaking') commands are described elsewhere; *note Stack
Basics:: and *note Normal Language Modes::, respectively.  Display
formats for vectors and matrices are also covered elsewhere; *note
Vector and Matrix Formats::.

   One thing all display modes have in common is their treatment of the
`H' prefix.  This prefix causes any mode command that would normally
refresh the stack to leave the stack display alone.  The word "Dirty"
will appear in the mode line when Calc thinks the stack display may not
reflect the latest mode settings.

   The `d <RET>' (`calc-refresh-top') command reformats the top stack
entry according to all the current modes.  Positive prefix arguments
reformat the top N entries; negative prefix arguments reformat the
specified entry, and a prefix of zero is equivalent to `d <SPC>'
(`calc-refresh'), which reformats the entire stack.  For example, `H d
s M-2 d <RET>' changes to scientific notation but reformats only the
top two stack entries in the new mode.

   The `I' prefix has another effect on the display modes.  The mode is
set only temporarily; the top stack entry is reformatted according to
that mode, then the original mode setting is restored.  In other words,
`I d s' is equivalent to `H d s d <RET> H d (OLD MODE)'.

* Menu:

* Radix Modes::
* Grouping Digits::
* Float Formats::
* Complex Formats::
* Fraction Formats::
* HMS Formats::
* Date Formats::
* Truncating the Stack::
* Justification::
* Labels::

File: calc,  Node: Radix Modes,  Next: Grouping Digits,  Prev: Display Modes,  Up: Display Modes

8.7.1 Radix Modes
-----------------

Calc normally displays numbers in decimal ("base-10" or "radix-10")
notation.  Calc can actually display in any radix from two (binary) to
36.  When the radix is above 10, the letters `A' to `Z' are used as
digits.  When entering such a number, letter keys are interpreted as
potential digits rather than terminating numeric entry mode.

   The key sequences `d 2', `d 8', `d 6', and `d 0' select binary,
octal, hexadecimal, and decimal as the current display radix,
respectively.  Numbers can always be entered in any radix, though the
current radix is used as a default if you press `#' without any initial
digits.  A number entered without a `#' is _always_ interpreted as
decimal.

   To set the radix generally, use `d r' (`calc-radix') and enter an
integer from 2 to 36.  You can specify the radix as a numeric prefix
argument; otherwise you will be prompted for it.

   Integers normally are displayed with however many digits are
necessary to represent the integer and no more.  The `d z'
(`calc-leading-zeros') command causes integers to be padded out with
leading zeros according to the current binary word size.  (*Note Binary
Functions::, for a discussion of word size.)  If the absolute value of
the word size is `w', all integers are displayed with at least enough
digits to represent `(2^w)-1' in the current radix.  (Larger integers
will still be displayed in their entirety.)

File: calc,  Node: Grouping Digits,  Next: Float Formats,  Prev: Radix Modes,  Up: Display Modes

8.7.2 Grouping Digits
---------------------

Long numbers can be hard to read if they have too many digits.  For
example, the factorial of 30 is 33 digits long!  Press `d g'
(`calc-group-digits') to enable "Grouping" mode, in which digits are
displayed in clumps of 3 or 4 (depending on the current radix)
separated by commas.

   The `d g' command toggles grouping on and off.  With a numeric
prefix of 0, this command displays the current state of the grouping
flag; with an argument of minus one it disables grouping; with a
positive argument `N' it enables grouping on every `N' digits.  For
floating-point numbers, grouping normally occurs only before the
decimal point.  A negative prefix argument `-N' enables grouping every
`N' digits both before and after the decimal point.

   The `d ,' (`calc-group-char') command allows you to choose any
character as the grouping separator.  The default is the comma
character.  If you find it difficult to read vectors of large integers
grouped with commas, you may wish to use spaces or some other character
instead.  This command takes the next character you type, whatever it
is, and uses it as the digit separator.  As a special case, `d , \'
selects `\,' (TeX's thin-space symbol) as the digit separator.

   Please note that grouped numbers will not generally be parsed
correctly if re-read in textual form, say by the use of `C-x * y' and
`C-x * g'.  (*Note Kill and Yank::, for details on these commands.)
One exception is the `\,' separator, which doesn't interfere with
parsing because it is ignored by TeX language mode.

File: calc,  Node: Float Formats,  Next: Complex Formats,  Prev: Grouping Digits,  Up: Display Modes

8.7.3 Float Formats
-------------------

Floating-point quantities are normally displayed in standard decimal
form, with scientific notation used if the exponent is especially high
or low.  All significant digits are normally displayed.  The commands
in this section allow you to choose among several alternative display
formats for floats.

   The `d n' (`calc-normal-notation') command selects the normal
display format.  All significant figures in a number are displayed.
With a positive numeric prefix, numbers are rounded if necessary to
that number of significant digits.  With a negative numerix prefix, the
specified number of significant digits less than the current precision
is used.  (Thus `C-u -2 d n' displays 10 digits if the current
precision is 12.)

   The `d f' (`calc-fix-notation') command selects fixed-point
notation.  The numeric argument is the number of digits after the
decimal point, zero or more.  This format will relax into scientific
notation if a nonzero number would otherwise have been rounded all the
way to zero.  Specifying a negative number of digits is the same as for
a positive number, except that small nonzero numbers will be rounded to
zero rather than switching to scientific notation.

   The `d s' (`calc-sci-notation') command selects scientific notation.
A positive argument sets the number of significant figures displayed,
of which one will be before and the rest after the decimal point.  A
negative argument works the same as for `d n' format.  The default is
to display all significant digits.

   The `d e' (`calc-eng-notation') command selects engineering
notation.  This is similar to scientific notation except that the
exponent is rounded down to a multiple of three, with from one to three
digits before the decimal point.  An optional numeric prefix sets the
number of significant digits to display, as for `d s'.

   It is important to distinguish between the current _precision_ and
the current _display format_.  After the commands `C-u 10 p' and `C-u 6
d n' the Calculator computes all results to ten significant figures but
displays only six.  (In fact, intermediate calculations are often
carried to one or two more significant figures, but values placed on
the stack will be rounded down to ten figures.)  Numbers are never
actually rounded to the display precision for storage, except by
commands like `C-k' and `C-x * y' which operate on the actual displayed
text in the Calculator buffer.

   The `d .' (`calc-point-char') command selects the character used as
a decimal point.  Normally this is a period; users in some countries
may wish to change this to a comma.  Note that this is only a display
style; on entry, periods must always be used to denote floating-point
numbers, and commas to separate elements in a list.

File: calc,  Node: Complex Formats,  Next: Fraction Formats,  Prev: Float Formats,  Up: Display Modes

8.7.4 Complex Formats
---------------------

There are three supported notations for complex numbers in rectangular
form.  The default is as a pair of real numbers enclosed in parentheses
and separated by a comma: `(a,b)'.  The `d c' (`calc-complex-notation')
command selects this style.

   The other notations are `d i' (`calc-i-notation'), in which numbers
are displayed in `a+bi' form, and `d j' (`calc-j-notation') which
displays the form `a+bj' preferred in some disciplines.

   Complex numbers are normally entered in `(a,b)' format.  If you
enter `2+3i' as an algebraic formula, it will be stored as the formula
`2 + 3 * i'.  However, if you use `=' to evaluate this formula and you
have not changed the variable `i', the `i' will be interpreted as
`(0,1)' and the formula will be simplified to `(2,3)'.  Other commands
(like `calc-sin') will _not_ interpret the formula `2 + 3 * i' as a
complex number.  *Note Variables::, under "special constants."

File: calc,  Node: Fraction Formats,  Next: HMS Formats,  Prev: Complex Formats,  Up: Display Modes

8.7.5 Fraction Formats
----------------------

Display of fractional numbers is controlled by the `d o'
(`calc-over-notation') command.  By default, a number like eight thirds
is displayed in the form `8:3'.  The `d o' command prompts for a one-
or two-character format.  If you give one character, that character is
used as the fraction separator.  Common separators are `:' and `/'.
(During input of numbers, the `:' key must be used regardless of the
display format; in particular, the `/' is used for RPN-style division,
_not_ for entering fractions.)

   If you give two characters, fractions use
"integer-plus-fractional-part" notation.  For example, the format `+/'
would display eight thirds as `2+2/3'.  If two colons are present in a
number being entered, the number is interpreted in this form (so that
the entries `2:2:3' and `8:3' are equivalent).

   It is also possible to follow the one- or two-character format with
a number.  For example:  `:10' or `+/3'.  In this case, Calc adjusts
all fractions that are displayed to have the specified denominator, if
possible.  Otherwise it adjusts the denominator to be a multiple of the
specified value.  For example, in `:6' mode the fraction `1:6' will be
unaffected, but `2:3' will be displayed as `4:6', `1:2' will be
displayed as `3:6', and `1:8' will be displayed as `3:24'.  Integers
are also affected by this mode:  3 is displayed as `18:6'.  Note that
the format `:1' writes fractions the same as `:', but it writes
integers as `n:1'.

   The fraction format does not affect the way fractions or integers are
stored, only the way they appear on the screen.  The fraction format
never affects floats.

File: calc,  Node: HMS Formats,  Next: Date Formats,  Prev: Fraction Formats,  Up: Display Modes

8.7.6 HMS Formats
-----------------

The `d h' (`calc-hms-notation') command controls the display of HMS
(hours-minutes-seconds) forms.  It prompts for a string which consists
basically of an "hours" marker, optional punctuation, a "minutes"
marker, more optional punctuation, and a "seconds" marker.  Punctuation
is zero or more spaces, commas, or semicolons.  The hours marker is one
or more non-punctuation characters.  The minutes and seconds markers
must be single non-punctuation characters.

   The default HMS format is `@ ' "', producing HMS values of the form
`23@ 30' 15.75"'.  The format `deg, ms' would display this same value
as `23deg, 30m15.75s'.  During numeric entry, the `h' or `o' keys are
recognized as synonyms for `@' regardless of display format.  The `m'
and `s' keys are recognized as synonyms for `'' and `"', respectively,
but only if an `@' (or `h' or `o') has already been typed; otherwise,
they have their usual meanings (`m-' prefix and `s-' prefix).  Thus, `5
"', `0 @ 5 "', and `0 h 5 s' are some of the ways to enter the quantity
"five seconds."  The `'' key is recognized as "minutes" only if `@' (or
`h' or `o') has already been pressed; otherwise it means to switch to
algebraic entry.

File: calc,  Node: Date Formats,  Next: Truncating the Stack,  Prev: HMS Formats,  Up: Display Modes

8.7.7 Date Formats
------------------

The `d d' (`calc-date-notation') command controls the display of date
forms (*note Date Forms::).  It prompts for a string which contains
letters that represent the various parts of a date and time.  To show
which parts should be omitted when the form represents a pure date with
no time, parts of the string can be enclosed in `< >' marks.  If you
don't include `< >' markers in the format, Calc guesses at which parts,
if any, should be omitted when formatting pure dates.

   The default format is:  `<H:mm:SSpp >Www Mmm D, YYYY'.  An example
string in this format is `3:32pm Wed Jan 9, 1991'.  If you enter a
blank format string, this default format is reestablished.

   Calc uses `< >' notation for nameless functions as well as for
dates.  *Note Specifying Operators::.  To avoid confusion with nameless
functions, your date formats should avoid using the `#' character.

* Menu:

* Date Formatting Codes::
* Free-Form Dates::
* Standard Date Formats::

File: calc,  Node: Date Formatting Codes,  Next: Free-Form Dates,  Prev: Date Formats,  Up: Date Formats

8.7.7.1 Date Formatting Codes
.............................

When displaying a date, the current date format is used.  All
characters except for letters and `<' and `>' are copied literally when
dates are formatted.  The portion between `< >' markers is omitted for
pure dates, or included for date/time forms.  Letters are interpreted
according to the table below.

   When dates are read in during algebraic entry, Calc first tries to
match the input string to the current format either with or without the
time part.  The punctuation characters (including spaces) must match
exactly; letter fields must correspond to suitable text in the input.
If this doesn't work, Calc checks if the input is a simple number; if
so, the number is interpreted as a number of days since Jan 1, 1 AD.
Otherwise, Calc tries a much more relaxed and flexible algorithm which
is described in the next section.

   Weekday names are ignored during reading.

   Two-digit year numbers are interpreted as lying in the range from
1941 to 2039.  Years outside that range are always entered and
displayed in full.  Year numbers with a leading `+' sign are always
interpreted exactly, allowing the entry and display of the years 1
through 99 AD.

   Here is a complete list of the formatting codes for dates:

Y
     Year:  "91" for 1991, "7" for 2007, "+23" for 23 AD.

YY
     Year:  "91" for 1991, "07" for 2007, "+23" for 23 AD.

BY
     Year:  "91" for 1991, " 7" for 2007, "+23" for 23 AD.

YYY
     Year:  "1991" for 1991, "23" for 23 AD.

YYYY
     Year:  "1991" for 1991, "+23" for 23 AD.

aa
     Year:  "ad" or blank.

AA
     Year:  "AD" or blank.

aaa
     Year:  "ad " or blank.  (Note trailing space.)

AAA
     Year:  "AD " or blank.

aaaa
     Year:  "a.d." or blank.

AAAA
     Year:  "A.D." or blank.

bb
     Year:  "bc" or blank.

BB
     Year:  "BC" or blank.

bbb
     Year:  " bc" or blank.  (Note leading space.)

BBB
     Year:  " BC" or blank.

bbbb
     Year:  "b.c." or blank.

BBBB
     Year:  "B.C." or blank.

M
     Month:  "8" for August.

MM
     Month:  "08" for August.

BM
     Month:  " 8" for August.

MMM
     Month:  "AUG" for August.

Mmm
     Month:  "Aug" for August.

mmm
     Month:  "aug" for August.

MMMM
     Month:  "AUGUST" for August.

Mmmm
     Month:  "August" for August.

D
     Day:  "7" for 7th day of month.

DD
     Day:  "07" for 7th day of month.

BD
     Day:  " 7" for 7th day of month.

W
     Weekday:  "0" for Sunday, "6" for Saturday.

WWW
     Weekday:  "SUN" for Sunday.

Www
     Weekday:  "Sun" for Sunday.

www
     Weekday:  "sun" for Sunday.

WWWW
     Weekday:  "SUNDAY" for Sunday.

Wwww
     Weekday:  "Sunday" for Sunday.

d
     Day of year:  "34" for Feb. 3.

ddd
     Day of year:  "034" for Feb. 3.

bdd
     Day of year:  " 34" for Feb. 3.

h
     Hour:  "5" for 5 AM; "17" for 5 PM.

hh
     Hour:  "05" for 5 AM; "17" for 5 PM.

bh
     Hour:  " 5" for 5 AM; "17" for 5 PM.

H
     Hour:  "5" for 5 AM and 5 PM.

HH
     Hour:  "05" for 5 AM and 5 PM.

BH
     Hour:  " 5" for 5 AM and 5 PM.

p
     AM/PM:  "a" or "p".

P
     AM/PM:  "A" or "P".

pp
     AM/PM:  "am" or "pm".

PP
     AM/PM:  "AM" or "PM".

pppp
     AM/PM:  "a.m." or "p.m.".

PPPP
     AM/PM:  "A.M." or "P.M.".

m
     Minutes:  "7" for 7.

mm
     Minutes:  "07" for 7.

bm
     Minutes:  " 7" for 7.

s
     Seconds:  "7" for 7;  "7.23" for 7.23.

ss
     Seconds:  "07" for 7;  "07.23" for 7.23.

bs
     Seconds:  " 7" for 7;  " 7.23" for 7.23.

SS
     Optional seconds:  "07" for 7;  blank for 0.

BS
     Optional seconds:  " 7" for 7;  blank for 0.

N
     Numeric date/time:  "726842.25" for 6:00am Wed Jan 9, 1991.

n
     Numeric date:  "726842" for any time on Wed Jan 9, 1991.

J
     Julian date/time:  "2448265.75" for 6:00am Wed Jan 9, 1991.

j
     Julian date:  "2448266" for any time on Wed Jan 9, 1991.

U
     Unix time:  "663400800" for 6:00am Wed Jan 9, 1991.

X
     Brackets suppression.  An "X" at the front of the format causes
     the surrounding `< >' delimiters to be omitted when formatting
     dates.  Note that the brackets are still required for algebraic
     entry.

   If "SS" or "BS" (optional seconds) is preceded by a colon, the colon
is also omitted if the seconds part is zero.

   If "bb," "bbb" or "bbbb" or their upper-case equivalents appear in
the format, then negative year numbers are displayed without a minus
sign.  Note that "aa" and "bb" are mutually exclusive.  Some typical
usages would be `YYYY AABB'; `AAAYYYYBBB'; `YYYYBBB'.

   The formats "YY," "YYYY," "MM," "DD," "ddd," "hh," "HH," "mm," "ss,"
and "SS" actually match any number of digits during reading unless
several of these codes are strung together with no punctuation in
between, in which case the input must have exactly as many digits as
there are letters in the format.

   The "j," "J," and "U" formats do not make any time zone adjustment.
They effectively use `julian(x,0)' and `unixtime(x,0)' to make the
conversion; *note Date Arithmetic::.

File: calc,  Node: Free-Form Dates,  Next: Standard Date Formats,  Prev: Date Formatting Codes,  Up: Date Formats

8.7.7.2 Free-Form Dates
.......................

When reading a date form during algebraic entry, Calc falls back on the
algorithm described here if the input does not exactly match the
current date format.  This algorithm generally "does the right thing"
and you don't have to worry about it, but it is described here in full
detail for the curious.

   Calc does not distinguish between upper- and lower-case letters
while interpreting dates.

   First, the time portion, if present, is located somewhere in the
text and then removed.  The remaining text is then interpreted as the
date.

   A time is of the form `hh:mm:ss', possibly with the seconds part
omitted and possibly with an AM/PM indicator added to indicate 12-hour
time.  If the AM/PM is present, the minutes may also be omitted.  The
AM/PM part may be any of the words `am', `pm', `noon', or `midnight';
each of these may be abbreviated to one letter, and the alternate forms
`a.m.', `p.m.', and `mid' are also understood.  Obviously `noon' and
`midnight' are allowed only on 12:00:00.  The words `noon', `mid', and
`midnight' are also recognized with no number attached.

   If there is no AM/PM indicator, the time is interpreted in 24-hour
format.

   To read the date portion, all words and numbers are isolated from
the string; other characters are ignored.  All words must be either
month names or day-of-week names (the latter of which are ignored).
Names can be written in full or as three-letter abbreviations.

   Large numbers, or numbers with `+' or `-' signs, are interpreted as
years.  If one of the other numbers is greater than 12, then that must
be the day and the remaining number in the input is therefore the
month.  Otherwise, Calc assumes the month, day and year are in the same
order that they appear in the current date format.  If the year is
omitted, the current year is taken from the system clock.

   If there are too many or too few numbers, or any unrecognizable
words, then the input is rejected.

   If there are any large numbers (of five digits or more) other than
the year, they are ignored on the assumption that they are something
like Julian dates that were included along with the traditional date
components when the date was formatted.

   One of the words `ad', `a.d.', `bc', or `b.c.' may optionally be
used; the latter two are equivalent to a minus sign on the year value.

   If you always enter a four-digit year, and use a name instead of a
number for the month, there is no danger of ambiguity.

File: calc,  Node: Standard Date Formats,  Prev: Free-Form Dates,  Up: Date Formats

8.7.7.3 Standard Date Formats
.............................

There are actually ten standard date formats, numbered 0 through 9.
Entering a blank line at the `d d' command's prompt gives you format
number 1, Calc's usual format.  You can enter any digit to select the
other formats.

   To create your own standard date formats, give a numeric prefix
argument from 0 to 9 to the `d d' command.  The format you enter will
be recorded as the new standard format of that number, as well as
becoming the new current date format.  You can save your formats
permanently with the `m m' command (*note Mode Settings::).

0
     `N'  (Numerical format)

1
     `<H:mm:SSpp >Www Mmm D, YYYY'  (American format)

2
     `D Mmm YYYY<, h:mm:SS>'  (European format)

3
     `Www Mmm BD< hh:mm:ss> YYYY'  (Unix written date format)

4
     `M/D/Y< H:mm:SSpp>'  (American slashed format)

5
     `D.M.Y< h:mm:SS>'  (European dotted format)

6
     `M-D-Y< H:mm:SSpp>'  (American dashed format)

7
     `D-M-Y< h:mm:SS>'  (European dashed format)

8
     `j<, h:mm:ss>'  (Julian day plus time)

9
     `YYddd< hh:mm:ss>'  (Year-day format)

File: calc,  Node: Truncating the Stack,  Next: Justification,  Prev: Date Formats,  Up: Display Modes

8.7.8 Truncating the Stack
--------------------------

The `d t' (`calc-truncate-stack') command moves the `.' line that marks
the top-of-stack up or down in the Calculator buffer.  The number right
above that line is considered to the be at the top of the stack.  Any
numbers below that line are "hidden" from all stack operations
(although still visible to the user).  This is similar to the Emacs
"narrowing" feature, except that the values below the `.' are
_visible_, just temporarily frozen.  This feature allows you to keep
several independent calculations running at once in different parts of
the stack, or to apply a certain command to an element buried deep in
the stack.

   Pressing `d t' by itself moves the `.' to the line the cursor is on.
Thus, this line and all those below it become hidden.  To un-hide these
lines, move down to the end of the buffer and press `d t'.  With a
positive numeric prefix argument `n', `d t' hides the bottom `n' values
in the buffer.  With a negative argument, it hides all but the top `n'
values.  With an argument of zero, it hides zero values, i.e., moves
the `.' all the way down to the bottom.

   The `d [' (`calc-truncate-up') and `d ]' (`calc-truncate-down')
commands move the `.' up or down one line at a time (or several lines
with a prefix argument).

File: calc,  Node: Justification,  Next: Labels,  Prev: Truncating the Stack,  Up: Display Modes

8.7.9 Justification
-------------------

Values on the stack are normally left-justified in the window.  You can
control this arrangement by typing `d <' (`calc-left-justify'), `d >'
(`calc-right-justify'), or `d =' (`calc-center-justify').  For example,
in Right-Justification mode, stack entries are displayed flush-right
against the right edge of the window.

   If you change the width of the Calculator window you may have to type
`d <SPC>' (`calc-refresh') to re-align right-justified or centered text.

   Right-justification is especially useful together with fixed-point
notation (see `d f'; `calc-fix-notation').  With these modes together,
the decimal points on numbers will always line up.

   With a numeric prefix argument, the justification commands give you
a little extra control over the display.  The argument specifies the
horizontal "origin" of a display line.  It is also possible to specify
a maximum line width using the `d b' command (*note Normal Language
Modes::).  For reference, the precise rules for formatting and breaking
lines are given below.  Notice that the interaction between origin and
line width is slightly different in each justification mode.

   In Left-Justified mode, the line is indented by a number of spaces
given by the origin (default zero).  If the result is longer than the
maximum line width, if given, or too wide to fit in the Calc window
otherwise, then it is broken into lines which will fit; each broken
line is indented to the origin.

   In Right-Justified mode, lines are shifted right so that the
rightmost character is just before the origin, or just before the
current window width if no origin was specified.  If the line is too
long for this, then it is broken; the current line width is used, if
specified, or else the origin is used as a width if that is specified,
or else the line is broken to fit in the window.

   In Centering mode, the origin is the column number of the center of
each stack entry.  If a line width is specified, lines will not be
allowed to go past that width; Calc will either indent less or break
the lines if necessary.  If no origin is specified, half the line width
or Calc window width is used.

   Note that, in each case, if line numbering is enabled the display is
indented an additional four spaces to make room for the line number.
The width of the line number is taken into account when positioning
according to the current Calc window width, but not when positioning by
explicit origins and widths.  In the latter case, the display is
formatted as specified, and then uniformly shifted over four spaces to
fit the line numbers.

File: calc,  Node: Labels,  Prev: Justification,  Up: Display Modes

8.7.10 Labels
-------------

The `d {' (`calc-left-label') command prompts for a string, then
displays that string to the left of every stack entry.  If the entries
are left-justified (*note Justification::), then they will appear
immediately after the label (unless you specified an origin greater
than the length of the label).  If the entries are centered or
right-justified, the label appears on the far left and does not affect
the horizontal position of the stack entry.

   Give a blank string (with `d { <RET>') to turn the label off.

   The `d }' (`calc-right-label') command similarly adds a label on the
righthand side.  It does not affect positioning of the stack entries
unless they are right-justified.  Also, if both a line width and an
origin are given in Right-Justified mode, the stack entry is justified
to the origin and the righthand label is justified to the line width.

   One application of labels would be to add equation numbers to
formulas you are manipulating in Calc and then copying into a document
(possibly using Embedded mode).  The equations would typically be
centered, and the equation numbers would be on the left or right as you
prefer.

File: calc,  Node: Language Modes,  Next: Modes Variable,  Prev: Display Modes,  Up: Mode Settings

8.8 Language Modes
==================

The commands in this section change Calc to use a different notation for
entry and display of formulas, corresponding to the conventions of some
other common language such as Pascal or LaTeX.  Objects displayed on the
stack or yanked from the Calculator to an editing buffer will be
formatted in the current language; objects entered in algebraic entry
or yanked from another buffer will be interpreted according to the
current language.

   The current language has no effect on things written to or read from
the trail buffer, nor does it affect numeric entry.  Only algebraic
entry is affected.  You can make even algebraic entry ignore the
current language and use the standard notation by giving a numeric
prefix, e.g., `C-u ''.

   For example, suppose the formula `2*a[1] + atan(a[2])' occurs in a C
program; elsewhere in the program you need the derivatives of this
formula with respect to `a[1]' and `a[2]'.  First, type `d C' to switch
to C notation.  Now use `C-u C-x * g' to grab the formula into the
Calculator, `a d a[1] <RET>' to differentiate with respect to the first
variable, and `C-x * y' to yank the formula for the derivative back
into your C program.  Press `U' to undo the differentiation and repeat
with `a d a[2] <RET>' for the other derivative.

   Without being switched into C mode first, Calc would have
misinterpreted the brackets in `a[1]' and `a[2]', would not have known
that `atan' was equivalent to Calc's built-in `arctan' function, and
would have written the formula back with notations (like implicit
multiplication) which would not have been valid for a C program.

   As another example, suppose you are maintaining a C program and a
LaTeX document, each of which needs a copy of the same formula.  You
can grab the formula from the program in C mode, switch to LaTeX mode,
and yank the formula into the document in LaTeX math-mode format.

   Language modes are selected by typing the letter `d' followed by a
shifted letter key.

* Menu:

* Normal Language Modes::
* C FORTRAN Pascal::
* TeX and LaTeX Language Modes::
* Eqn Language Mode::
* Yacas Language Mode::
* Maxima Language Mode::
* Giac Language Mode::
* Mathematica Language Mode::
* Maple Language Mode::
* Compositions::
* Syntax Tables::

File: calc,  Node: Normal Language Modes,  Next: C FORTRAN Pascal,  Prev: Language Modes,  Up: Language Modes

8.8.1 Normal Language Modes
---------------------------

The `d N' (`calc-normal-language') command selects the usual notation
for Calc formulas, as described in the rest of this manual.  Matrices
are displayed in a multi-line tabular format, but all other objects are
written in linear form, as they would be typed from the keyboard.

   The `d O' (`calc-flat-language') command selects a language
identical with the normal one, except that matrices are written in
one-line form along with everything else.  In some applications this
form may be more suitable for yanking data into other buffers.

   Even in one-line mode, long formulas or vectors will still be split
across multiple lines if they exceed the width of the Calculator window.
The `d b' (`calc-line-breaking') command turns this line-breaking
feature on and off.  (It works independently of the current language.)
If you give a numeric prefix argument of five or greater to the `d b'
command, that argument will specify the line width used when breaking
long lines.

   The `d B' (`calc-big-language') command selects a language which
uses textual approximations to various mathematical notations, such as
powers, quotients, and square roots:

       ____________
      | a + 1    2
      | ----- + c
     \|   b

in place of `sqrt((a+1)/b + c^2)'.

   Subscripts like `a_i' are displayed as actual subscripts in Big
mode.  Double subscripts, `a_i_j' (`subscr(subscr(a, i), j)') are
displayed as `a' with subscripts separated by commas: `i, j'.  They
must still be entered in the usual underscore notation.

   One slight ambiguity of Big notation is that

       3
     - -
       4

can represent either the negative rational number `-3:4', or the actual
expression `-(3/4)'; but the latter formula would normally never be
displayed because it would immediately be evaluated to `-3:4' or
`-0.75', so this ambiguity is not a problem in typical use.

   Non-decimal numbers are displayed with subscripts.  Thus there is no
way to tell the difference between `16#C2' and `C2_16', though
generally you will know which interpretation is correct.  Logarithms
`log(x,b)' and `log10(x)' also use subscripts in Big mode.

   In Big mode, stack entries often take up several lines.  To aid
readability, stack entries are separated by a blank line in this mode.
You may find it useful to expand the Calc window's height using `C-x ^'
(`enlarge-window') or to make the Calc window the only one on the
screen with `C-x 1' (`delete-other-windows').

   Long lines are currently not rearranged to fit the window width in
Big mode, so you may need to use the `<' and `>' keys to scroll across
a wide formula.  For really big formulas, you may even need to use `{'
and `}' to scroll up and down.

   The `d U' (`calc-unformatted-language') command altogether disables
the use of operator notation in formulas.  In this mode, the formula
shown above would be displayed:

     sqrt(add(div(add(a, 1), b), pow(c, 2)))

   These four modes differ only in display format, not in the format
expected for algebraic entry.  The standard Calc operators work in all
four modes, and unformatted notation works in any language mode (except
that Mathematica mode expects square brackets instead of parentheses).

File: calc,  Node: C FORTRAN Pascal,  Next: TeX and LaTeX Language Modes,  Prev: Normal Language Modes,  Up: Language Modes

8.8.2 C, FORTRAN, and Pascal Modes
----------------------------------

The `d C' (`calc-c-language') command selects the conventions of the C
language for display and entry of formulas.  This differs from the
normal language mode in a variety of (mostly minor) ways.  In
particular, C language operators and operator precedences are used in
place of Calc's usual ones.  For example, `a^b' means `xor(a,b)' in C
mode; a value raised to a power is written as a function call,
`pow(a,b)'.

   In C mode, vectors and matrices use curly braces instead of brackets.
Octal and hexadecimal values are written with leading `0' or `0x'
rather than using the `#' symbol.  Array subscripting is translated
into `subscr' calls, so that `a[i]' in C mode is the same as `a_i' in
Normal mode.  Assignments turn into the `assign' function, which Calc
normally displays using the `:=' symbol.

   The variables `pi' and `e' would be displayed `pi' and `e' in Normal
mode, but in C mode they are displayed as `M_PI' and `M_E',
corresponding to the names of constants typically provided in the
`<math.h>' header.  Functions whose names are different in C are
translated automatically for entry and display purposes.  For example,
entering `asin(x)' will push the formula `arcsin(x)' onto the stack;
this formula will be displayed as `asin(x)' as long as C mode is in
effect.

   The `d P' (`calc-pascal-language') command selects Pascal
conventions.  Like C mode, Pascal mode interprets array brackets and
uses a different table of operators.  Hexadecimal numbers are entered
and displayed with a preceding dollar sign.  (Thus the regular meaning
of `$2' during algebraic entry does not work in Pascal mode, though `$'
(and `$$', etc.) not followed by digits works the same as always.)  No
special provisions are made for other non-decimal numbers, vectors, and
so on, since there is no universally accepted standard way of handling
these in Pascal.

   The `d F' (`calc-fortran-language') command selects FORTRAN
conventions.  Various function names are transformed into FORTRAN
equivalents.  Vectors are written as `/1, 2, 3/', and may be entered
this way or using square brackets.  Since FORTRAN uses round
parentheses for both function calls and array subscripts, Calc displays
both in the same way; `a(i)' is interpreted as a function call upon
reading, and subscripts must be entered as `subscr(a, i)'.  If the
variable `a' has been declared to have type `vector' or `matrix',
however,  then `a(i)' will be parsed as a subscript.  (*Note
Declarations::.)  Usually it doesn't matter, though; if you enter the
subscript expression `a(i)' and Calc interprets it as a function call,
you'll never know the difference unless you switch to another language
mode or replace `a' with an actual vector (or unless `a' happens to be
the name of a built-in function!).

   Underscores are allowed in variable and function names in all of
these language modes.  The underscore here is equivalent to the `#' in
Normal mode, or to hyphens in the underlying Emacs Lisp variable names.

   FORTRAN and Pascal modes normally do not adjust the case of letters
in formulas.  Most built-in Calc names use lower-case letters.  If you
use a positive numeric prefix argument with `d P' or `d F', these modes
will use upper-case letters exclusively for display, and will convert
to lower-case on input.  With a negative prefix, these modes convert to
lower-case for display and input.

File: calc,  Node: TeX and LaTeX Language Modes,  Next: Eqn Language Mode,  Prev: C FORTRAN Pascal,  Up: Language Modes

8.8.3 TeX and LaTeX Language Modes
----------------------------------

The `d T' (`calc-tex-language') command selects the conventions of
"math mode" in Donald Knuth's TeX typesetting language, and the `d L'
(`calc-latex-language') command selects the conventions of "math mode"
in LaTeX, a typesetting language that uses TeX as its formatting
engine.  Calc's LaTeX language mode can read any formula that the TeX
language mode can, although LaTeX mode may display it differently.

   Formulas are entered and displayed in the appropriate notation;
`sin(a/b)' will appear as `\sin\left( {a \over b} \right)' in TeX mode
and `\sin\left(\frac{a}{b}\right)' in LaTeX mode.  Math formulas are
often enclosed by `$ $' signs in TeX and LaTeX; these should be omitted
when interfacing with Calc.  To Calc, the `$' sign has the same meaning
it always does in algebraic formulas (a reference to an existing entry
on the stack).

   Complex numbers are displayed as in `3 + 4i'.  Fractions and
quotients are written using `\over' in TeX mode (as in `{a \over b}')
and `\frac' in LaTeX mode (as in `\frac{a}{b}');  binomial coefficients
are written with `\choose' in TeX mode (as in `{a \choose b}') and
`\binom' in LaTeX mode (as in `\binom{a}{b}').  Interval forms are
written with `\ldots', and error forms are written with `\pm'. Absolute
values are written as in `|x + 1|', and the floor and ceiling functions
are written with `\lfloor', `\rfloor', etc. The words `\left' and
`\right' are ignored when reading formulas in TeX and LaTeX modes.
Both `inf' and `uinf' are written as `\infty'; when read, `\infty'
always translates to `inf'.

   Function calls are written the usual way, with the function name
followed by the arguments in parentheses.  However, functions for which
TeX and LaTeX have special names (like `\sin') will use curly braces
instead of parentheses for very simple arguments.  During input, curly
braces and parentheses work equally well for grouping, but when the
document is formatted the curly braces will be invisible.  Thus the
printed result is `sin 2x' but `sin(2 + x)'.

   Function and variable names not treated specially by TeX and LaTeX
are simply written out as-is, which will cause them to come out in
italic letters in the printed document.  If you invoke `d T' or `d L'
with a positive numeric prefix argument, names of more than one
character will instead be enclosed in a protective commands that will
prevent them from being typeset in the math italics; they will be
written `\hbox{NAME}' in TeX mode and `\text{NAME}' in LaTeX mode.  The
`\hbox{ }' and `\text{ }' notations are ignored during reading.  If you
use a negative prefix argument, such function names are written
`\NAME', and function names that begin with `\' during reading have the
`\' removed.  (Note that in this mode, long variable names are still
written with `\hbox' or `\text'.  However, you can always make an
actual variable name like `\bar' in any TeX mode.)

   During reading, text of the form `\matrix{ ... }' is replaced by `[
... ]'.  The same also applies to `\pmatrix' and `\bmatrix'.  In LaTeX
mode this also applies to `\begin{matrix} ... \end{matrix}',
`\begin{bmatrix} ... \end{bmatrix}', `\begin{pmatrix} ...
\end{pmatrix}', as well as `\begin{smallmatrix} ... \end{smallmatrix}'.
The symbol `&' is interpreted as a comma, and the symbols `\cr' and
`\\' are interpreted as semicolons.  During output, matrices are
displayed in `\matrix{ a & b \\ c & d}' format in TeX mode and in
`\begin{pmatrix} a & b \\ c & d \end{pmatrix}' format in LaTeX mode;
you may need to edit this afterwards to change to your preferred matrix
form.  If you invoke `d T' or `d L' with an argument of 2 or -2, then
matrices will be displayed in two-dimensional form, such as

     \begin{pmatrix}
     a & b \\
     c & d
     \end{pmatrix}

This may be convenient for isolated matrices, but could lead to
expressions being displayed like

     \begin{pmatrix} \times x
     a & b \\
     c & d
     \end{pmatrix}

While this wouldn't bother Calc, it is incorrect LaTeX.  (Similarly for
TeX.)

   Accents like `\tilde' and `\bar' translate into function calls
internally (`tilde(x)', `bar(x)').  The `\underline' sequence is
treated as an accent.  The `\vec' accent corresponds to the function
name `Vec', because `vec' is the name of a built-in Calc function.  The
following table shows the accents in Calc, TeX, LaTeX and "eqn"
(described in the next section):

     Calc      TeX           LaTeX         eqn
     ----      ---           -----         ---
     acute     \acute        \acute
     Acute                   \Acute
     bar       \bar          \bar          bar
     Bar                     \Bar
     breve     \breve        \breve
     Breve                   \Breve
     check     \check        \check
     Check                   \Check
     dddot                   \dddot
     ddddot                  \ddddot
     dot       \dot          \dot          dot
     Dot                     \Dot
     dotdot    \ddot         \ddot         dotdot
     DotDot                  \Ddot
     dyad                                  dyad
     grave     \grave        \grave
     Grave                   \Grave
     hat       \hat          \hat          hat
     Hat                     \Hat
     Prime                                 prime
     tilde     \tilde        \tilde        tilde
     Tilde                   \Tilde
     under     \underline    \underline    under
     Vec       \vec          \vec          vec
     VEC                     \Vec

   The `=>' (evaluates-to) operator appears as a `\to' symbol: `{A \to
B}'.  TeX defines `\to' as an alias for `\rightarrow'.  However, if the
`=>' is the top-level expression being formatted, a slightly different
notation is used:  `\evalto A \to B'.  The `\evalto' word is ignored by
Calc's input routines, and is undefined in TeX.  You will typically
want to include one of the following definitions at the top of a TeX
file that uses `\evalto':

     \def\evalto{}
     \def\evalto#1\to{}

   The first definition formats evaluates-to operators in the usual
way.  The second causes only the B part to appear in the printed
document; the A part and the arrow are hidden.  Another definition you
may wish to use is `\let\to=\Rightarrow' which causes `\to' to appear
more like Calc's `=>' symbol.  *Note Evaluates-To Operator::, for a
discussion of `evalto'.

   The complete set of TeX control sequences that are ignored during
reading is:

     \hbox  \mbox  \text  \left  \right
     \,  \>  \:  \;  \!  \quad  \qquad  \hfil  \hfill
     \displaystyle  \textstyle  \dsize  \tsize
     \scriptstyle  \scriptscriptstyle  \ssize  \ssize
     \rm  \bf  \it  \sl  \roman  \bold  \italic  \slanted
     \cal  \mit  \Cal  \Bbb  \frak  \goth
     \evalto

   Note that, because these symbols are ignored, reading a TeX or LaTeX
formula into Calc and writing it back out may lose spacing and font
information.

   Also, the "discretionary multiplication sign" `\*' is read the same
as `*'.

   The TeX version of this manual includes some printed examples at the
end of this section.

File: calc,  Node: Eqn Language Mode,  Next: Yacas Language Mode,  Prev: TeX and LaTeX Language Modes,  Up: Language Modes

8.8.4 Eqn Language Mode
-----------------------

"Eqn" is another popular formatter for math formulas.  It is designed
for use with the TROFF text formatter, and comes standard with many
versions of Unix.  The `d E' (`calc-eqn-language') command selects
"eqn" notation.

   The "eqn" language's main idiosyncrasy is that whitespace plays a
significant part in the parsing of the language.  For example, `sqrt
x+1 + y' treats `x+1' as the argument of the `sqrt' operator.  "Eqn"
also understands more conventional grouping using curly braces:
`sqrt{x+1} + y'.  Braces are required only when the argument contains
spaces.

   In Calc's "eqn" mode, however, curly braces are required to delimit
arguments of operators like `sqrt'.  The first of the above examples
would treat only the `x' as the argument of `sqrt', and in fact `sin
x+1' would be interpreted as `sin * x + 1', because `sin' is not a
special operator in the "eqn" language.  If you always surround the
argument with curly braces, Calc will never misunderstand.

   Calc also understands parentheses as grouping characters.  Another
peculiarity of "eqn"'s syntax makes it advisable to separate words with
spaces from any surrounding characters that aren't curly braces, so
Calc writes `sin ( x + y )' in "eqn" mode.  (The spaces around `sin'
are important to make "eqn" recognize that `sin' should be typeset in a
roman font, and the spaces around `x' and `y' are a good idea just in
case the "eqn" document has defined special meanings for these names,
too.)

   Powers and subscripts are written with the `sub' and `sup'
operators, respectively.  Note that the caret symbol `^' is treated the
same as a space in "eqn" mode, as is the `~' symbol (these are used to
introduce spaces of various widths into the typeset output of "eqn").

   As in LaTeX mode, Calc's formatter omits parentheses around the
arguments of functions like `ln' and `sin' if they are
"simple-looking"; in this case Calc surrounds the argument with braces,
separated by a `~' from the function name: `sin~{x}'.

   Font change codes (like `roman X') and positioning codes (like `~'
and `down N X') are ignored by the "eqn" reader.  Also ignored are the
words `left', `right', `mark', and `lineup'.  Quotation marks in "eqn"
mode input are treated the same as curly braces: `sqrt "1+x"' is
equivalent to `sqrt {1+x}'; this is only an approximation to the true
meaning of quotes in "eqn", but it is good enough for most uses.

   Accent codes (`X dot') are handled by treating them as function
calls (`dot(X)') internally.  *Note TeX and LaTeX Language Modes::, for
a table of these accent functions.  The `prime' accent is treated
specially if it occurs on a variable or function name: `f prime prime
( x prime )' is stored internally as `f''(x')'.  For example, taking the
derivative of `f(2 x)' with `a d x' will produce `2 f'(2 x)', which
"eqn" mode will display as `2 f prime ( 2 x )'.

   Assignments are written with the `<-' (left-arrow) symbol, and
`evalto' operators are written with `->' or `evalto ... ->' (*note TeX
and LaTeX Language Modes::, for a discussion of this).  The regular
Calc symbols `:=' and `=>' are also recognized for these operators
during reading.

   Vectors in "eqn" mode use regular Calc square brackets, but matrices
are formatted as `matrix { ccol { a above b } ... }'.  The words `lcol'
and `rcol' are recognized as synonyms for `ccol' during input, and are
generated instead of `ccol' if the matrix justification mode so
specifies.

File: calc,  Node: Yacas Language Mode,  Next: Maxima Language Mode,  Prev: Eqn Language Mode,  Up: Language Modes

8.8.5 Yacas Language Mode
-------------------------

The `d Y' (`calc-yacas-language') command selects the conventions of
Yacas, a free computer algebra system.  While the operators and
functions in Yacas are similar to those of Calc, the names of built-in
functions in Yacas are capitalized.  The Calc formula `sin(2 x)', for
example, is entered and displayed `Sin(2 x)' in Yacas mode,  and
``arcsin(x^2)' is `ArcSin(x^2)' in Yacas mode.  Complex numbers are
written  are written `3 + 4 I'.  The standard special constants are
written `Pi', `E', `I', `GoldenRatio' and `Gamma'.  `Infinity'
represents both `inf' and `uinf', and `Undefined' represents `nan'.

   Certain operators on functions, such as `D' for differentiation and
`Integrate' for integration, take a prefix form in Yacas.  For example,
the derivative of `e^x sin(x)' can be computed with
`D(x) Exp(x)*Sin(x)'.

   Other notable differences between Yacas and standard Calc expressions
are that vectors and matrices use curly braces in Yacas, and subscripts
use square brackets.  If, for example, `A' represents the list
`{a,2,c,4}', then `A[3]' would equal `c'.

File: calc,  Node: Maxima Language Mode,  Next: Giac Language Mode,  Prev: Yacas Language Mode,  Up: Language Modes

8.8.6 Maxima Language Mode
--------------------------

The `d X' (`calc-maxima-language') command selects the conventions of
Maxima, another free computer algebra system.  The function names in
Maxima are similar, but not always identical, to Calc.  For example,
instead of `arcsin(x)', Maxima will use `asin(x)'.  Complex numbers are
written `3 + 4 %i'.  The standard special constants are written `%pi',
`%e', `%i', `%phi' and `%gamma'.  In Maxima,  `inf' means the same as
in Calc, but `infinity' represents Calc's `uinf'.

   Underscores as well as percent signs are allowed in function and
variable names in Maxima mode.  The underscore again is equivalent to
the `#' in Normal mode, and the percent sign is equivalent to `o'o'.

   Maxima uses square brackets for lists and vectors, and matrices are
written as calls to the function `matrix', given the row vectors of the
matrix as arguments.  Square brackets are also used as subscripts.

File: calc,  Node: Giac Language Mode,  Next: Mathematica Language Mode,  Prev: Maxima Language Mode,  Up: Language Modes

8.8.7 Giac Language Mode
------------------------

The `d A' (`calc-giac-language') command selects the conventions of
Giac, another free computer algebra system.  The function names in Giac
are similar to Maxima.  Complex numbers are written `3 + 4 i'.  The
standard special constants in Giac are the same as in Calc, except that
`infinity' represents both Calc's `inf' and `uinf'.

   Underscores are allowed in function and variable names in Giac mode.
Brackets are used for subscripts.  In Giac, indexing of lists begins at
0, instead of 1 as in Calc.  So if  `A' represents the list
`[a,2,c,4]', then `A[2]' would equal `c'.  In general, `A[n]' in Giac
mode corresponds to `A_(n+1)' in Normal mode.

   The Giac interval notation `2 .. 3' has no surrounding brackets;
Calc reads `2 .. 3' as the closed interval `[2 .. 3]' and writes any
kind of interval as `2 .. 3'.  This means you cannot see the difference
between an open and a closed interval while in Giac mode.

File: calc,  Node: Mathematica Language Mode,  Next: Maple Language Mode,  Prev: Giac Language Mode,  Up: Language Modes

8.8.8 Mathematica Language Mode
-------------------------------

The `d M' (`calc-mathematica-language') command selects the conventions
of Mathematica.  Notable differences in Mathematica mode are that the
names of built-in functions are capitalized, and function calls use
square brackets instead of parentheses.  Thus the Calc formula `sin(2
x)' is entered and displayed `Sin[2 x]' in Mathematica mode.

   Vectors and matrices use curly braces in Mathematica.  Complex
numbers are written `3 + 4 I'.  The standard special constants in Calc
are written `Pi', `E', `I', `GoldenRatio', `EulerGamma', `Infinity',
`ComplexInfinity', and `Indeterminate' in Mathematica mode.
Non-decimal numbers are written, e.g., `16^^7fff'.  Floating-point
numbers in scientific notation are written `1.23*10.^3'.  Subscripts
use double square brackets: `a[[i]]'.

File: calc,  Node: Maple Language Mode,  Next: Compositions,  Prev: Mathematica Language Mode,  Up: Language Modes

8.8.9 Maple Language Mode
-------------------------

The `d W' (`calc-maple-language') command selects the conventions of
Maple.

   Maple's language is much like C.  Underscores are allowed in symbol
names; square brackets are used for subscripts; explicit `*'s for
multiplications are required.  Use either `^' or `**' to denote powers.

   Maple uses square brackets for lists and curly braces for sets.  Calc
interprets both notations as vectors, and displays vectors with square
brackets.  This means Maple sets will be converted to lists when they
pass through Calc.  As a special case, matrices are written as calls to
the function `matrix', given a list of lists as the argument, and can
be read in this form or with all-capitals `MATRIX'.

   The Maple interval notation `2 .. 3' is like Giac's interval
notation, and is handled the same by Calc.

   Maple writes complex numbers as `3 + 4*I'.  Its special constants
are `Pi', `E', `I', and `infinity' (all three of `inf', `uinf', and
`nan' display as `infinity').  Floating-point numbers are written
`1.23*10.^3'.

   Among things not currently handled by Calc's Maple mode are the
various quote symbols, procedures and functional operators, and inert
(`&') operators.

File: calc,  Node: Compositions,  Next: Syntax Tables,  Prev: Maple Language Mode,  Up: Language Modes

8.8.10 Compositions
-------------------

There are several "composition functions" which allow you to get
displays in a variety of formats similar to those in Big language mode.
Most of these functions do not evaluate to anything; they are
placeholders which are left in symbolic form by Calc's evaluator but
are recognized by Calc's display formatting routines.

   Two of these, `string' and `bstring', are described elsewhere.
*Note Strings::.  For example, `string("ABC")' is displayed as `ABC'.
When viewed on the stack it will be indistinguishable from the variable
`ABC', but internally it will be stored as `string([65, 66, 67])' and
can still be manipulated this way; for example, the selection and
vector commands `j 1 v v j u' would select the vector portion of this
object and reverse the elements, then deselect to reveal a string whose
characters had been reversed.

   The composition functions do the same thing in all language modes
(although their components will of course be formatted in the current
language mode).  The one exception is Unformatted mode (`d U'), which
does not give the composition functions any special treatment.  The
functions are discussed here because of their relationship to the
language modes.

* Menu:

* Composition Basics::
* Horizontal Compositions::
* Vertical Compositions::
* Other Compositions::
* Information about Compositions::
* User-Defined Compositions::

File: calc,  Node: Composition Basics,  Next: Horizontal Compositions,  Prev: Compositions,  Up: Compositions

8.8.10.1 Composition Basics
...........................

Compositions are generally formed by stacking formulas together
horizontally or vertically in various ways.  Those formulas are
themselves compositions.  TeX users will find this analogous to TeX's
"boxes."  Each multi-line composition has a "baseline"; horizontal
compositions use the baselines to decide how formulas should be
positioned relative to one another.  For example, in the Big mode
formula

               2
          a + b
     17 + ------
            c

the second term of the sum is four lines tall and has line three as its
baseline.  Thus when the term is combined with 17, line three is placed
on the same level as the baseline of 17.

   Another important composition concept is "precedence".  This is an
integer that represents the binding strength of various operators.  For
example, `*' has higher precedence (195) than `+' (180), which means
that `(a * b) + c' will be formatted without the parentheses, but `a *
(b + c)' will keep the parentheses.

   The operator table used by normal and Big language modes has the
following precedences:

     _     1200    (subscripts)
     %     1100    (as in n%)
     !     1000    (as in !n)
     mod    400
     +/-    300
     !!     210    (as in n!!)
     !      210    (as in n!)
     ^      200
     -      197    (as in -n)
     *      195    (or implicit multiplication)
     / % \  190
     + -    180    (as in a+b)
     |      170
     < =    160    (and other relations)
     &&     110
     ||     100
     ? :     90
     !!!     85
     &&&     80
     |||     75
     :=      50
     ::      45
     =>      40

   The general rule is that if an operator with precedence `n' occurs
as an argument to an operator with precedence `m', then the argument is
enclosed in parentheses if `n < m'.  Top-level expressions and
expressions which are function arguments, vector components, etc., are
formatted with precedence zero (so that they normally never get
additional parentheses).

   For binary left-associative operators like `+', the righthand
argument is actually formatted with one-higher precedence than shown in
the table.  This makes sure `(a + b) + c' omits the parentheses, but
the unnatural form `a + (b + c)' keeps its parentheses.
Right-associative operators like `^' format the lefthand argument with
one-higher precedence.

   The `cprec' function formats an expression with an arbitrary
precedence.  For example, `cprec(abc, 185)' will combine into sums and
products as follows:  `7 + abc', `7 (abc)' (because this `cprec' form
has higher precedence than addition, but lower precedence than
multiplication).

   A final composition issue is "line breaking".  Calc uses two
different strategies for "flat" and "non-flat" compositions.  A
non-flat composition is anything that appears on multiple lines (not
counting line breaking).  Examples would be matrices and Big mode
powers and quotients.  Non-flat compositions are displayed exactly as
specified.  If they come out wider than the current window, you must
use horizontal scrolling (`<' and `>') to view them.

   Flat compositions, on the other hand, will be broken across several
lines if they are too wide to fit the window.  Certain points in a
composition are noted internally as "break points".  Calc's general
strategy is to fill each line as much as possible, then to move down to
the next line starting at the first break point that didn't fit.
However, the line breaker understands the hierarchical structure of
formulas.  It will not break an "inner" formula if it can use an
earlier break point from an "outer" formula instead.  For example, a
vector of sums might be formatted as:

     [ a + b + c, d + e + f,
       g + h + i, j + k + l, m ]

If the `m' can fit, then so, it seems, could the `g'.  But Calc prefers
to break at the comma since the comma is part of a "more outer"
formula.  Calc would break at a plus sign only if it had to, say, if
the very first sum in the vector had itself been too large to fit.

   Of the composition functions described below, only `choriz'
generates break points.  The `bstring' function (*note Strings::) also
generates breakable items:  A break point is added after every space
(or group of spaces) except for spaces at the very beginning or end of
the string.

   Composition functions themselves count as levels in the formula
hierarchy, so a `choriz' that is a component of a larger `choriz' will
be less likely to be broken.  As a special case, if a `bstring' occurs
as a component of a `choriz' or `choriz'-like object (such as a vector
or a list of arguments in a function call), then the break points in
that `bstring' will be on the same level as the break points of the
surrounding object.

File: calc,  Node: Horizontal Compositions,  Next: Vertical Compositions,  Prev: Composition Basics,  Up: Compositions

8.8.10.2 Horizontal Compositions
................................

The `choriz' function takes a vector of objects and composes them
horizontally.  For example, `choriz([17, a b/c, d])' formats as
`17a b / cd' in Normal language mode, or as

       a b
     17---d
        c

in Big language mode.  This is actually one case of the general
function `choriz(VEC, SEP, PREC)', where either or both of SEP and PREC
may be omitted.  PREC gives the "precedence" to use when formatting
each of the components of VEC.  The default precedence is the
precedence from the surrounding environment.

   SEP is a string (i.e., a vector of character codes as might be
entered with `" "' notation) which should separate components of the
composition.  Also, if SEP is given, the line breaker will allow lines
to be broken after each occurrence of SEP.  If SEP is omitted, the
composition will not be breakable (unless any of its component
compositions are breakable).

   For example, `2 choriz([a, b c, d = e], " + ", 180)' is formatted as
`2 a + b c + (d = e)'.  To get the `choriz' to have precedence 180
"outwards" as well as "inwards," enclose it in a `cprec' form:  `2
cprec(choriz(...), 180)' formats as `2 (a + b c + (d = e))'.

   The baseline of a horizontal composition is the same as the
baselines of the component compositions, which are all aligned.

File: calc,  Node: Vertical Compositions,  Next: Other Compositions,  Prev: Horizontal Compositions,  Up: Compositions

8.8.10.3 Vertical Compositions
..............................

The `cvert' function makes a vertical composition.  Each component of
the vector is centered in a column.  The baseline of the result is by
default the top line of the resulting composition.  For example,
`f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))' formats in Big mode as

     f( a ,  2    )
       bb   a  + 1
       ccc     2
              b

   There are several special composition functions that work only as
components of a vertical composition.  The `cbase' function controls
the baseline of the vertical composition; the baseline will be the same
as the baseline of whatever component is enclosed in `cbase'.  Thus
`f(cvert([a, cbase(bb), ccc]), cvert([a^2 + 1, cbase(b^2)]))' displays
as

             2
            a  + 1
        a      2
     f(bb ,   b   )
       ccc

   There are also `ctbase' and `cbbase' functions which make the
baseline of the vertical composition equal to the top or bottom line
(rather than the baseline) of that component.  Thus `cvert([cbase(a /
b)]) + cvert([ctbase(a / b)]) + cvert([cbbase(a / b)])' gives

             a
     a       -
     - + a + b
     b   -
         b

   There should be only one `cbase', `ctbase', or `cbbase' function in
a given vertical composition.  These functions can also be written with
no arguments:  `ctbase()' is a zero-height object which means the
baseline is the top line of the following item, and `cbbase()' means
the baseline is the bottom line of the preceding item.

   The `crule' function builds a "rule," or horizontal line, across a
vertical composition.  By itself `crule()' uses `-' characters to build
the rule.  You can specify any other character, e.g., `crule("=")'.
The argument must be a character code or vector of exactly one
character code.  It is repeated to match the width of the widest item
in the stack.  For example, a quotient with a thick line is `cvert([a +
1, cbase(crule("=")), b^2])':

     a + 1
     =====
       2
      b

   Finally, the functions `clvert' and `crvert' act exactly like
`cvert' except that the items are left- or right-justified in the
stack.  Thus `clvert([a, bb, ccc]) + crvert([a, bb, ccc])' gives:

     a   +   a
     bb     bb
     ccc   ccc

   Like `choriz', the vertical compositions accept a second argument
which gives the precedence to use when formatting the components.
Vertical compositions do not support separator strings.

File: calc,  Node: Other Compositions,  Next: Information about Compositions,  Prev: Vertical Compositions,  Up: Compositions

8.8.10.4 Other Compositions
...........................

The `csup' function builds a superscripted expression.  For example,
`csup(a, b)' looks the same as `a^b' does in Big language mode.  This
is essentially a horizontal composition of `a' and `b', where `b' is
shifted up so that its bottom line is one above the baseline.

   Likewise, the `csub' function builds a subscripted expression.  This
shifts `b' down so that its top line is one below the bottom line of
`a' (note that this is not quite analogous to `csup').  Other
arrangements can be obtained by using `choriz' and `cvert' directly.

   The `cflat' function formats its argument in "flat" mode, as
obtained by `d O', if the current language mode is normal or Big.  It
has no effect in other language modes.  For example, `a^(b/c)' is
formatted by Big mode like `csup(a, cflat(b/c))' to improve its
readability.

   The `cspace' function creates horizontal space.  For example,
`cspace(4)' is effectively the same as `string("    ")'.  A second
string (i.e., vector of characters) argument is repeated instead of the
space character.  For example, `cspace(4, "ab")' looks like `abababab'.
If the second argument is not a string, it is formatted in the normal
way and then several copies of that are composed together:  `cspace(4,
a^2)' yields

      2 2 2 2
     a a a a

If the number argument is zero, this is a zero-width object.

   The `cvspace' function creates vertical space, or a vertical stack
of copies of a certain string or formatted object.  The baseline is the
center line of the resulting stack.  A numerical argument of zero will
produce an object which contributes zero height if used in a vertical
composition.

   There are also `ctspace' and `cbspace' functions which create
vertical space with the baseline the same as the baseline of the top or
bottom copy, respectively, of the second argument.  Thus `cvspace(2,
a/b) + ctspace(2, a/b) + cbspace(2, a/b)' displays as:

             a
             -
     a       b
     -   a   a
     b + - + -
     a   b   b
     -   a
     b   -
         b

File: calc,  Node: Information about Compositions,  Next: User-Defined Compositions,  Prev: Other Compositions,  Up: Compositions

8.8.10.5 Information about Compositions
.......................................

The functions in this section are actual functions; they compose their
arguments according to the current language and other display modes,
then return a certain measurement of the composition as an integer.

   The `cwidth' function measures the width, in characters, of a
composition.  For example, `cwidth(a + b)' is 5, and `cwidth(a / b)' is
5 in Normal mode, 1 in Big mode, and 11 in TeX mode (for `{a \over
b}').  The argument may involve the composition functions described in
this section.

   The `cheight' function measures the height of a composition.  This
is the total number of lines in the argument's printed form.

   The functions `cascent' and `cdescent' measure the amount of the
height that is above (and including) the baseline, or below the
baseline, respectively.  Thus `cascent(X) + cdescent(X)' always equals
`cheight(X)'.  For a one-line formula like `a + b', `cascent' returns 1
and `cdescent' returns 0.  For `a / b' in Big mode, `cascent' returns 2
and `cdescent' returns 1.  The only formula for which `cascent' will
return zero is `cvspace(0)' or equivalents.

File: calc,  Node: User-Defined Compositions,  Prev: Information about Compositions,  Up: Compositions

8.8.10.6 User-Defined Compositions
..................................

The `Z C' (`calc-user-define-composition') command lets you define the
display format for any algebraic function.  You provide a formula
containing a certain number of argument variables on the stack.  Any
time Calc formats a call to the specified function in the current
language mode and with that number of arguments, Calc effectively
replaces the function call with that formula with the arguments
replaced.

   Calc builds the default argument list by sorting all the variable
names that appear in the formula into alphabetical order.  You can edit
this argument list before pressing <RET> if you wish.  Any variables in
the formula that do not appear in the argument list will be displayed
literally; any arguments that do not appear in the formula will not
affect the display at all.

   You can define formats for built-in functions, for functions you have
defined with `Z F' (*note Algebraic Definitions::), or for functions
which have no definitions but are being used as purely syntactic
objects.  You can define different formats for each language mode, and
for each number of arguments, using a succession of `Z C' commands.
When Calc formats a function call, it first searches for a format
defined for the current language mode (and number of arguments); if
there is none, it uses the format defined for the Normal language mode.
If neither format exists, Calc uses its built-in standard format for
that function (usually just `FUNC(ARGS)').

   If you execute `Z C' with the number 0 on the stack instead of a
formula, any defined formats for the function in the current language
mode will be removed.  The function will revert to its standard format.

   For example, the default format for the binomial coefficient function
`choose(n, m)' in the Big language mode is

      n
     ( )
      m

You might prefer the notation,

      C
     n m

To define this notation, first make sure you are in Big mode, then put
the formula

     choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])

on the stack and type `Z C'.  Answer the first prompt with `choose'.
The second prompt will be the default argument list of `(C m n)'.  Edit
this list to be `(n m)' and press <RET>.  Now, try it out:  For
example, turn simplification off with `m O' and enter `choose(a,b) +
choose(7,3)' as an algebraic entry.

      C  +  C
     a b   7 3

   As another example, let's define the usual notation for Stirling
numbers of the first kind, `stir1(n, m)'.  This is just like the
regular format for binomial coefficients but with square brackets
instead of parentheses.

     choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])

   Now type `Z C stir1 <RET>', edit the argument list to `(n m)', and
type <RET>.

   The formula provided to `Z C' usually will involve composition
functions, but it doesn't have to.  Putting the formula `a + b + c'
onto the stack and typing `Z C foo <RET> <RET>' would define the
function `foo(x,y,z)' to display like `x + y + z'.  This "sum" will act
exactly like a real sum for all formatting purposes (it will be
parenthesized the same, and so on).  However it will be computationally
unrelated to a sum.  For example, the formula `2 * foo(1, 2, 3)' will
display as `2 (1 + 2 + 3)'.  Operator precedences have caused the "sum"
to be written in parentheses, but the arguments have not actually been
summed.  (Generally a display format like this would be undesirable,
since it can easily be confused with a real sum.)

   The special function `eval' can be used inside a `Z C' composition
formula to cause all or part of the formula to be evaluated at display
time.  For example, if the formula is `a + eval(b + c)', then `foo(1,
2, 3)' will be displayed as `1 + 5'.  Evaluation will use the default
simplifications, regardless of the current simplification mode.  There
are also `evalsimp' and `evalextsimp' which simplify as if by `a s' and
`a e' (respectively).  Note that these "functions" operate only in the
context of composition formulas (and also in rewrite rules, where they
serve a similar purpose; *note Rewrite Rules::).  On the stack, a call
to `eval' will be left in symbolic form.

   It is not a good idea to use `eval' except as a last resort.  It can
cause the display of formulas to be extremely slow.  For example, while
`eval(a + b)' might seem quite fast and simple, there are several
situations where it could be slow.  For example, `a' and/or `b' could
be polar complex numbers, in which case doing the sum requires
trigonometry.  Or, `a' could be the factorial `fact(100)' which is
unevaluated because you have typed `m O'; `eval' will evaluate it
anyway to produce a large, unwieldy integer.

   You can save your display formats permanently using the `Z P'
command (*note Creating User Keys::).

File: calc,  Node: Syntax Tables,  Prev: Compositions,  Up: Language Modes

8.8.11 Syntax Tables
--------------------

Syntax tables do for input what compositions do for output:  They allow
you to teach custom notations to Calc's formula parser.  Calc keeps a
separate syntax table for each language mode.

   (Note that the Calc "syntax tables" discussed here are completely
unrelated to the syntax tables described in the Emacs manual.)

   The `Z S' (`calc-edit-user-syntax') command edits the syntax table
for the current language mode.  If you want your syntax to work in any
language, define it in the Normal language mode.  Type `C-c C-c' to
finish editing the syntax table, or `C-x k' to cancel the edit.  The `m
m' command saves all the syntax tables along with the other mode
settings; *note General Mode Commands::.

* Menu:

* Syntax Table Basics::
* Precedence in Syntax Tables::
* Advanced Syntax Patterns::
* Conditional Syntax Rules::

File: calc,  Node: Syntax Table Basics,  Next: Precedence in Syntax Tables,  Prev: Syntax Tables,  Up: Syntax Tables

8.8.11.1 Syntax Table Basics
............................

"Parsing" is the process of converting a raw string of characters, such
as you would type in during algebraic entry, into a Calc formula.
Calc's parser works in two stages.  First, the input is broken down
into "tokens", such as words, numbers, and punctuation symbols like
`+', `:=', and `+/-'.  Space between tokens is ignored (except when it
serves to separate adjacent words).  Next, the parser matches this
string of tokens against various built-in syntactic patterns, such as
"an expression followed by `+' followed by another expression" or "a
name followed by `(', zero or more expressions separated by commas, and
`)'."

   A "syntax table" is a list of user-defined "syntax rules", which
allow you to specify new patterns to define your own favorite input
notations.  Calc's parser always checks the syntax table for the
current language mode, then the table for the Normal language mode,
before it uses its built-in rules to parse an algebraic formula you
have entered.  Each syntax rule should go on its own line; it consists
of a "pattern", a `:=' symbol, and a Calc formula with an optional
"condition".  (Syntax rules resemble algebraic rewrite rules, but the
notation for patterns is completely different.)

   A syntax pattern is a list of tokens, separated by spaces.  Except
for a few special symbols, tokens in syntax patterns are matched
literally, from left to right.  For example, the rule,

     foo ( ) := 2+3

would cause Calc to parse the formula `4+foo()*5' as if it were
`4+(2+3)*5'.  Notice that the parentheses were written as two separate
tokens in the rule.  As a result, the rule works for both `foo()' and
`foo (  )'.  If we had written the rule as `foo () := 2+3', then Calc
would treat `()' as a single, indivisible token, so that `foo( )' would
not be recognized by the rule.  (It would be parsed as a regular
zero-argument function call instead.)  In fact, this rule would also
make trouble for the rest of Calc's parser:  An unrelated formula like
`bar()' would now be tokenized into `bar ()' instead of `bar ( )', so
that the standard parser for function calls would no longer recognize
it!

   While it is possible to make a token with a mixture of letters and
punctuation symbols, this is not recommended.  It is better to break it
into several tokens, as we did with `foo()' above.

   The symbol `#' in a syntax pattern matches any Calc expression.  On
the righthand side, the things that matched the `#'s can be referred to
as `#1', `#2', and so on (where `#1' matches the leftmost `#' in the
pattern).  For example, these rules match a user-defined function,
prefix operator, infix operator, and postfix operator, respectively:

     foo ( # ) := myfunc(#1)
     foo # := myprefix(#1)
     # foo # := myinfix(#1,#2)
     # foo := mypostfix(#1)

   Thus `foo(3)' will parse as `myfunc(3)', and `2+3 foo' will parse as
`mypostfix(2+3)'.

   It is important to write the first two rules in the order shown,
because Calc tries rules in order from first to last.  If the pattern
`foo #' came first, it would match anything that could match the `foo (
# )' rule, since an expression in parentheses is itself a valid
expression.  Thus the `foo ( # )' rule would never get to match
anything.  Likewise, the last two rules must be written in the order
shown or else `3 foo 4' will be parsed as `mypostfix(3) * 4'.  (Of
course, the best way to avoid these ambiguities is not to use the same
symbol in more than one way at the same time!  In case you're not
convinced, try the following exercise:  How will the above rules parse
the input `foo(3,4)', if at all?  Work it out for yourself, then try it
in Calc and see.)

   Calc is quite flexible about what sorts of patterns are allowed.
The only rule is that every pattern must begin with a literal token
(like `foo' in the first two patterns above), or with a `#' followed by
a literal token (as in the last two patterns).  After that, any mixture
is allowed, although putting two `#'s in a row will not be very useful
since two expressions with nothing between them will be parsed as one
expression that uses implicit multiplication.

   As a more practical example, Maple uses the notation `sum(a(i),
i=1..10)' for sums, which Calc's Maple mode doesn't recognize at
present.  To handle this syntax, we simply add the rule,

     sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)

to the Maple mode syntax table.  As another example, C mode can't read
assignment operators like `++' and `*='.  We can define these operators
quite easily:

     # *= # := muleq(#1,#2)
     # ++ := postinc(#1)
     ++ # := preinc(#1)

To complete the job, we would use corresponding composition functions
and `Z C' to cause these functions to display in their respective Maple
and C notations.  (Note that the C example ignores issues of operator
precedence, which are discussed in the next section.)

   You can enclose any token in quotes to prevent its usual
interpretation in syntax patterns:

     # ":=" # := becomes(#1,#2)

   Quotes also allow you to include spaces in a token, although once
again it is generally better to use two tokens than one token with an
embedded space.  To include an actual quotation mark in a quoted token,
precede it with a backslash.  (This also works to include backslashes
in tokens.)

     # "bad token" # "/\"\\" # := silly(#1,#2,#3)

This will parse `3 bad token 4 /"\ 5' to `silly(3,4,5)'.

   The token `#' has a predefined meaning in Calc's formula parser; it
is not valid to use `"#"' in a syntax rule.  However, longer tokens
that include the `#' character are allowed.  Also, while `"$"' and
`"\""' are allowed as tokens, their presence in the syntax table will
prevent those characters from working in their usual ways (referring to
stack entries and quoting strings, respectively).

   Finally, the notation `%%' anywhere in a syntax table causes the
rest of the line to be ignored as a comment.

File: calc,  Node: Precedence in Syntax Tables,  Next: Advanced Syntax Patterns,  Prev: Syntax Table Basics,  Up: Syntax Tables

8.8.11.2 Precedence
...................

Different operators are generally assigned different "precedences".  By
default, an operator defined by a rule like

     # foo # := foo(#1,#2)

will have an extremely low precedence, so that `2*3+4 foo 5 == 6' will
be parsed as `(2*3+4) foo (5 == 6)'.  To change the precedence of an
operator, use the notation `#/P' in place of `#', where P is an integer
precedence level.  For example, 185 lies between the precedences for
`+' and `*', so if we change this rule to

     #/185 foo #/186 := foo(#1,#2)

then `2+3 foo 4*5' will be parsed as `2+(3 foo (4*5))'.  Also, because
we've given the righthand expression slightly higher precedence, our
new operator will be left-associative: `1 foo 2 foo 3' will be parsed
as `(1 foo 2) foo 3'.  By raising the precedence of the lefthand
expression instead, we can create a right-associative operator.

   *Note Composition Basics::, for a table of precedences of the
standard Calc operators.  For the precedences of operators in other
language modes, look in the Calc source file `calc-lang.el'.

File: calc,  Node: Advanced Syntax Patterns,  Next: Conditional Syntax Rules,  Prev: Precedence in Syntax Tables,  Up: Syntax Tables

8.8.11.3 Advanced Syntax Patterns
.................................

To match a function with a variable number of arguments, you could write

     foo ( # ) := myfunc(#1)
     foo ( # , # ) := myfunc(#1,#2)
     foo ( # , # , # ) := myfunc(#1,#2,#3)

but this isn't very elegant.  To match variable numbers of items, Calc
uses some notations inspired regular expressions and the "extended BNF"
style used by some language designers.

     foo ( { # }*, ) := apply(myfunc,#1)

   The token `{' introduces a repeated or optional portion.  One of the
three tokens `}*', `}+', or `}?' ends the portion.  These will match
zero or more, one or more, or zero or one copies of the enclosed
pattern, respectively.  In addition, `}*' and `}+' can be followed by a
separator token (with no space in between, as shown above).  Thus `{ #
}*,' matches nothing, or one expression, or several expressions
separated by commas.

   A complete `{ ... }' item matches as a vector of the items that
matched inside it.  For example, the above rule will match `foo(1,2,3)'
to get `apply(myfunc,[1,2,3])'.  The Calc `apply' function takes a
function name and a vector of arguments and builds a call to the
function with those arguments, so the net result is the formula
`myfunc(1,2,3)'.

   If the body of a `{ ... }' contains several `#'s (or nested `{ ...
}' constructs), then the items will be strung together into the
resulting vector.  If the body does not contain anything but literal
tokens, the result will always be an empty vector.

     foo ( { # , # }+, ) := bar(#1)
     foo ( { { # }*, }*; ) := matrix(#1)

will parse `foo(1, 2, 3, 4)' as `bar([1, 2, 3, 4])', and `foo(1, 2; 3,
4)' as `matrix([[1, 2], [3, 4]])'.  Also, after some thought it's easy
to see how this pair of rules will parse `foo(1, 2, 3)' as `matrix([[1,
2, 3]])', since the first rule will only match an even number of
arguments.  The rule

     foo ( # { , # , # }? ) := bar(#1,#2)

will parse `foo(2,3,4)' as `bar(2,[3,4])', and `foo(2)' as `bar(2,[])'.

   The notation `{ ... }?.' (note the trailing period) works just the
same as regular `{ ... }?', except that it does not count as an
argument; the following two rules are equivalent:

     foo ( # , { also }? # ) := bar(#1,#3)
     foo ( # , { also }?. # ) := bar(#1,#2)

Note that in the first case the optional text counts as `#2', which
will always be an empty vector, but in the second case no empty vector
is produced.

   Another variant is `{ ... }?$', which means the body is optional
only at the end of the input formula.  All built-in syntax rules in
Calc use this for closing delimiters, so that during algebraic entry
you can type `[sqrt(2), sqrt(3 <RET>', omitting the closing parenthesis
and bracket.  Calc does this automatically for trailing `)', `]', and
`>' tokens in syntax rules, but you can use `{ ... }?$' explicitly to
get this effect with any token (such as `"}"' or `end').  Like `{ ...
}?.', this notation does not count as an argument.  Conversely, you can
use quotes, as in `")"', to prevent a closing-delimiter token from
being automatically treated as optional.

   Calc's parser does not have full backtracking, which means some
patterns will not work as you might expect:

     foo ( { # , }? # , # ) := bar(#1,#2,#3)

Here we are trying to make the first argument optional, so that
`foo(2,3)' parses as `bar([],2,3)'.  Unfortunately, Calc first tries to
match `2,' against the optional part of the pattern, finds a match, and
so goes ahead to match the rest of the pattern.  Later on it will fail
to match the second comma, but it doesn't know how to go back and try
the other alternative at that point.  One way to get around this would
be to use two rules:

     foo ( # , # , # ) := bar([#1],#2,#3)
     foo ( # , # ) := bar([],#1,#2)

   More precisely, when Calc wants to match an optional or repeated
part of a pattern, it scans forward attempting to match that part.  If
it reaches the end of the optional part without failing, it "finalizes"
its choice and proceeds.  If it fails, though, it backs up and tries
the other alternative.  Thus Calc has "partial" backtracking.  A fully
backtracking parser would go on to make sure the rest of the pattern
matched before finalizing the choice.

File: calc,  Node: Conditional Syntax Rules,  Prev: Advanced Syntax Patterns,  Up: Syntax Tables

8.8.11.4 Conditional Syntax Rules
.................................

It is possible to attach a "condition" to a syntax rule.  For example,
the rules

     foo ( # ) := ifoo(#1) :: integer(#1)
     foo ( # ) := gfoo(#1)

will parse `foo(3)' as `ifoo(3)', but will parse `foo(3.5)' and
`foo(x)' as calls to `gfoo'.  Any number of conditions may be attached;
all must be true for the rule to succeed.  A condition is "true" if it
evaluates to a nonzero number.  *Note Logical Operations::, for a list
of Calc functions like `integer' that perform logical tests.

   The exact sequence of events is as follows:  When Calc tries a rule,
it first matches the pattern as usual.  It then substitutes `#1', `#2',
etc., in the conditions, if any.  Next, the conditions are simplified
and evaluated in order from left to right, as if by the `a s' algebra
command (*note Simplifying Formulas::).  Each result is true if it is a
nonzero number, or an expression that can be proven to be nonzero
(*note Declarations::).  If the results of all conditions are true, the
expression (such as `ifoo(#1)') has its `#'s substituted, and that is
the result of the parse.  If the result of any condition is false, Calc
goes on to try the next rule in the syntax table.

   Syntax rules also support `let' conditions, which operate in exactly
the same way as they do in algebraic rewrite rules.  *Note Other
Features of Rewrite Rules::, for details.  A `let' condition is always
true, but as a side effect it defines a variable which can be used in
later conditions, and also in the expression after the `:=' sign:

     foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)

The `dnumint' function tests if a value is numerically an integer,
i.e., either a true integer or an integer-valued float.  This rule will
parse `foo' with a half-integer argument, like `foo(3.5)', to a call
like `hifoo(4.)'.

   The lefthand side of a syntax rule `let' must be a simple variable,
not the arbitrary pattern that is allowed in rewrite rules.

   The `matches' function is also treated specially in syntax rule
conditions (again, in the same way as in rewrite rules).  *Note
Matching Commands::.  If the matching pattern contains meta-variables,
then those meta-variables may be used in later conditions and in the
result expression.  The arguments to `matches' are not evaluated in
this situation.

     sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])

This is another way to implement the Maple mode `sum' notation.  In
this approach, we allow `#2' to equal the whole expression `i=1..10'.
Then, we use `matches' to break it apart into its components.  If the
expression turns out not to match the pattern, the syntax rule will
fail.  Note that `Z S' always uses Calc's Normal language mode for
editing expressions in syntax rules, so we must use regular Calc
notation for the interval `[b..c]' that will correspond to the Maple
mode interval `1..10'.

File: calc,  Node: Modes Variable,  Next: Calc Mode Line,  Prev: Language Modes,  Up: Mode Settings

8.9 The `Modes' Variable
========================

The `m g' (`calc-get-modes') command pushes onto the stack a vector of
numbers that describes the various mode settings that are in effect.
With a numeric prefix argument, it pushes only the Nth mode, i.e., the
Nth element of this vector.  Keyboard macros can use the `m g' command
to modify their behavior based on the current mode settings.

   The modes vector is also available in the special variable `Modes'.
In other words, `m g' is like `s r Modes <RET>'.  It will not work to
store into this variable; in fact, if you do, `Modes' will cease to
track the current modes.  (The `m g' command will continue to work,
however.)

   In general, each number in this vector is suitable as a numeric
prefix argument to the associated mode-setting command.  (Recall that
the `~' key takes a number from the stack and gives it as a numeric
prefix to the next command.)

   The elements of the modes vector are as follows:

  1. Current precision.  Default is 12; associated command is `p'.

  2. Binary word size.  Default is 32; associated command is `b w'.

  3. Stack size (not counting the value about to be pushed by `m g').
     This is zero if `m g' is executed with an empty stack.

  4. Number radix.  Default is 10; command is `d r'.

  5. Floating-point format.  This is the number of digits, plus the
     constant 0 for normal notation, 10000 for scientific notation,
     20000 for engineering notation, or 30000 for fixed-point notation.
     These codes are acceptable as prefix arguments to the `d n'
     command, but note that this may lose information:  For example, `d
     s' and `C-u 12 d s' have similar (but not quite identical) effects
     if the current precision is 12, but they both produce a code of
     10012, which will be treated by `d n' as `C-u 12 d s'.  If the
     precision then changes, the float format will still be frozen at
     12 significant figures.

  6. Angular mode.  Default is 1 (degrees).  Other values are 2
     (radians) and 3 (HMS).  The `m d' command accepts these prefixes.

  7. Symbolic mode.  Value is 0 or 1; default is 0.  Command is `m s'.

  8. Fraction mode.  Value is 0 or 1; default is 0.  Command is `m f'.

  9. Polar mode.  Value is 0 (rectangular) or 1 (polar); default is 0.
     Command is `m p'.

 10. Matrix/Scalar mode.  Default value is -1.  Value is 0 for Scalar
     mode, -2 for Matrix mode, -3 for square Matrix mode, or N for NxN
     Matrix mode.  Command is `m v'.

 11. Simplification mode.  Default is 1.  Value is -1 for off (`m O'),
     0 for `m N', 2 for `m B', 3 for `m A', 4 for `m E', or 5 for
     `m U'.  The `m D' command accepts these prefixes.

 12. Infinite mode.  Default is -1 (off).  Value is 1 if the mode is on,
     or 0 if the mode is on with positive zeros.  Command is `m i'.

   For example, the sequence `M-1 m g <RET> 2 + ~ p' increases the
precision by two, leaving a copy of the old precision on the stack.
Later, `~ p' will restore the original precision using that stack
value.  (This sequence might be especially useful inside a keyboard
macro.)

   As another example, `M-3 m g 1 - ~ <DEL>' deletes all but the oldest
(bottommost) stack entry.

   Yet another example:  The HP-48 "round" command rounds a number to
the current displayed precision.  You could roughly emulate this in
Calc with the sequence `M-5 m g 10000 % ~ c c'.  (This would not work
for fixed-point mode, but it wouldn't be hard to do a full emulation
with the help of the `Z [' and `Z ]' programming commands.  *Note
Conditionals in Macros::.)

File: calc,  Node: Calc Mode Line,  Prev: Modes Variable,  Up: Mode Settings

8.10 The Calc Mode Line
=======================

This section is a summary of all symbols that can appear on the Calc
mode line, the highlighted bar that appears under the Calc stack window
(or under an editing window in Embedded mode).

   The basic mode line format is:

     --%*-Calc: 12 Deg OTHER MODES       (Calculator)

   The `%*' indicates that the buffer is "read-only"; it shows that
regular Emacs commands are not allowed to edit the stack buffer as if
it were text.

   The word `Calc:' changes to `CalcEmbed:' if Embedded mode is
enabled.  The words after this describe the various Calc modes that are
in effect.

   The first mode is always the current precision, an integer.  The
second mode is always the angular mode, either `Deg', `Rad', or `Hms'.

   Here is a complete list of the remaining symbols that can appear on
the mode line:

`Alg'
     Algebraic mode (`m a'; *note Algebraic Entry::).

`Alg[('
     Incomplete algebraic mode (`C-u m a').

`Alg*'
     Total algebraic mode (`m t').

`Symb'
     Symbolic mode (`m s'; *note Symbolic Mode::).

`Matrix'
     Matrix mode (`m v'; *note Matrix Mode::).

`MatrixN'
     Dimensioned Matrix mode (`C-u N m v'; *note Matrix Mode::).

`SqMatrix'
     Square Matrix mode (`C-u m v'; *note Matrix Mode::).

`Scalar'
     Scalar mode (`m v'; *note Matrix Mode::).

`Polar'
     Polar complex mode (`m p'; *note Polar Mode::).

`Frac'
     Fraction mode (`m f'; *note Fraction Mode::).

`Inf'
     Infinite mode (`m i'; *note Infinite Mode::).

`+Inf'
     Positive Infinite mode (`C-u 0 m i').

`NoSimp'
     Default simplifications off (`m O'; *note Simplification Modes::).

`NumSimp'
     Default simplifications for numeric arguments only (`m N').

`BinSimpW'
     Binary-integer simplification mode; word size W (`m B', `b w').

`AlgSimp'
     Algebraic simplification mode (`m A').

`ExtSimp'
     Extended algebraic simplification mode (`m E').

`UnitSimp'
     Units simplification mode (`m U').

`Bin'
     Current radix is 2 (`d 2'; *note Radix Modes::).

`Oct'
     Current radix is 8 (`d 8').

`Hex'
     Current radix is 16 (`d 6').

`RadixN'
     Current radix is N (`d r').

`Zero'
     Leading zeros (`d z'; *note Radix Modes::).

`Big'
     Big language mode (`d B'; *note Normal Language Modes::).

`Flat'
     One-line normal language mode (`d O').

`Unform'
     Unformatted language mode (`d U').

`C'
     C language mode (`d C'; *note C FORTRAN Pascal::).

`Pascal'
     Pascal language mode (`d P').

`Fortran'
     FORTRAN language mode (`d F').

`TeX'
     TeX language mode (`d T'; *note TeX and LaTeX Language Modes::).

`LaTeX'
     LaTeX language mode (`d L'; *note TeX and LaTeX Language Modes::).

`Eqn'
     "Eqn" language mode (`d E'; *note Eqn Language Mode::).

`Math'
     Mathematica language mode (`d M'; *note Mathematica Language
     Mode::).

`Maple'
     Maple language mode (`d W'; *note Maple Language Mode::).

`NormN'
     Normal float mode with N digits (`d n'; *note Float Formats::).

`FixN'
     Fixed point mode with N digits after the point (`d f').

`Sci'
     Scientific notation mode (`d s').

`SciN'
     Scientific notation with N digits (`d s').

`Eng'
     Engineering notation mode (`d e').

`EngN'
     Engineering notation with N digits (`d e').

`LeftN'
     Left-justified display indented by N (`d <'; *note
     Justification::).

`Right'
     Right-justified display (`d >').

`RightN'
     Right-justified display with width N (`d >').

`Center'
     Centered display (`d =').

`CenterN'
     Centered display with center column N (`d =').

`WidN'
     Line breaking with width N (`d b'; *note Normal Language Modes::).

`Wide'
     No line breaking (`d b').

`Break'
     Selections show deep structure (`j b'; *note Making Selections::).

`Save'
     Record modes in `~/.calc.el' (`m R'; *note General Mode
     Commands::).

`Local'
     Record modes in Embedded buffer (`m R').

`LocEdit'
     Record modes as editing-only in Embedded buffer (`m R').

`LocPerm'
     Record modes as permanent-only in Embedded buffer (`m R').

`Global'
     Record modes as global in Embedded buffer (`m R').

`Manual'
     Automatic recomputation turned off (`m C'; *note Automatic
     Recomputation::).

`Graph'
     GNUPLOT process is alive in background (*note Graphics::).

`Sel'
     Top-of-stack has a selection (Embedded only; *note Making
     Selections::).

`Dirty'
     The stack display may not be up-to-date (*note Display Modes::).

`Inv'
     "Inverse" prefix was pressed (`I'; *note Inverse and Hyperbolic::).

`Hyp'
     "Hyperbolic" prefix was pressed (`H').

`Keep'
     "Keep-arguments" prefix was pressed (`K').

`Narrow'
     Stack is truncated (`d t'; *note Truncating the Stack::).

   In addition, the symbols `Active' and `~Active' can appear as minor
modes on an Embedded buffer's mode line.  *Note Embedded Mode::.

File: calc,  Node: Arithmetic,  Next: Scientific Functions,  Prev: Mode Settings,  Up: Top

9 Arithmetic Functions
**********************

This chapter describes the Calc commands for doing simple calculations
on numbers, such as addition, absolute value, and square roots.  These
commands work by removing the top one or two values from the stack,
performing the desired operation, and pushing the result back onto the
stack.  If the operation cannot be performed, the result pushed is a
formula instead of a number, such as `2/0' (because division by zero is
invalid) or `sqrt(x)' (because the argument `x' is a formula).

   Most of the commands described here can be invoked by a single
keystroke.  Some of the more obscure ones are two-letter sequences
beginning with the `f' ("functions") prefix key.

   *Note Prefix Arguments::, for a discussion of the effect of numeric
prefix arguments on commands in this chapter which do not otherwise
interpret a prefix argument.

* Menu:

* Basic Arithmetic::
* Integer Truncation::
* Complex Number Functions::
* Conversions::
* Date Arithmetic::
* Financial Functions::
* Binary Functions::

File: calc,  Node: Basic Arithmetic,  Next: Integer Truncation,  Prev: Arithmetic,  Up: Arithmetic

9.1 Basic Arithmetic
====================

The `+' (`calc-plus') command adds two numbers.  The numbers may be any
of the standard Calc data types.  The resulting sum is pushed back onto
the stack.

   If both arguments of `+' are vectors or matrices (of matching
dimensions), the result is a vector or matrix sum.  If one argument is
a vector and the other a scalar (i.e., a non-vector), the scalar is
added to each of the elements of the vector to form a new vector.  If
the scalar is not a number, the operation is left in symbolic form:
Suppose you added `x' to the vector `[1,2]'.  You may want the result
`[1+x,2+x]', or you may plan to substitute a 2-vector for `x' in the
future.  Since the Calculator can't tell which interpretation you want,
it makes the safest assumption.  *Note Reducing and Mapping::, for a
way to add `x' to every element of a vector.

   If either argument of `+' is a complex number, the result will in
general be complex.  If one argument is in rectangular form and the
other polar, the current Polar mode determines the form of the result.
If Symbolic mode is enabled, the sum may be left as a formula if the
necessary conversions for polar addition are non-trivial.

   If both arguments of `+' are HMS forms, the forms are added
according to the usual conventions of hours-minutes-seconds notation.
If one argument is an HMS form and the other is a number, that number
is converted from degrees or radians (depending on the current Angular
mode) to HMS format and then the two HMS forms are added.

   If one argument of `+' is a date form, the other can be either a
real number, which advances the date by a certain number of days, or an
HMS form, which advances the date by a certain amount of time.
Subtracting two date forms yields the number of days between them.
Adding two date forms is meaningless, but Calc interprets it as the
subtraction of one date form and the negative of the other.  (The
negative of a date form can be understood by remembering that dates are
stored as the number of days before or after Jan 1, 1 AD.)

   If both arguments of `+' are error forms, the result is an error form
with an appropriately computed standard deviation.  If one argument is
an error form and the other is a number, the number is taken to have
zero error.  Error forms may have symbolic formulas as their mean
and/or error parts; adding these will produce a symbolic error form
result.  However, adding an error form to a plain symbolic formula (as
in `(a +/- b) + c') will not work, for the same reasons just mentioned
for vectors.  Instead you must write `(a +/- b) + (c +/- 0)'.

   If both arguments of `+' are modulo forms with equal values of `M',
or if one argument is a modulo form and the other a plain number, the
result is a modulo form which represents the sum, modulo `M', of the
two values.

   If both arguments of `+' are intervals, the result is an interval
which describes all possible sums of the possible input values.  If one
argument is a plain number, it is treated as the interval `[x .. x]'.

   If one argument of `+' is an infinity and the other is not, the
result is that same infinity.  If both arguments are infinite and in
the same direction, the result is the same infinity, but if they are
infinite in different directions the result is `nan'.

   The `-' (`calc-minus') command subtracts two values.  The top number
on the stack is subtracted from the one behind it, so that the
computation `5 <RET> 2 -' produces 3, not -3.  All options available
for `+' are available for `-' as well.

   The `*' (`calc-times') command multiplies two numbers.  If one
argument is a vector and the other a scalar, the scalar is multiplied by
the elements of the vector to produce a new vector.  If both arguments
are vectors, the interpretation depends on the dimensions of the
vectors:  If both arguments are matrices, a matrix multiplication is
done.  If one argument is a matrix and the other a plain vector, the
vector is interpreted as a row vector or column vector, whichever is
dimensionally correct.  If both arguments are plain vectors, the result
is a single scalar number which is the dot product of the two vectors.

   If one argument of `*' is an HMS form and the other a number, the
HMS form is multiplied by that amount.  It is an error to multiply two
HMS forms together, or to attempt any multiplication involving date
forms.  Error forms, modulo forms, and intervals can be multiplied; see
the comments for addition of those forms.  When two error forms or
intervals are multiplied they are considered to be statistically
independent; thus, `[-2 .. 3] * [-2 .. 3]' is `[-6 .. 9]', whereas
`[-2 .. 3] ^ 2' is `[0 .. 9]'.

   The `/' (`calc-divide') command divides two numbers.

   When combining multiplication and division in an algebraic formula,
it is good style to use parentheses to distinguish between possible
interpretations; the expression `a/b*c' should be written `(a/b)*c' or
`a/(b*c)', as appropriate.  Without the parentheses, Calc will
interpret `a/b*c' as `a/(b*c)', since in algebraic entry Calc gives
division a lower precedence than multiplication. (This is not standard
across all computer languages, and Calc may change the precedence
depending on the language mode being used.  *Note Language Modes::.)
This default ordering can be changed by setting the customizable
variable `calc-multiplication-has-precedence' to `nil' (*note
Customizing Calc::); this will give multiplication and division equal
precedences.  Note that Calc's default choice of precedence allows `a b
/ c d' to be used as a shortcut for
     a b
     ---.
     c d

   When dividing a scalar `B' by a square matrix `A', the computation
performed is `B' times the inverse of `A'.  This also occurs if `B' is
itself a vector or matrix, in which case the effect is to solve the set
of linear equations represented by `B'.  If `B' is a matrix with the
same number of rows as `A', or a plain vector (which is interpreted
here as a column vector), then the equation `A X = B' is solved for the
vector or matrix `X'.  Otherwise, if `B' is a non-square matrix with
the same number of _columns_ as `A', the equation `X A = B' is solved.
If you wish a vector `B' to be interpreted as a row vector to be solved
as `X A = B', make it into a one-row matrix with `C-u 1 v p' first.  To
force a left-handed solution with a square matrix `B', transpose `A'
and `B' before dividing, then transpose the result.

   HMS forms can be divided by real numbers or by other HMS forms.
Error forms can be divided in any combination of ways.  Modulo forms
where both values and the modulo are integers can be divided to get an
integer modulo form result.  Intervals can be divided; dividing by an
interval that encompasses zero or has zero as a limit will result in an
infinite interval.

   The `^' (`calc-power') command raises a number to a power.  If the
power is an integer, an exact result is computed using repeated
multiplications.  For non-integer powers, Calc uses Newton's method or
logarithms and exponentials.  Square matrices can be raised to integer
powers.  If either argument is an error (or interval or modulo) form,
the result is also an error (or interval or modulo) form.

   If you press the `I' (inverse) key first, the `I ^' command computes
an Nth root:  `125 <RET> 3 I ^' computes the number 5.  (This is
entirely equivalent to `125 <RET> 1:3 ^'.)

   The `\' (`calc-idiv') command divides two numbers on the stack to
produce an integer result.  It is equivalent to dividing with </>, then
rounding down with `F' (`calc-floor'), only a bit more convenient and
efficient.  Also, since it is an all-integer operation when the
arguments are integers, it avoids problems that `/ F' would have with
floating-point roundoff.

   The `%' (`calc-mod') command performs a "modulo" (or "remainder")
operation.  Mathematically, `a%b = a - (a\b)*b', and is defined for all
real numbers `a' and `b' (except `b=0').  For positive `b', the result
will always be between 0 (inclusive) and `b' (exclusive).  Modulo does
not work for HMS forms and error forms.  If `a' is a modulo form, its
modulo is changed to `b', which must be positive real number.

   The `:' (`calc-fdiv') [`fdiv'] command divides the two integers on
the top of the stack to produce a fractional result.  This is a
convenient shorthand for enabling Fraction mode (with `m f')
temporarily and using `/'.  Note that during numeric entry the `:' key
is interpreted as a fraction separator, so to divide 8 by 6 you would
have to type `8 <RET> 6 <RET> :'.  (Of course, in this case, it would
be much easier simply to enter the fraction directly as `8:6 <RET>'!)

   The `n' (`calc-change-sign') command negates the number on the top
of the stack.  It works on numbers, vectors and matrices, HMS forms,
date forms, error forms, intervals, and modulo forms.

   The `A' (`calc-abs') [`abs'] command computes the absolute value of
a number.  The result of `abs' is always a nonnegative real number:
With a complex argument, it computes the complex magnitude.  With a
vector or matrix argument, it computes the Frobenius norm, i.e., the
square root of the sum of the squares of the absolute values of the
elements.  The absolute value of an error form is defined by replacing
the mean part with its absolute value and leaving the error part the
same.  The absolute value of a modulo form is undefined.  The absolute
value of an interval is defined in the obvious way.

   The `f A' (`calc-abssqr') [`abssqr'] command computes the absolute
value squared of a number, vector or matrix, or error form.

   The `f s' (`calc-sign') [`sign'] command returns 1 if its argument
is positive, -1 if its argument is negative, or 0 if its argument is
zero.  In algebraic form, you can also write `sign(a,x)' which
evaluates to `x * sign(a)', i.e., either `x', `-x', or zero depending
on the sign of `a'.

   The `&' (`calc-inv') [`inv'] command computes the reciprocal of a
number, i.e., `1 / x'.  Operating on a square matrix, it computes the
inverse of that matrix.

   The `Q' (`calc-sqrt') [`sqrt'] command computes the square root of a
number.  For a negative real argument, the result will be a complex
number whose form is determined by the current Polar mode.

   The `f h' (`calc-hypot') [`hypot'] command computes the square root
of the sum of the squares of two numbers.  That is, `hypot(a,b)' is the
length of the hypotenuse of a right triangle with sides `a' and `b'.
If the arguments are complex numbers, their squared magnitudes are used.

   The `f Q' (`calc-isqrt') [`isqrt'] command computes the integer
square root of an integer.  This is the true square root of the number,
rounded down to an integer.  For example, `isqrt(10)' produces 3.  Note
that, like `\' [`idiv'], this uses exact integer arithmetic throughout
to avoid roundoff problems.  If the input is a floating-point number or
other non-integer value, this is exactly the same as `floor(sqrt(x))'.

   The `f n' (`calc-min') [`min'] and `f x' (`calc-max') [`max']
commands take the minimum or maximum of two real numbers, respectively.
These commands also work on HMS forms, date forms, intervals, and
infinities.  (In algebraic expressions, these functions take any number
of arguments and return the maximum or minimum among all the arguments.)

   The `f M' (`calc-mant-part') [`mant'] function extracts the
"mantissa" part `m' of its floating-point argument; `f X'
(`calc-xpon-part') [`xpon'] extracts the "exponent" part `e'.  The
original number is equal to `m * 10^e', where `m' is in the interval
`[1.0 .. 10.0)' except that `m=e=0' if the original number is zero.
For integers and fractions, `mant' returns the number unchanged and
`xpon' returns zero.  The `v u' (`calc-unpack') command can also be
used to "unpack" a floating-point number; this produces an integer
mantissa and exponent, with the constraint that the mantissa is not a
multiple of ten (again except for the `m=e=0' case).

   The `f S' (`calc-scale-float') [`scf'] function scales a number by a
given power of ten.  Thus, `scf(mant(x), xpon(x)) = x' for any real
`x'.  The second argument must be an integer, but the first may
actually be any numeric value.  For example, `scf(5,-2) = 0.05' or
`1:20' depending on the current Fraction mode.

   The `f [' (`calc-decrement') [`decr'] and `f ]' (`calc-increment')
[`incr'] functions decrease or increase a number by one unit.  For
integers, the effect is obvious.  For floating-point numbers, the
change is by one unit in the last place.  For example, incrementing
`12.3456' when the current precision is 6 digits yields `12.3457'.  If
the current precision had been 8 digits, the result would have been
`12.345601'.  Incrementing `0.0' produces `10^-p', where `p' is the
current precision.  These operations are defined only on integers and
floats.  With numeric prefix arguments, they change the number by `n'
units.

   Note that incrementing followed by decrementing, or vice-versa, will
almost but not quite always cancel out.  Suppose the precision is 6
digits and the number `9.99999' is on the stack.  Incrementing will
produce `10.0000'; decrementing will produce `9.9999'.  One digit has
been dropped.  This is an unavoidable consequence of the way
floating-point numbers work.

   Incrementing a date/time form adjusts it by a certain number of
seconds.  Incrementing a pure date form adjusts it by a certain number
of days.

File: calc,  Node: Integer Truncation,  Next: Complex Number Functions,  Prev: Basic Arithmetic,  Up: Arithmetic

9.2 Integer Truncation
======================

There are four commands for truncating a real number to an integer,
differing mainly in their treatment of negative numbers.  All of these
commands have the property that if the argument is an integer, the
result is the same integer.  An integer-valued floating-point argument
is converted to integer form.

   If you press `H' (`calc-hyperbolic') first, the result will be
expressed as an integer-valued floating-point number.

   The `F' (`calc-floor') [`floor' or `ffloor'] command truncates a
real number to the next lower integer, i.e., toward minus infinity.
Thus `3.6 F' produces 3, but `_3.6 F' produces -4.

   The `I F' (`calc-ceiling') [`ceil' or `fceil'] command truncates
toward positive infinity.  Thus `3.6 I F' produces 4, and `_3.6 I F'
produces -3.

   The `R' (`calc-round') [`round' or `fround'] command rounds to the
nearest integer.  When the fractional part is .5 exactly, this command
rounds away from zero.  (All other rounding in the Calculator uses this
convention as well.)  Thus `3.5 R' produces 4 but `3.4 R' produces 3;
`_3.5 R' produces -4.

   The `I R' (`calc-trunc') [`trunc' or `ftrunc'] command truncates
toward zero.  In other words, it "chops off" everything after the
decimal point.  Thus `3.6 I R' produces 3 and `_3.6 I R' produces -3.

   These functions may not be applied meaningfully to error forms, but
they do work for intervals.  As a convenience, applying `floor' to a
modulo form floors the value part of the form.  Applied to a vector,
these functions operate on all elements of the vector one by one.
Applied to a date form, they operate on the internal numerical
representation of dates, converting a date/time form into a pure date.

   There are two more rounding functions which can only be entered in
algebraic notation.  The `roundu' function is like `round' except that
it rounds up, toward plus infinity, when the fractional part is .5.
This distinction matters only for negative arguments.  Also, `rounde'
rounds to an even number in the case of a tie, rounding up or down as
necessary.  For example, `rounde(3.5)' and `rounde(4.5)' both return 4,
but `rounde(5.5)' returns 6.  The advantage of round-to-even is that
the net error due to rounding after a long calculation tends to cancel
out to zero.  An important subtle point here is that the number being
fed to `rounde' will already have been rounded to the current precision
before `rounde' begins.  For example, `rounde(2.500001)' with a current
precision of 6 will incorrectly, or at least surprisingly, yield 2
because the argument will first have been rounded down to `2.5' (which
`rounde' sees as an exact tie between 2 and 3).

   Each of these functions, when written in algebraic formulas, allows
a second argument which specifies the number of digits after the
decimal point to keep.  For example, `round(123.4567, 2)' will produce
the answer 123.46, and `round(123.4567, -1)' will produce 120 (i.e.,
the cutoff is one digit to the _left_ of the decimal point).  A second
argument of zero is equivalent to no second argument at all.

   To compute the fractional part of a number (i.e., the amount which,
when added to `floor(N)', will produce N) just take N modulo 1 using
the `%' command.

   Note also the `\' (integer quotient), `f I' (integer logarithm), and
`f Q' (integer square root) commands, which are analogous to `/', `B',
and `Q', respectively, except that they take integer arguments and
return the result rounded down to an integer.

File: calc,  Node: Complex Number Functions,  Next: Conversions,  Prev: Integer Truncation,  Up: Arithmetic

9.3 Complex Number Functions
============================

The `J' (`calc-conj') [`conj'] command computes the complex conjugate
of a number.  For complex number `a+bi', the complex conjugate is
`a-bi'.  If the argument is a real number, this command leaves it the
same.  If the argument is a vector or matrix, this command replaces
each element by its complex conjugate.

   The `G' (`calc-argument') [`arg'] command computes the "argument" or
polar angle of a complex number.  For a number in polar notation, this
is simply the second component of the pair `(R;THETA)'.  The result is
expressed according to the current angular mode and will be in the
range -180 degrees (exclusive) to +180 degrees (inclusive), or the
equivalent range in radians.

   The `calc-imaginary' command multiplies the number on the top of the
stack by the imaginary number `i = (0,1)'.  This command is not
normally bound to a key in Calc, but it is available on the <IMAG>
button in Keypad mode.

   The `f r' (`calc-re') [`re'] command replaces a complex number by
its real part.  This command has no effect on real numbers.  (As an
added convenience, `re' applied to a modulo form extracts the value
part.)

   The `f i' (`calc-im') [`im'] command replaces a complex number by
its imaginary part; real numbers are converted to zero.  With a vector
or matrix argument, these functions operate element-wise.

   The `v p' (`calc-pack') command can pack the top two numbers on the
stack into a composite object such as a complex number.  With a prefix
argument of -1, it produces a rectangular complex number; with an
argument of -2, it produces a polar complex number.  (Also, *note
Building Vectors::.)

   The `v u' (`calc-unpack') command takes the complex number (or other
composite object) on the top of the stack and unpacks it into its
separate components.

File: calc,  Node: Conversions,  Next: Date Arithmetic,  Prev: Complex Number Functions,  Up: Arithmetic

9.4 Conversions
===============

The commands described in this section convert numbers from one form to
another; they are two-key sequences beginning with the letter `c'.

   The `c f' (`calc-float') [`pfloat'] command converts the number on
the top of the stack to floating-point form.  For example, `23' is
converted to `23.0', `3:2' is converted to `1.5', and `2.3' is left the
same.  If the value is a composite object such as a complex number or
vector, each of the components is converted to floating-point.  If the
value is a formula, all numbers in the formula are converted to
floating-point.  Note that depending on the current floating-point
precision, conversion to floating-point format may lose information.

   As a special exception, integers which appear as powers or subscripts
are not floated by `c f'.  If you really want to float a power, you can
use a `j s' command to select the power followed by `c f'.  Because `c
f' cannot examine the formula outside of the selection, it does not
notice that the thing being floated is a power.  *Note Selecting
Subformulas::.

   The normal `c f' command is "pervasive" in the sense that it applies
to all numbers throughout the formula.  The `pfloat' algebraic function
never stays around in a formula; `pfloat(a + 1)' changes to `a + 1.0'
as soon as it is evaluated.

   With the Hyperbolic flag, `H c f' [`float'] operates only on the
number or vector of numbers at the top level of its argument.  Thus,
`float(1)' is 1.0, but `float(a + 1)' is left unevaluated because its
argument is not a number.

   You should use `H c f' if you wish to guarantee that the final
value, once all the variables have been assigned, is a float; you would
use `c f' if you wish to do the conversion on the numbers that appear
right now.

   The `c F' (`calc-fraction') [`pfrac'] command converts a
floating-point number into a fractional approximation.  By default, it
produces a fraction whose decimal representation is the same as the
input number, to within the current precision.  You can also give a
numeric prefix argument to specify a tolerance, either directly, or, if
the prefix argument is zero, by using the number on top of the stack as
the tolerance.  If the tolerance is a positive integer, the fraction is
correct to within that many significant figures.  If the tolerance is a
non-positive integer, it specifies how many digits fewer than the
current precision to use.  If the tolerance is a floating-point number,
the fraction is correct to within that absolute amount.

   The `pfrac' function is pervasive, like `pfloat'.  There is also a
non-pervasive version, `H c F' [`frac'], which is analogous to `H c f'
discussed above.

   The `c d' (`calc-to-degrees') [`deg'] command converts a number into
degrees form.  The value on the top of the stack may be an HMS form
(interpreted as degrees-minutes-seconds), or a real number which will
be interpreted in radians regardless of the current angular mode.

   The `c r' (`calc-to-radians') [`rad'] command converts an HMS form
or angle in degrees into an angle in radians.

   The `c h' (`calc-to-hms') [`hms'] command converts a real number,
interpreted according to the current angular mode, to an HMS form
describing the same angle.  In algebraic notation, the `hms' function
also accepts three arguments: `hms(H, M, S)'.  (The three-argument
version is independent of the current angular mode.)

   The `calc-from-hms' command converts the HMS form on the top of the
stack into a real number according to the current angular mode.

   The `c p' (`calc-polar') command converts the complex number on the
top of the stack from polar to rectangular form, or from rectangular to
polar form, whichever is appropriate.  Real numbers are left the same.
This command is equivalent to the `rect' or `polar' functions in
algebraic formulas, depending on the direction of conversion.  (It uses
`polar', except that if the argument is already a polar complex number,
it uses `rect' instead.  The `I c p' command always uses `rect'.)

   The `c c' (`calc-clean') [`pclean'] command "cleans" the number on
the top of the stack.  Floating point numbers are re-rounded according
to the current precision.  Polar numbers whose angular components have
strayed from the -180 to +180 degree range are normalized.  (Note that
results will be undesirable if the current angular mode is different
from the one under which the number was produced!)  Integers and
fractions are generally unaffected by this operation.  Vectors and
formulas are cleaned by cleaning each component number (i.e.,
pervasively).

   If the simplification mode is set below the default level, it is
raised to the default level for the purposes of this command.  Thus, `c
c' applies the default simplifications even if their automatic
application is disabled.  *Note Simplification Modes::.

   A numeric prefix argument to `c c' sets the floating-point precision
to that value for the duration of the command.  A positive prefix (of at
least 3) sets the precision to the specified value; a negative or zero
prefix decreases the precision by the specified amount.

   The keystroke sequences `c 0' through `c 9' are equivalent to `c c'
with the corresponding negative prefix argument.  If roundoff errors
have changed 2.0 into 1.999999, typing `c 1' to clip off one decimal
place often conveniently does the trick.

   The `c c' command with a numeric prefix argument, and the `c 0'
through `c 9' commands, also "clip" very small floating-point numbers
to zero.  If the exponent is less than or equal to the negative of the
specified precision, the number is changed to 0.0.  For example, if the
current precision is 12, then `c 2' changes the vector `[1e-8, 1e-9,
1e-10, 1e-11]' to `[1e-8, 1e-9, 0, 0]'.  Numbers this small generally
arise from roundoff noise.

   If the numbers you are using really are legitimately this small, you
should avoid using the `c 0' through `c 9' commands.  (The plain `c c'
command rounds to the current precision but does not clip small
numbers.)

   One more property of `c 0' through `c 9', and of `c c' with a prefix
argument, is that integer-valued floats are converted to plain
integers, so that `c 1' on `[1., 1.5, 2., 2.5, 3.]' produces `[1, 1.5,
2, 2.5, 3]'.  This is not done for huge numbers (`1e100' is technically
an integer-valued float, but you wouldn't want it automatically
converted to a 100-digit integer).

   With the Hyperbolic flag, `H c c' and `H c 0' through `H c 9'
operate non-pervasively [`clean'].

File: calc,  Node: Date Arithmetic,  Next: Financial Functions,  Prev: Conversions,  Up: Arithmetic

9.5 Date Arithmetic
===================

The commands described in this section perform various conversions and
calculations involving date forms (*note Date Forms::).  They use the
`t' (for time/date) prefix key followed by shifted letters.

   The simplest date arithmetic is done using the regular `+' and `-'
commands.  In particular, adding a number to a date form advances the
date form by a certain number of days; adding an HMS form to a date
form advances the date by a certain amount of time; and subtracting two
date forms produces a difference measured in days.  The commands
described here provide additional, more specialized operations on dates.

   Many of these commands accept a numeric prefix argument; if you give
plain `C-u' as the prefix, these commands will instead take the
additional argument from the top of the stack.

* Menu:

* Date Conversions::
* Date Functions::
* Time Zones::
* Business Days::

File: calc,  Node: Date Conversions,  Next: Date Functions,  Prev: Date Arithmetic,  Up: Date Arithmetic

9.5.1 Date Conversions
----------------------

The `t D' (`calc-date') [`date'] command converts a date form into a
number, measured in days since Jan 1, 1 AD.  The result will be an
integer if DATE is a pure date form, or a fraction or float if DATE is
a date/time form.  Or, if its argument is a number, it converts this
number into a date form.

   With a numeric prefix argument, `t D' takes that many objects (up to
six) from the top of the stack and interprets them in one of the
following ways:

   The `date(YEAR, MONTH, DAY)' function builds a pure date form out of
the specified year, month, and day, which must all be integers.  YEAR
is a year number, such as 1991 (_not_ the same as 91!).  MONTH must be
an integer in the range 1 to 12; DAY must be in the range 1 to 31.  If
the specified month has fewer than 31 days and DAY is too large, the
equivalent day in the following month will be used.

   The `date(MONTH, DAY)' function builds a pure date form using the
current year, as determined by the real-time clock.

   The `date(YEAR, MONTH, DAY, HMS)' function builds a date/time form
using an HMS form.

   The `date(YEAR, MONTH, DAY, HOUR, MINUTE, SECOND)' function builds a
date/time form.  HOUR should be an integer in the range 0 to 23; MINUTE
should be an integer in the range 0 to 59; SECOND should be any real
number in the range `[0 .. 60)'.  The last two arguments default to
zero if omitted.

   The `t J' (`calc-julian') [`julian'] command converts a date form
into a Julian day count, which is the number of days since noon (GMT)
on Jan 1, 4713 BC.  A pure date is converted to an integer Julian count
representing noon of that day.  A date/time form is converted to an
exact floating-point Julian count, adjusted to interpret the date form
in the current time zone but the Julian day count in Greenwich Mean
Time.  A numeric prefix argument allows you to specify the time zone;
*note Time Zones::.  Use a prefix of zero to suppress the time zone
adjustment.  Note that pure date forms are never time-zone adjusted.

   This command can also do the opposite conversion, from a Julian day
count (either an integer day, or a floating-point day and time in the
GMT zone), into a pure date form or a date/time form in the current or
specified time zone.

   The `t U' (`calc-unix-time') [`unixtime'] command converts a date
form into a Unix time value, which is the number of seconds since
midnight on Jan 1, 1970, or vice-versa.  The numeric result will be an
integer if the current precision is 12 or less; for higher precisions,
the result may be a float with (PRECISION-12) digits after the decimal.
Just as for `t J', the numeric time is interpreted in the GMT time zone
and the date form is interpreted in the current or specified zone.
Some systems use Unix-like numbering but with the local time zone; give
a prefix of zero to suppress the adjustment if so.

   The `t C' (`calc-convert-time-zones') [`tzconv'] command converts a
date form from one time zone to another.  You are prompted for each
time zone name in turn; you can answer with any suitable Calc time zone
expression (*note Time Zones::).  If you answer either prompt with a
blank line, the local time zone is used for that prompt.  You can also
answer the first prompt with `$' to take the two time zone names from
the stack (and the date to be converted from the third stack level).

File: calc,  Node: Date Functions,  Next: Business Days,  Prev: Date Conversions,  Up: Date Arithmetic

9.5.2 Date Functions
--------------------

The `t N' (`calc-now') [`now'] command pushes the current date and time
on the stack as a date form.  The time is reported in terms of the
specified time zone; with no numeric prefix argument, `t N' reports for
the current time zone.

   The `t P' (`calc-date-part') command extracts one part of a date
form.  The prefix argument specifies the part; with no argument, this
command prompts for a part code from 1 to 9.  The various part codes
are described in the following paragraphs.

   The `M-1 t P' [`year'] function extracts the year number from a date
form as an integer, e.g., 1991.  This and the following functions will
also accept a real number for an argument, which is interpreted as a
standard Calc day number.  Note that this function will never return
zero, since the year 1 BC immediately precedes the year 1 AD.

   The `M-2 t P' [`month'] function extracts the month number from a
date form as an integer in the range 1 to 12.

   The `M-3 t P' [`day'] function extracts the day number from a date
form as an integer in the range 1 to 31.

   The `M-4 t P' [`hour'] function extracts the hour from a date form
as an integer in the range 0 (midnight) to 23.  Note that 24-hour time
is always used.  This returns zero for a pure date form.  This function
(and the following two) also accept HMS forms as input.

   The `M-5 t P' [`minute'] function extracts the minute from a date
form as an integer in the range 0 to 59.

   The `M-6 t P' [`second'] function extracts the second from a date
form.  If the current precision is 12 or less, the result is an integer
in the range 0 to 59.  For higher precisions, the result may instead be
a floating-point number.

   The `M-7 t P' [`weekday'] function extracts the weekday number from
a date form as an integer in the range 0 (Sunday) to 6 (Saturday).

   The `M-8 t P' [`yearday'] function extracts the day-of-year number
from a date form as an integer in the range 1 (January 1) to 366
(December 31 of a leap year).

   The `M-9 t P' [`time'] function extracts the time portion of a date
form as an HMS form.  This returns `0@ 0' 0"' for a pure date form.

   The `t M' (`calc-new-month') [`newmonth'] command computes a new
date form that represents the first day of the month specified by the
input date.  The result is always a pure date form; only the year and
month numbers of the input are retained.  With a numeric prefix
argument N in the range from 1 to 31, `t M' computes the Nth day of the
month.  (If N is greater than the actual number of days in the month,
or if N is zero, the last day of the month is used.)

   The `t Y' (`calc-new-year') [`newyear'] command computes a new pure
date form that represents the first day of the year specified by the
input.  The month, day, and time of the input date form are lost.  With
a numeric prefix argument N in the range from 1 to 366, `t Y' computes
the Nth day of the year (366 is treated as 365 in non-leap years).  A
prefix argument of 0 computes the last day of the year (December 31).
A negative prefix argument from -1 to -12 computes the first day of the
Nth month of the year.

   The `t W' (`calc-new-week') [`newweek'] command computes a new pure
date form that represents the Sunday on or before the input date.  With
a numeric prefix argument, it can be made to use any day of the week as
the starting day; the argument must be in the range from 0 (Sunday) to
6 (Saturday).  This function always subtracts between 0 and 6 days from
the input date.

   Here's an example use of `newweek':  Find the date of the next
Wednesday after a given date.  Using `M-3 t W' or `newweek(d, 3)' will
give you the _preceding_ Wednesday, so `newweek(d+7, 3)' will give you
the following Wednesday.  A further look at the definition of `newweek'
shows that if the input date is itself a Wednesday, this formula will
return the Wednesday one week in the future.  An exercise for the
reader is to modify this formula to yield the same day if the input is
already a Wednesday.  Another interesting exercise is to preserve the
time-of-day portion of the input (`newweek' resets the time to
midnight; hint: how can `newweek' be defined in terms of the `weekday'
function?).

   The `pwday(DATE)' function (not on any key) computes the
day-of-month number of the Sunday on or before DATE.  With two
arguments, `pwday(DATE, DAY)' computes the day number of the Sunday on
or before day number DAY of the month specified by DATE.  The DAY must
be in the range from 7 to 31; if the day number is greater than the
actual number of days in the month, the true number of days is used
instead.  Thus `pwday(DATE, 7)' finds the first Sunday of the month, and
`pwday(DATE, 31)' finds the last Sunday of the month.  With a third
WEEKDAY argument, `pwday' can be made to look for any day of the week
instead of Sunday.

   The `t I' (`calc-inc-month') [`incmonth'] command increases a date
form by one month, or by an arbitrary number of months specified by a
numeric prefix argument.  The time portion, if any, of the date form
stays the same.  The day also stays the same, except that if the new
month has fewer days the day number may be reduced to lie in the valid
range.  For example, `incmonth(<Jan 31, 1991>)' produces `<Feb 28,
1991>'.  Because of this, `t I t I' and `M-2 t I' do not always give
the same results (`<Mar 28, 1991>' versus `<Mar 31, 1991>' in this
case).

   The `incyear(DATE, STEP)' function increases a date form by the
specified number of years, which may be any positive or negative
integer.  Note that `incyear(d, n)' is equivalent to
`incmonth(d, 12*n)', but these do not have simple equivalents in terms
of day arithmetic because months and years have varying lengths.  If
the STEP argument is omitted, 1 year is assumed.  There is no keyboard
command for this function; use `C-u 12 t I' instead.

   There is no `newday' function at all because `F' [`floor'] serves
this purpose.  Similarly, instead of `incday' and `incweek' simply use
`d + n' or `d + 7 n'.

   *Note Basic Arithmetic::, for the `f ]' [`incr'] command which can
adjust a date/time form by a certain number of seconds.

File: calc,  Node: Time Zones,  Prev: Business Days,  Up: Date Arithmetic

9.5.4 Time Zones
----------------

Time zones and daylight saving time are a complicated business.  The
conversions to and from Julian and Unix-style dates automatically
compute the correct time zone and daylight saving adjustment to use,
provided they can figure out this information.  This section describes
Calc's time zone adjustment algorithm in detail, in case you want to do
conversions in different time zones or in case Calc's algorithms can't
determine the right correction to use.

   Adjustments for time zones and daylight saving time are done by `t
U', `t J', `t N', and `t C', but not by any other commands.  In
particular, `<may 1 1991> - <apr 1 1991>' evaluates to exactly 30 days
even though there is a daylight-saving transition in between.  This is
also true for Julian pure dates: `julian(<may 1 1991>) - julian(<apr 1
1991>)'.  But Julian and Unix date/times will adjust for daylight
saving time:  using Calc's default daylight saving time rule (see the
explanation below), `julian(<12am may 1 1991>) - julian(<12am apr 1
1991>)' evaluates to `29.95833' (that's 29 days and 23 hours) because
one hour was lost when daylight saving commenced on April 7, 1991.

   In brief, the idiom `julian(DATE1) - julian(DATE2)' computes the
actual number of 24-hour periods between two dates, whereas `DATE1 -
DATE2' computes the number of calendar days between two dates without
taking daylight saving into account.

   The `calc-time-zone' [`tzone'] command converts the time zone
specified by its numeric prefix argument into a number of seconds
difference from Greenwich mean time (GMT).  If the argument is a
number, the result is simply that value multiplied by 3600.  Typical
arguments for North America are 5 (Eastern) or 8 (Pacific).  If
Daylight Saving time is in effect, one hour should be subtracted from
the normal difference.

   If you give a prefix of plain `C-u', `calc-time-zone' (like other
date arithmetic commands that include a time zone argument) takes the
zone argument from the top of the stack.  (In the case of `t J' and `t
U', the normal argument is then taken from the second-to-top stack
position.)  This allows you to give a non-integer time zone adjustment.
The time-zone argument can also be an HMS form, or it can be a variable
which is a time zone name in upper- or lower-case.  For example
`tzone(PST) = tzone(8)' and `tzone(pdt) = tzone(7)' (for Pacific
standard and daylight saving times, respectively).

   North American and European time zone names are defined as follows;
note that for each time zone there is one name for standard time,
another for daylight saving time, and a third for "generalized" time in
which the daylight saving adjustment is computed from context.

     YST  PST  MST  CST  EST  AST    NST    GMT   WET     MET    MEZ
      9    8    7    6    5    4     3.5     0     -1      -2     -2

     YDT  PDT  MDT  CDT  EDT  ADT    NDT    BST  WETDST  METDST  MESZ
      8    7    6    5    4    3     2.5     -1    -2      -3     -3

     YGT  PGT  MGT  CGT  EGT  AGT    NGT    BGT   WEGT    MEGT   MEGZ
     9/8  8/7  7/6  6/5  5/4  4/3  3.5/2.5  0/-1 -1/-2   -2/-3  -2/-3

   To define time zone names that do not appear in the above table, you
must modify the Lisp variable `math-tzone-names'.  This is a list of
lists describing the different time zone names; its structure is best
explained by an example.  The three entries for Pacific Time look like
this:

     ( ( "PST" 8 0 )    ; Name as an upper-case string, then standard
       ( "PDT" 8 -1 )   ; adjustment, then daylight saving adjustment.
       ( "PGT" 8 "PST" "PDT" ) )   ; Generalized time zone.

   With no arguments, `calc-time-zone' or `tzone()' will by default get
the time zone and daylight saving information from the calendar (*note
Calendar/Diary: (emacs)Daylight Saving.).  To use a different time
zone, or if the calendar does not give the desired result, you can set
the Calc variable `TimeZone' (which is by default `nil') to an
appropriate time zone name.  (The easiest way to do this is to edit the
`TimeZone' variable using Calc's `s T' command, then use the `s p'
(`calc-permanent-variable') command to save the value of `TimeZone'
permanently.)  If the time zone given by `TimeZone' is a generalized
time zone, e.g., `EGT', Calc examines the date being converted to tell
whether to use standard or daylight saving time.  But if the current
time zone is explicit, e.g., `EST' or `EDT', then that adjustment is
used exactly and Calc's daylight saving algorithm is not consulted.
The special time zone name `local' is equivalent to no argument; i.e.,
it uses the information obtained from the calendar.

   The `t J' and `t U' commands with no numeric prefix arguments do the
same thing as `tzone()'; namely, use the information from the calendar
if `TimeZone' is `nil', otherwise use the time zone given by `TimeZone'.

   When Calc computes the daylight saving information itself (i.e., when
the `TimeZone' variable is set), it will by default consider daylight
saving time to begin at 2 a.m. on the second Sunday of March (for years
from 2007 on) or on the last Sunday in April (for years before 2007),
and to end at 2 a.m. on the first Sunday of November. (for years from
2007 on) or the last Sunday in October (for years before 2007).  These
are the rules that have been in effect in much of North America since
1966 and take into account the rule change that began in 2007.  If you
are in a country that uses different rules for computing daylight
saving time, you have two choices: Write your own daylight saving hook,
or control time zones explicitly by setting the `TimeZone' variable
and/or always giving a time-zone argument for the conversion functions.

   The Lisp variable `math-daylight-savings-hook' holds the name of a
function that is used to compute the daylight saving adjustment for a
given date.  The default is `math-std-daylight-savings', which computes
an adjustment (either 0 or -1) using the North American rules given
above.

   The daylight saving hook function is called with four arguments: The
date, as a floating-point number in standard Calc format; a six-element
list of the date decomposed into year, month, day, hour, minute, and
second, respectively; a string which contains the generalized time zone
name in upper-case, e.g., `"WEGT"'; and a special adjustment to be
applied to the hour value when converting into a generalized time zone
(see below).

   The Lisp function `math-prev-weekday-in-month' is useful for
daylight saving computations.  This is an internal version of the
user-level `pwday' function described in the previous section. It takes
four arguments:  The floating-point date value, the corresponding
six-element date list, the day-of-month number, and the weekday number
(0-6).

   The default daylight saving hook ignores the time zone name, but a
more sophisticated hook could use different algorithms for different
time zones.  It would also be possible to use different algorithms
depending on the year number, but the default hook always uses the
algorithm for 1987 and later.  Here is a listing of the default
daylight saving hook:

     (defun math-std-daylight-savings (date dt zone bump)
       (cond ((< (nth 1 dt) 4) 0)
             ((= (nth 1 dt) 4)
              (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
                (cond ((< (nth 2 dt) sunday) 0)
                      ((= (nth 2 dt) sunday)
                       (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
                      (t -1))))
             ((< (nth 1 dt) 10) -1)
             ((= (nth 1 dt) 10)
              (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
                (cond ((< (nth 2 dt) sunday) -1)
                      ((= (nth 2 dt) sunday)
                       (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
                      (t 0))))
             (t 0))
     )

The `bump' parameter is equal to zero when Calc is converting from a
date form in a generalized time zone into a GMT date value.  It is -1
when Calc is converting in the other direction.  The adjustments shown
above ensure that the conversion behaves correctly and reasonably
around the 2 a.m. transition in each direction.

   There is a "missing" hour between 2 a.m. and 3 a.m. at the beginning
of daylight saving time; converting a date/time form that falls in this
hour results in a time value for the following hour, from 3 a.m. to 4
a.m.  At the end of daylight saving time, the hour from 1 a.m. to 2
a.m. repeats itself; converting a date/time form that falls in this
hour results in a time value for the first manifestation of that time
(_not_ the one that occurs one hour later).

   If `math-daylight-savings-hook' is `nil', then the daylight saving
adjustment is always taken to be zero.

   In algebraic formulas, `tzone(ZONE, DATE)' computes the time zone
adjustment for a given zone name at a given date.  The DATE is ignored
unless ZONE is a generalized time zone.  If DATE is a date form, the
daylight saving computation is applied to it as it appears.  If DATE is
a numeric date value, it is adjusted for the daylight-saving version of
ZONE before being given to the daylight saving hook.  This odd-sounding
rule ensures that the daylight-saving computation is always done in
local time, not in the GMT time that a numeric DATE is typically
represented in.

   The `dsadj(DATE, ZONE)' function computes the daylight saving
adjustment that is appropriate for DATE in time zone ZONE.  If ZONE is
explicitly in or not in daylight saving time (e.g., `PDT' or `PST') the
DATE is ignored.  If ZONE is a generalized time zone, the algorithms
described above are used.  If ZONE is omitted, the computation is done
for the current time zone.

File: calc,  Node: Business Days,  Next: Time Zones,  Prev: Date Functions,  Up: Date Arithmetic

9.5.3 Business Days
-------------------

Often time is measured in "business days" or "working days," where
weekends and holidays are skipped.  Calc's normal date arithmetic
functions use calendar days, so that subtracting two consecutive
Mondays will yield a difference of 7 days.  By contrast, subtracting
two consecutive Mondays would yield 5 business days (assuming two-day
weekends and the absence of holidays).

   The `t +' (`calc-business-days-plus') [`badd'] and `t -'
(`calc-business-days-minus') [`bsub'] commands perform arithmetic using
business days.  For `t +', one argument must be a date form and the
other must be a real number (positive or negative).  If the number is
not an integer, then a certain amount of time is added as well as a
number of days; for example, adding 0.5 business days to a time in
Friday evening will produce a time in Monday morning.  It is also
possible to add an HMS form; adding `12@ 0' 0"' also adds half a
business day.  For `t -', the arguments are either a date form and a
number or HMS form, or two date forms, in which case the result is the
number of business days between the two dates.

   By default, Calc considers any day that is not a Saturday or Sunday
to be a business day.  You can define any number of additional holidays
by editing the variable `Holidays'.  (There is an `s H' convenience
command for editing this variable.)  Initially, `Holidays' contains the
vector `[sat, sun]'.  Entries in the `Holidays' vector may be any of
the following kinds of objects:

   * Date forms (pure dates, not date/time forms).  These specify
     particular days which are to be treated as holidays.

   * Intervals of date forms.  These specify a range of days, all of
     which are holidays (e.g., Christmas week).  *Note Interval Forms::.

   * Nested vectors of date forms.  Each date form in the vector is
     considered to be a holiday.

   * Any Calc formula which evaluates to one of the above three things.
     If the formula involves the variable `y', it stands for a yearly
     repeating holiday; `y' will take on various year numbers like
     1992.  For example, `date(y, 12, 25)' specifies Christmas day, and
     `newweek(date(y, 11, 7), 4) + 21' specifies Thanksgiving (which is
     held on the fourth Thursday of November).  If the formula involves
     the variable `m', that variable takes on month numbers from 1 to
     12:  `date(y, m, 15)' is a holiday that takes place on the 15th of
     every month.

   * A weekday name, such as `sat' or `sun'.  This is really a variable
     whose name is a three-letter, lower-case day name.

   * An interval of year numbers (integers).  This specifies the span of
     years over which this holiday list is to be considered valid.  Any
     business-day arithmetic that goes outside this range will result
     in an error message.  Use this if you are including an explicit
     list of holidays, rather than a formula to generate them, and you
     want to make sure you don't accidentally go beyond the last point
     where the holidays you entered are complete.  If there is no
     limiting interval in the `Holidays' vector, the default `[1 ..
     2737]' is used.  (This is the absolute range of years for which
     Calc's business-day algorithms will operate.)

   * An interval of HMS forms.  This specifies the span of hours that
     are to be considered one business day.  For example, if this range
     is `[9@ 0' 0" .. 17@ 0' 0"]' (i.e., 9am to 5pm), then the business
     day is only eight hours long, so that `1.5 t +' on `<4:00pm Fri
     Dec 13, 1991>' will add one business day and four business hours
     to produce `<12:00pm Tue Dec 17, 1991>'.  Likewise, `t -' will now
     express differences in time as fractions of an eight-hour day.
     Times before 9am will be treated as 9am by business date
     arithmetic, and times at or after 5pm will be treated as
     4:59:59pm.  If there is no HMS interval in `Holidays', the full
     24-hour day `[0 0' 0" .. 24 0' 0"]' is assumed.  (Regardless of
     the type of bounds you specify, the interval is treated as
     inclusive on the low end and exclusive on the high end, so that
     the work day goes from 9am up to, but not including, 5pm.)

   If the `Holidays' vector is empty, then `t +' and `t -' will act
just like `+' and `-' because there will then be no difference between
business days and calendar days.

   Calc expands the intervals and formulas you give into a complete
list of holidays for internal use.  This is done mainly to make sure it
can detect multiple holidays.  (For example, `<Jan 1, 1989>' is both
New Year's Day and a Sunday, but Calc's algorithms take care to count
it only once when figuring the number of holidays between two dates.)

   Since the complete list of holidays for all the years from 1 to 2737
would be huge, Calc actually computes only the part of the list between
the smallest and largest years that have been involved in business-day
calculations so far.  Normally, you won't have to worry about this.
Keep in mind, however, that if you do one calculation for 1992, and
another for 1792, even if both involve only a small range of years,
Calc will still work out all the holidays that fall in that 200-year
span.

   If you add a (positive) number of days to a date form that falls on a
weekend or holiday, the date form is treated as if it were the most
recent business day.  (Thus adding one business day to a Friday,
Saturday, or Sunday will all yield the following Monday.)  If you
subtract a number of days from a weekend or holiday, the date is
effectively on the following business day.  (So subtracting one business
day from Saturday, Sunday, or Monday yields the preceding Friday.)  The
difference between two dates one or both of which fall on holidays
equals the number of actual business days between them.  These
conventions are consistent in the sense that, if you add N business
days to any date, the difference between the result and the original
date will come out to N business days.  (It can't be completely
consistent though; a subtraction followed by an addition might come out
a bit differently, since `t +' is incapable of producing a date that
falls on a weekend or holiday.)

   There is a `holiday' function, not on any keys, that takes any date
form and returns 1 if that date falls on a weekend or holiday, as
defined in `Holidays', or 0 if the date is a business day.

File: calc,  Node: Financial Functions,  Next: Binary Functions,  Prev: Date Arithmetic,  Up: Arithmetic

9.6 Financial Functions
=======================

Calc's financial or business functions use the `b' prefix key followed
by a shifted letter.  (The `b' prefix followed by a lower-case letter
is used for operations on binary numbers.)

   Note that the rate and the number of intervals given to these
functions must be on the same time scale, e.g., both months or both
years.  Mixing an annual interest rate with a time expressed in months
will give you very wrong answers!

   It is wise to compute these functions to a higher precision than you
really need, just to make sure your answer is correct to the last
penny; also, you may wish to check the definitions at the end of this
section to make sure the functions have the meaning you expect.

* Menu:

* Percentages::
* Future Value::
* Present Value::
* Related Financial Functions::
* Depreciation Functions::
* Definitions of Financial Functions::

File: calc,  Node: Percentages,  Next: Future Value,  Prev: Financial Functions,  Up: Financial Functions

9.6.1 Percentages
-----------------

The `M-%' (`calc-percent') command takes a percentage value, say 5.4,
and converts it to an equivalent actual number.  For example, `5.4 M-%'
enters 0.054 on the stack.  (That's the <META> or <ESC> key combined
with `%'.)

   Actually, `M-%' creates a formula of the form `5.4%'.  You can enter
`5.4%' yourself during algebraic entry.  The `%' operator simply means,
"the preceding value divided by 100."  The `%' operator has very high
precedence, so that `1+8%' is interpreted as `1+(8%)', not as `(1+8)%'.
(The `%' operator is just a postfix notation for the `percent'
function, just like `20!' is the notation for `fact(20)', or
twenty-factorial.)

   The formula `5.4%' would normally evaluate immediately to 0.054, but
the `M-%' command suppresses evaluation as it puts the formula onto the
stack.  However, the next Calc command that uses the formula `5.4%'
will evaluate it as its first step.  The net effect is that you get to
look at `5.4%' on the stack, but Calc commands see it as `0.054', which
is what they expect.

   In particular, `5.4%' and `0.054' are suitable values for the RATE
arguments of the various financial functions, but the number `5.4' is
probably _not_ suitable--it represents a rate of 540 percent!

   The key sequence `M-% *' effectively means "percent-of."  For
example, `68 <RET> 25 M-% *' computes 17, which is 25% of 68 (and also
68% of 25, which comes out to the same thing).

   The `c %' (`calc-convert-percent') command converts the value on the
top of the stack from numeric to percentage form.  For example, if 0.08
is on the stack, `c %' converts it to `8%'.  The quantity is the same,
it's just represented differently.  (Contrast this with `M-%', which
would convert this number to `0.08%'.)  The `=' key is a convenient way
to convert a formula like `8%' back to numeric form, 0.08.

   To compute what percentage one quantity is of another quantity, use
`/ c %'.  For example, `17 <RET> 68 / c %' displays `25%'.

   The `b %' (`calc-percent-change') [`relch'] command calculates the
percentage change from one number to another.  For example, `40 <RET>
50 b %' produces the answer `25%', since 50 is 25% larger than 40.  A
negative result represents a decrease:  `50 <RET> 40 b %' produces
`-20%', since 40 is 20% smaller than 50.  (The answers are different in
magnitude because, in the first case, we're increasing by 25% of 40, but
in the second case, we're decreasing by 20% of 50.)  The effect of `40
<RET> 50 b %' is to compute `(50-40)/40', converting the answer to
percentage form as if by `c %'.

File: calc,  Node: Future Value,  Next: Present Value,  Prev: Percentages,  Up: Financial Functions

9.6.2 Future Value
------------------

The `b F' (`calc-fin-fv') [`fv'] command computes the future value of
an investment.  It takes three arguments from the stack:  `fv(RATE, N,
PAYMENT)'.  If you give payments of PAYMENT every year for N years, and
the money you have paid earns interest at RATE per year, then this
function tells you what your investment would be worth at the end of
the period.  (The actual interval doesn't have to be years, as long as
N and RATE are expressed in terms of the same intervals.)  This
function assumes payments occur at the _end_ of each interval.

   The `I b F' [`fvb'] command does the same computation, but assuming
your payments are at the beginning of each interval.  Suppose you plan
to deposit $1000 per year in a savings account earning 5.4% interest,
starting right now.  How much will be in the account after five years?
`fvb(5.4%, 5, 1000) = 5870.73'.  Thus you will have earned $870 worth
of interest over the years.  Using the stack, this calculation would
have been `5.4 M-% 5 <RET> 1000 I b F'.  Note that the rate is expressed
as a number between 0 and 1, _not_ as a percentage.

   The `H b F' [`fvl'] command computes the future value of an initial
lump sum investment.  Suppose you could deposit those five thousand
dollars in the bank right now; how much would they be worth in five
years?  `fvl(5.4%, 5, 5000) = 6503.89'.

   The algebraic functions `fv' and `fvb' accept an optional fourth
argument, which is used as an initial lump sum in the sense of `fvl'.
In other words, `fv(RATE, N, PAYMENT, INITIAL) = fv(RATE, N, PAYMENT) +
fvl(RATE, N, INITIAL)'.

   To illustrate the relationships between these functions, we could do
the `fvb' calculation "by hand" using `fvl'.  The final balance will be
the sum of the contributions of our five deposits at various times.
The first deposit earns interest for five years:  `fvl(5.4%, 5, 1000) =
1300.78'.  The second deposit only earns interest for four years:
`fvl(5.4%, 4, 1000) = 1234.13'.  And so on down to the last deposit,
which earns one year's interest:  `fvl(5.4%, 1, 1000) = 1054.00'.  The
sum of these five values is, sure enough, $5870.73, just as was computed
by `fvb' directly.

   What does `fv(5.4%, 5, 1000) = 5569.96' mean?  The payments are now
at the ends of the periods.  The end of one year is the same as the
beginning of the next, so what this really means is that we've lost the
payment at year zero (which contributed $1300.78), but we're now
counting the payment at year five (which, since it didn't have a chance
to earn interest, counts as $1000).  Indeed, `5569.96 = 5870.73 -
1300.78 + 1000' (give or take a bit of roundoff error).

File: calc,  Node: Present Value,  Next: Related Financial Functions,  Prev: Future Value,  Up: Financial Functions

9.6.3 Present Value
-------------------

The `b P' (`calc-fin-pv') [`pv'] command computes the present value of
an investment.  Like `fv', it takes three arguments:  `pv(RATE, N,
PAYMENT)'.  It computes the present value of a series of regular
payments.  Suppose you have the chance to make an investment that will
pay $2000 per year over the next four years; as you receive these
payments you can put them in the bank at 9% interest.  You want to know
whether it is better to make the investment, or to keep the money in
the bank where it earns 9% interest right from the start.  The
calculation `pv(9%, 4, 2000)' gives the result 6479.44.  If your
initial investment must be less than this, say, $6000, then the
investment is worthwhile.  But if you had to put up $7000, then it
would be better just to leave it in the bank.

   Here is the interpretation of the result of `pv':  You are trying to
compare the return from the investment you are considering, which is
`fv(9%, 4, 2000) = 9146.26', with the return from leaving the money in
the bank, which is `fvl(9%, 4, X)' where X is the amount of money you
would have to put up in advance.  The `pv' function finds the
break-even point, `x = 6479.44', at which `fvl(9%, 4, 6479.44)' is also
equal to 9146.26.  This is the largest amount you should be willing to
invest.

   The `I b P' [`pvb'] command solves the same problem, but with
payments occurring at the beginning of each interval.  It has the same
relationship to `fvb' as `pv' has to `fv'.  For example `pvb(9%, 4,
2000) = 7062.59', a larger number than `pv' produced because we get to
start earning interest on the return from our investment sooner.

   The `H b P' [`pvl'] command computes the present value of an
investment that will pay off in one lump sum at the end of the period.
For example, if we get our $8000 all at the end of the four years,
`pvl(9%, 4, 8000) = 5667.40'.  This is much less than `pv' reported,
because we don't earn any interest on the return from this investment.
Note that `pvl' and `fvl' are simple inverses:  `fvl(9%, 4, 5667.40) =
8000'.

   You can give an optional fourth lump-sum argument to `pv' and `pvb';
this is handled in exactly the same way as the fourth argument for `fv'
and `fvb'.

   The `b N' (`calc-fin-npv') [`npv'] command computes the net present
value of a series of irregular investments.  The first argument is the
interest rate.  The second argument is a vector which represents the
expected return from the investment at the end of each interval.  For
example, if the rate represents a yearly interest rate, then the vector
elements are the return from the first year, second year, and so on.

   Thus, `npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44'.
Obviously this function is more interesting when the payments are not
all the same!

   The `npv' function can actually have two or more arguments.
Multiple arguments are interpreted in the same way as for the vector
statistical functions like `vsum'.  *Note Single-Variable Statistics::.
Basically, if there are several payment arguments, each either a vector
or a plain number, all these values are collected left-to-right into
the complete list of payments.  A numeric prefix argument on the `b N'
command says how many payment values or vectors to take from the stack.

   The `I b N' [`npvb'] command computes the net present value where
payments occur at the beginning of each interval rather than at the end.

File: calc,  Node: Related Financial Functions,  Next: Depreciation Functions,  Prev: Present Value,  Up: Financial Functions

9.6.4 Related Financial Functions
---------------------------------

The functions in this section are basically inverses of the present
value functions with respect to the various arguments.

   The `b M' (`calc-fin-pmt') [`pmt'] command computes the amount of
periodic payment necessary to amortize a loan.  Thus `pmt(RATE, N,
AMOUNT)' equals the value of PAYMENT such that `pv(RATE, N, PAYMENT) =
AMOUNT'.

   The `I b M' [`pmtb'] command does the same computation but using
`pvb' instead of `pv'.  Like `pv' and `pvb', these functions can also
take a fourth argument which represents an initial lump-sum investment.

   The `H b M' key just invokes the `fvl' function, which is the
inverse of `pvl'.  There is no explicit `pmtl' function.

   The `b #' (`calc-fin-nper') [`nper'] command computes the number of
regular payments necessary to amortize a loan.  Thus `nper(RATE,
PAYMENT, AMOUNT)' equals the value of N such that `pv(RATE, N, PAYMENT)
= AMOUNT'.  If PAYMENT is too small ever to amortize a loan for AMOUNT
at interest rate RATE, the `nper' function is left in symbolic form.

   The `I b #' [`nperb'] command does the same computation but using
`pvb' instead of `pv'.  You can give a fourth lump-sum argument to
these functions, but the computation will be rather slow in the
four-argument case.

   The `H b #' [`nperl'] command does the same computation using `pvl'.
By exchanging PAYMENT and AMOUNT you can also get the solution for
`fvl'.  For example, `nperl(8%, 2000, 1000) = 9.006', so if you place
$1000 in a bank account earning 8%, it will take nine years to grow to
$2000.

   The `b T' (`calc-fin-rate') [`rate'] command computes the rate of
return on an investment.  This is also an inverse of `pv': `rate(N,
PAYMENT, AMOUNT)' computes the value of RATE such that `pv(RATE, N,
PAYMENT) = AMOUNT'.  The result is expressed as a formula like `6.3%'.

   The `I b T' [`rateb'] and `H b T' [`ratel'] commands solve the
analogous equations with `pvb' or `pvl' in place of `pv'.  Also, `rate'
and `rateb' can accept an optional fourth argument just like `pv' and
`pvb'.  To redo the above example from a different perspective,
`ratel(9, 2000, 1000) = 8.00597%', which says you will need an interest
rate of 8% in order to double your account in nine years.

   The `b I' (`calc-fin-irr') [`irr'] command is the analogous function
to `rate' but for net present value.  Its argument is a vector of
payments.  Thus `irr(PAYMENTS)' computes the RATE such that `npv(RATE,
PAYMENTS) = 0'; this rate is known as the "internal rate of return".

   The `I b I' [`irrb'] command computes the internal rate of return
assuming payments occur at the beginning of each period.

File: calc,  Node: Depreciation Functions,  Next: Definitions of Financial Functions,  Prev: Related Financial Functions,  Up: Financial Functions

9.6.5 Depreciation Functions
----------------------------

The functions in this section calculate "depreciation", which is the
amount of value that a possession loses over time.  These functions are
characterized by three parameters:  COST, the original cost of the
asset; SALVAGE, the value the asset will have at the end of its
expected "useful life"; and LIFE, the number of years (or other
periods) of the expected useful life.

   There are several methods for calculating depreciation that differ in
the way they spread the depreciation over the lifetime of the asset.

   The `b S' (`calc-fin-sln') [`sln'] command computes the
"straight-line" depreciation.  In this method, the asset depreciates by
the same amount every year (or period).  For example, `sln(12000, 2000,
5)' returns 2000.  The asset costs $12000 initially and will be worth
$2000 after five years; it loses $2000 per year.

   The `b Y' (`calc-fin-syd') [`syd'] command computes the accelerated
"sum-of-years'-digits" depreciation.  Here the depreciation is higher
during the early years of the asset's life.  Since the depreciation is
different each year, `b Y' takes a fourth PERIOD parameter which
specifies which year is requested, from 1 to LIFE.  If PERIOD is
outside this range, the `syd' function will return zero.

   The `b D' (`calc-fin-ddb') [`ddb'] command computes an accelerated
depreciation using the double-declining balance method.  It also takes
a fourth PERIOD parameter.

   For symmetry, the `sln' function will accept a PERIOD parameter as
well, although it will ignore its value except that the return value
will as usual be zero if PERIOD is out of range.

   For example, pushing the vector `[1,2,3,4,5]' (perhaps with `v x 5')
and then mapping `V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
ddb(12000,2000,5,$)] <RET>' produces a matrix that allows us to compare
the three depreciation methods:

     [ [ 2000, 3333, 4800 ]
       [ 2000, 2667, 2880 ]
       [ 2000, 2000, 1728 ]
       [ 2000, 1333,  592 ]
       [ 2000,  667,   0  ] ]

(Values have been rounded to nearest integers in this figure.)  We see
that `sln' depreciates by the same amount each year, `syd' depreciates
more at the beginning and less at the end, and `ddb' weights the
depreciation even more toward the beginning.

   Summing columns with `V R : +' yields `[10000, 10000, 10000]'; the
total depreciation in any method is (by definition) the difference
between the cost and the salvage value.

File: calc,  Node: Definitions of Financial Functions,  Prev: Depreciation Functions,  Up: Financial Functions

9.6.6 Definitions
-----------------

For your reference, here are the actual formulas used to compute Calc's
financial functions.

   Calc will not evaluate a financial function unless the RATE or N
argument is known.  However, PAYMENT or AMOUNT can be a variable.  Calc
expands these functions according to the formulas below for symbolic
arguments only when you use the `a "' (`calc-expand-formula') command,
or when taking derivatives or integrals or solving equations involving
the functions.

   These formulas are shown using the conventions of Big display mode
(`d B'); for example, the formula for `fv' written linearly is `pmt *
((1 + rate)^n) - 1) / rate'.

                                             n
                                   (1 + rate)  - 1
     fv(rate, n, pmt) =      pmt * ---------------
                                        rate

                                              n
                                   ((1 + rate)  - 1) (1 + rate)
     fvb(rate, n, pmt) =     pmt * ----------------------------
                                              rate

                                             n
     fvl(rate, n, pmt) =     pmt * (1 + rate)

                                                 -n
                                   1 - (1 + rate)
     pv(rate, n, pmt) =      pmt * ----------------
                                         rate

                                                  -n
                                   (1 - (1 + rate)  ) (1 + rate)
     pvb(rate, n, pmt) =     pmt * -----------------------------
                                              rate

                                             -n
     pvl(rate, n, pmt) =     pmt * (1 + rate)

                                         -1               -2               -3
     npv(rate, [a, b, c]) =  a*(1 + rate)   + b*(1 + rate)   + c*(1 + rate)

                                             -1               -2
     npvb(rate, [a, b, c]) = a + b*(1 + rate)   + c*(1 + rate)

                                                  -n
                             (amt - x * (1 + rate)  ) * rate
     pmt(rate, n, amt, x) =  -------------------------------
                                                  -n
                                    1 - (1 + rate)

                                                  -n
                             (amt - x * (1 + rate)  ) * rate
     pmtb(rate, n, amt, x) = -------------------------------
                                             -n
                              (1 - (1 + rate)  ) (1 + rate)

                                        amt * rate
     nper(rate, pmt, amt) =  - log(1 - ------------, 1 + rate)
                                           pmt

                                         amt * rate
     nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
                                       pmt * (1 + rate)

                                   amt
     nperl(rate, pmt, amt) = - log(---, 1 + rate)
                                   pmt

                                1/n
                             pmt
     ratel(n, pmt, amt) =    ------ - 1
                                1/n
                             amt

                             cost - salv
     sln(cost, salv, life) = -----------
                                life

                                  (cost - salv) * (life - per + 1)
     syd(cost, salv, life, per) = --------------------------------
                                       life * (life + 1) / 2

                                  book * 2
     ddb(cost, salv, life, per) = --------,  book = cost - depreciation so far
                                    life

In `pmt' and `pmtb', `x=0' if omitted.

   These functions accept any numeric objects, including error forms,
intervals, and even (though not very usefully) complex numbers.  The
above formulas specify exactly the behavior of these functions with all
sorts of inputs.

   Note that if the first argument to the `log' in `nper' is negative,
`nper' leaves itself in symbolic form rather than returning a
(financially meaningless) complex number.

   `rate(num, pmt, amt)' solves the equation `pv(rate, num, pmt) = amt'
for `rate' using `H a R' (`calc-find-root'), with the interval `[.01%
.. 100%]' for an initial guess.  The `rateb' function is the same except
that it uses `pvb'.  Note that `ratel' can be solved directly; its
formula is shown in the above list.

   Similarly, `irr(pmts)' solves the equation `npv(rate, pmts) = 0' for
`rate'.

   If you give a fourth argument to `nper' or `nperb', Calc will also
use `H a R' to solve the equation using an initial guess interval of
`[0 .. 100]'.

   A fourth argument to `fv' simply sums the two components calculated
from the above formulas for `fv' and `fvl'.  The same is true of `fvb',
`pv', and `pvb'.

   The `ddb' function is computed iteratively; the "book" value starts
out equal to COST, and decreases according to the above formula for the
specified number of periods.  If the book value would decrease below
SALVAGE, it only decreases to SALVAGE and the depreciation is zero for
all subsequent periods.  The `ddb' function returns the amount the book
value decreased in the specified period.

File: calc,  Node: Binary Functions,  Prev: Financial Functions,  Up: Arithmetic

9.7 Binary Number Functions
===========================

The commands in this chapter all use two-letter sequences beginning with
the `b' prefix.

   The "binary" operations actually work regardless of the currently
displayed radix, although their results make the most sense in a radix
like 2, 8, or 16 (as obtained by the `d 2', `d 8', or `d 6' commands,
respectively).  You may also wish to enable display of leading zeros
with `d z'.  *Note Radix Modes::.

   The Calculator maintains a current "word size" `w', an arbitrary
positive or negative integer.  For a positive word size, all of the
binary operations described here operate modulo `2^w'.  In particular,
negative arguments are converted to positive integers modulo `2^w' by
all binary functions.

   If the word size is negative, binary operations produce 2's
complement integers from `-(2^(-w-1))' to `2^(-w-1)-1' inclusive.
Either mode accepts inputs in any range; the sign of `w' affects only
the results produced.

   The `b c' (`calc-clip') [`clip'] command can be used to clip a
number by reducing it modulo `2^w'.  The commands described in this
chapter automatically clip their results to the current word size.
Note that other operations like addition do not use the current word
size, since integer addition generally is not "binary."  (However,
*note Simplification Modes::, `calc-bin-simplify-mode'.)  For example,
with a word size of 8 bits `b c' converts a number to the range 0 to
255; with a word size of -8 `b c' converts to the range -128 to 127.

   The default word size is 32 bits.  All operations except the shifts
and rotates allow you to specify a different word size for that one
operation by giving a numeric prefix argument:  `C-u 8 b c' clips the
top of stack to the range 0 to 255 regardless of the current word size.
To set the word size permanently, use `b w' (`calc-word-size').  This
command displays a prompt with the current word size; press <RET>
immediately to keep this word size, or type a new word size at the
prompt.

   When the binary operations are written in symbolic form, they take an
optional second (or third) word-size parameter.  When a formula like
`and(a,b)' is finally evaluated, the word size current at that time
will be used, but when `and(a,b,-8)' is evaluated, a word size of -8
will always be used.  A symbolic binary function will be left in
symbolic form unless the all of its argument(s) are integers or
integer-valued floats.

   If either or both arguments are modulo forms for which `M' is a
power of two, that power of two is taken as the word size unless a
numeric prefix argument overrides it.  The current word size is never
consulted when modulo-power-of-two forms are involved.

   The `b a' (`calc-and') [`and'] command computes the bitwise AND of
the two numbers on the top of the stack.  In other words, for each of
the `w' binary digits of the two numbers (pairwise), the corresponding
bit of the result is 1 if and only if both input bits are 1:
`and(2#1100, 2#1010) = 2#1000'.

   The `b o' (`calc-or') [`or'] command computes the bitwise inclusive
OR of two numbers.  A bit is 1 if either of the input bits, or both,
are 1:  `or(2#1100, 2#1010) = 2#1110'.

   The `b x' (`calc-xor') [`xor'] command computes the bitwise
exclusive OR of two numbers.  A bit is 1 if exactly one of the input
bits is 1:  `xor(2#1100, 2#1010) = 2#0110'.

   The `b d' (`calc-diff') [`diff'] command computes the bitwise
difference of two numbers; this is defined by `diff(a,b) =
and(a,not(b))', so that `diff(2#1100, 2#1010) = 2#0100'.

   The `b n' (`calc-not') [`not'] command computes the bitwise NOT of a
number.  A bit is 1 if the input bit is 0 and vice-versa.

   The `b l' (`calc-lshift-binary') [`lsh'] command shifts a number
left by one bit, or by the number of bits specified in the numeric
prefix argument.  A negative prefix argument performs a logical right
shift, in which zeros are shifted in on the left.  In symbolic form,
`lsh(a)' is short for `lsh(a,1)', which in turn is short for
`lsh(a,n,w)'.  Bits shifted "off the end," according to the current
word size, are lost.

   The `H b l' command also does a left shift, but it takes two
arguments from the stack (the value to shift, and, at top-of-stack, the
number of bits to shift).  This version interprets the prefix argument
just like the regular binary operations, i.e., as a word size.  The
Hyperbolic flag has a similar effect on the rest of the binary shift
and rotate commands.

   The `b r' (`calc-rshift-binary') [`rsh'] command shifts a number
right by one bit, or by the number of bits specified in the numeric
prefix argument:  `rsh(a,n) = lsh(a,-n)'.

   The `b L' (`calc-lshift-arith') [`ash'] command shifts a number
left.  It is analogous to `lsh', except that if the shift is rightward
(the prefix argument is negative), an arithmetic shift is performed as
described below.

   The `b R' (`calc-rshift-arith') [`rash'] command performs an
"arithmetic" shift to the right, in which the leftmost bit (according
to the current word size) is duplicated rather than shifting in zeros.
This corresponds to dividing by a power of two where the input is
interpreted as a signed, twos-complement number.  (The distinction
between the `rsh' and `rash' operations is totally independent from
whether the word size is positive or negative.)  With a negative prefix
argument, this performs a standard left shift.

   The `b t' (`calc-rotate-binary') [`rot'] command rotates a number
one bit to the left.  The leftmost bit (according to the current word
size) is dropped off the left and shifted in on the right.  With a
numeric prefix argument, the number is rotated that many bits to the
left or right.

   *Note Set Operations::, for the `b p' and `b u' commands that pack
and unpack binary integers into sets.  (For example, `b u' unpacks the
number `2#11001' to the set of bit-numbers `[0, 3, 4]'.)  Type `b u V
#' to count the number of "1" bits in a binary integer.

   Another interesting use of the set representation of binary integers
is to reverse the bits in, say, a 32-bit integer.  Type `b u' to
unpack; type `31 <TAB> -' to replace each bit-number in the set with 31
minus that bit-number; type `b p' to pack the set back into a binary
integer.

File: calc,  Node: Scientific Functions,  Next: Matrix Functions,  Prev: Arithmetic,  Up: Top

10 Scientific Functions
***********************

The functions described here perform trigonometric and other
transcendental calculations.  They generally produce floating-point
answers correct to the full current precision.  The `H' (Hyperbolic)
and `I' (Inverse) flag keys must be used to get some of these functions
from the keyboard.

   One miscellaneous command is shift-`P' (`calc-pi'), which pushes the
value of `pi' (at the current precision) onto the stack.  With the
Hyperbolic flag, it pushes the value `e', the base of natural
logarithms.  With the Inverse flag, it pushes Euler's constant `gamma'
(about 0.5772).  With both Inverse and Hyperbolic, it pushes the
"golden ratio" `phi' (about 1.618).  (At present, Euler's constant is
not available to unlimited precision; Calc knows only the first 100
digits.)  In Symbolic mode, these commands push the actual variables
`pi', `e', `gamma', and `phi', respectively, instead of their values;
*note Symbolic Mode::.

   The `Q' (`calc-sqrt') [`sqrt'] function is described elsewhere;
*note Basic Arithmetic::.  With the Inverse flag [`sqr'], this command
computes the square of the argument.

   *Note Prefix Arguments::, for a discussion of the effect of numeric
prefix arguments on commands in this chapter which do not otherwise
interpret a prefix argument.

* Menu:

* Logarithmic Functions::
* Trigonometric and Hyperbolic Functions::
* Advanced Math Functions::
* Branch Cuts::
* Random Numbers::
* Combinatorial Functions::
* Probability Distribution Functions::

File: calc,  Node: Logarithmic Functions,  Next: Trigonometric and Hyperbolic Functions,  Prev: Scientific Functions,  Up: Scientific Functions

10.1 Logarithmic Functions
==========================

The shift-`L' (`calc-ln') [`ln'] command computes the natural logarithm
of the real or complex number on the top of the stack.  With the
Inverse flag it computes the exponential function instead, although
this is redundant with the `E' command.

   The shift-`E' (`calc-exp') [`exp'] command computes the exponential,
i.e., `e' raised to the power of the number on the stack.  The meanings
of the Inverse and Hyperbolic flags follow from those for the `calc-ln'
command.

   The `H L' (`calc-log10') [`log10'] command computes the common
(base-10) logarithm of a number.  (With the Inverse flag [`exp10'], it
raises ten to a given power.)  Note that the common logarithm of a
complex number is computed by taking the natural logarithm and dividing
by `ln(10)'.

   The `B' (`calc-log') [`log'] command computes a logarithm to any
base.  For example, `1024 <RET> 2 B' produces 10, since `2^10 = 1024'.
In certain cases like `log(3,9)', the result will be either `1:2' or
`0.5' depending on the current Fraction mode setting.  With the Inverse
flag [`alog'], this command is similar to `^' except that the order of
the arguments is reversed.

   The `f I' (`calc-ilog') [`ilog'] command computes the integer
logarithm of a number to any base.  The number and the base must
themselves be positive integers.  This is the true logarithm, rounded
down to an integer.  Thus `ilog(x,10)' is 3 for all `x' in the range
from 1000 to 9999.  If both arguments are positive integers, exact
integer arithmetic is used; otherwise, this is equivalent to
`floor(log(x,b))'.

   The `f E' (`calc-expm1') [`expm1'] command computes `exp(x)-1', but
using an algorithm that produces a more accurate answer when the result
is close to zero, i.e., when `exp(x)' is close to one.

   The `f L' (`calc-lnp1') [`lnp1'] command computes `ln(x+1)',
producing a more accurate answer when `x' is close to zero.

File: calc,  Node: Trigonometric and Hyperbolic Functions,  Next: Advanced Math Functions,  Prev: Logarithmic Functions,  Up: Scientific Functions

10.2 Trigonometric/Hyperbolic Functions
=======================================

The shift-`S' (`calc-sin') [`sin'] command computes the sine of an
angle or complex number.  If the input is an HMS form, it is interpreted
as degrees-minutes-seconds; otherwise, the input is interpreted
according to the current angular mode.  It is best to use Radians mode
when operating on complex numbers.

   Calc's "units" mechanism includes angular units like `deg', `rad',
and `grad'.  While `sin(45 deg)' is not evaluated all the time, the `u
s' (`calc-simplify-units') command will simplify `sin(45 deg)' by
taking the sine of 45 degrees, regardless of the current angular mode.
*Note Basic Operations on Units::.

   Also, the symbolic variable `pi' is not ordinarily recognized in
arguments to trigonometric functions, as in `sin(3 pi / 4)', but the `a
s' (`calc-simplify') command recognizes many such formulas when the
current angular mode is Radians _and_ Symbolic mode is enabled; this
example would be replaced by `sqrt(2) / 2'.  *Note Symbolic Mode::.
Beware, this simplification occurs even if you have stored a different
value in the variable `pi'; this is one reason why changing built-in
variables is a bad idea.  Arguments of the form `x' plus a multiple of
`pi/2' are also simplified.  Calc includes similar formulas for `cos'
and `tan'.

   The `a s' command knows all angles which are integer multiples of
`pi/12', `pi/10', or `pi/8' radians.  In Degrees mode, analogous
simplifications occur for integer multiples of 15 or 18 degrees, and
for arguments plus multiples of 90 degrees.

   With the Inverse flag, `calc-sin' computes an arcsine.  This is also
available as the `calc-arcsin' command or `arcsin' algebraic function.
The returned argument is converted to degrees, radians, or HMS notation
depending on the current angular mode.

   With the Hyperbolic flag, `calc-sin' computes the hyperbolic sine,
also available as `calc-sinh' [`sinh'].  With the Hyperbolic and
Inverse flags, it computes the hyperbolic arcsine (`calc-arcsinh')
[`arcsinh'].

   The shift-`C' (`calc-cos') [`cos'] command computes the cosine of an
angle or complex number, and shift-`T' (`calc-tan') [`tan'] computes
the tangent, along with all the various inverse and hyperbolic variants
of these functions.

   The `f T' (`calc-arctan2') [`arctan2'] command takes two numbers
from the stack and computes the arc tangent of their ratio.  The result
is in the full range from -180 (exclusive) to +180 (inclusive) degrees,
or the analogous range in radians.  A similar result would be obtained
with `/' followed by `I T', but the value would only be in the range
from -90 to +90 degrees since the division loses information about the
signs of the two components, and an error might result from an explicit
division by zero which `arctan2' would avoid.  By (arbitrary)
definition, `arctan2(0,0)=0'.

   The `calc-sincos' [`sincos'] command computes the sine and cosine of
a number, returning them as a vector of the form `[COS, SIN]'.  With
the Inverse flag [`arcsincos'], this command takes a two-element vector
as an argument and computes `arctan2' of the elements.  (This command
does not accept the Hyperbolic flag.)

   The remaining trigonometric functions, `calc-sec' [`sec'],
`calc-csc' [`csc'] and `calc-cot' [`cot'], are also available.  With
the Hyperbolic flag, these compute their hyperbolic counterparts, which
are also available separately as `calc-sech' [`sech'], `calc-csch'
[`csch'] and `calc-coth' [`coth'].  (These commands do not accept the
Inverse flag.)

File: calc,  Node: Advanced Math Functions,  Next: Branch Cuts,  Prev: Trigonometric and Hyperbolic Functions,  Up: Scientific Functions

10.3 Advanced Mathematical Functions
====================================

Calc can compute a variety of less common functions that arise in
various branches of mathematics.  All of the functions described in
this section allow arbitrary complex arguments and, except as noted,
will work to arbitrarily large precisions.  They can not at present
handle error forms or intervals as arguments.

   NOTE:  These functions are still experimental.  In particular, their
accuracy is not guaranteed in all domains.  It is advisable to set the
current precision comfortably higher than you actually need when using
these functions.  Also, these functions may be impractically slow for
some values of the arguments.

   The `f g' (`calc-gamma') [`gamma'] command computes the Euler gamma
function.  For positive integer arguments, this is related to the
factorial function:  `gamma(n+1) = fact(n)'.  For general complex
arguments the gamma function can be defined by the following definite
integral: `gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)'.  (The actual
implementation uses far more efficient computational methods.)

   The `f G' (`calc-inc-gamma') [`gammaP'] command computes the
incomplete gamma function, denoted `P(a,x)'.  This is defined by the
integral, `gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)'.
This implies that `gammaP(a,inf) = 1' for any `a' (see the definition
of the normal gamma function).

   Several other varieties of incomplete gamma function are defined.
The complement of `P(a,x)', called `Q(a,x) = 1-P(a,x)' by some authors,
is computed by the `I f G' [`gammaQ'] command.  You can think of this
as taking the other half of the integral, from `x' to infinity.

   The functions corresponding to the integrals that define `P(a,x)'
and `Q(a,x)' but without the normalizing `1/gamma(a)' factor are called
`g(a,x)' and `G(a,x)', respectively (where `g' and `G' represent the
lower- and upper-case Greek letter gamma).  You can obtain these using
the `H f G' [`gammag'] and `H I f G' [`gammaG'] commands.

   The `f b' (`calc-beta') [`beta'] command computes the Euler beta
function, which is defined in terms of the gamma function as `beta(a,b)
= gamma(a) gamma(b) / gamma(a+b)', or by `beta(a,b) = integ(t^(a-1)
(1-t)^(b-1), t, 0, 1)'.

   The `f B' (`calc-inc-beta') [`betaI'] command computes the
incomplete beta function `I(x,a,b)'.  It is defined by `betaI(x,a,b) =
integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)'.  Once again, the `H'
(hyperbolic) prefix gives the corresponding un-normalized version
[`betaB'].

   The `f e' (`calc-erf') [`erf'] command computes the error function
`erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)'.  The complementary
error function `I f e' (`calc-erfc') [`erfc'] is the corresponding
integral from `x' to infinity; the sum `erf(x) + erfc(x) = 1'.

   The `f j' (`calc-bessel-J') [`besJ'] and `f y' (`calc-bessel-Y')
[`besY'] commands compute the Bessel functions of the first and second
kinds, respectively.  In `besJ(n,x)' and `besY(n,x)' the "order"
parameter `n' is often an integer, but is not required to be one.
Calc's implementation of the Bessel functions currently limits the
precision to 8 digits, and may not be exact even to that precision.
Use with care!

File: calc,  Node: Branch Cuts,  Next: Random Numbers,  Prev: Advanced Math Functions,  Up: Scientific Functions

10.4 Branch Cuts and Principal Values
=====================================

All of the logarithmic, trigonometric, and other scientific functions
are defined for complex numbers as well as for reals.  This section
describes the values returned in cases where the general result is a
family of possible values.  Calc follows section 12.5.3 of Steele's
"Common Lisp, the Language", second edition, in these matters.  This
section will describe each function briefly; for a more detailed
discussion (including some nifty diagrams), consult Steele's book.

   Note that the branch cuts for `arctan' and `arctanh' were changed
between the first and second editions of Steele.  Recent versions of
Calc follow the second edition.

   The new branch cuts exactly match those of the HP-28/48 calculators.
They also match those of Mathematica 1.2, except that Mathematica's
`arctan' cut is always in the right half of the complex plane, and its
`arctanh' cut is always in the top half of the plane.  Calc's cuts are
continuous with quadrants I and III for `arctan', or II and IV for
`arctanh'.

   Note:  The current implementations of these functions with complex
arguments are designed with proper behavior around the branch cuts in
mind, _not_ efficiency or accuracy.  You may need to increase the
floating precision and wait a while to get suitable answers from them.

   For `sqrt(a+bi)':  When `a<0' and `b' is small but positive or zero,
the result is close to the `+i' axis.  For `b' small and negative, the
result is close to the `-i' axis.  The result always lies in the right
half of the complex plane.

   For `ln(a+bi)':  The real part is defined as `ln(abs(a+bi))'.  The
imaginary part is defined as `arg(a+bi) = arctan2(b,a)'.  Thus the
branch cuts for `sqrt' and `ln' both lie on the negative real axis.

   The following table describes these branch cuts in another way.  If
the real and imaginary parts of `z' are as shown, then the real and
imaginary parts of `f(z)' will be as shown.  Here `eps' stands for a
small positive value; each occurrence of `eps' may stand for a
different small value.

          z           sqrt(z)       ln(z)
     ----------------------------------------
        +,   0         +,  0       any, 0
        -,   0         0,  +       any, pi
        -, +eps      +eps, +      +eps, +
        -, -eps      +eps, -      +eps, -

   For `z1^z2':  This is defined by `exp(ln(z1)*z2)'.  One interesting
consequence of this is that `(-8)^1:3' does not evaluate to -2 as you
might expect, but to the complex number `(1., 1.732)'.  Both of these
are valid cube roots of -8 (as is `(1., -1.732)'); Calc chooses a
perhaps less-obvious root for the sake of mathematical consistency.

   For `arcsin(z)':  This is defined by `-i*ln(i*z + sqrt(1-z^2))'.
The branch cuts are on the real axis, less than -1 and greater than 1.

   For `arccos(z)':  This is defined by `-i*ln(z + i*sqrt(1-z^2))', or
equivalently by `pi/2 - arcsin(z)'.  The branch cuts are on the real
axis, less than -1 and greater than 1.

   For `arctan(z)':  This is defined by `(ln(1+i*z) - ln(1-i*z)) /
(2*i)'.  The branch cuts are on the imaginary axis, below `-i' and
above `i'.

   For `arcsinh(z)':  This is defined by `ln(z + sqrt(1+z^2))'.  The
branch cuts are on the imaginary axis, below `-i' and above `i'.

   For `arccosh(z)':  This is defined by `ln(z +
(z+1)*sqrt((z-1)/(z+1)))'.  The branch cut is on the real axis less
than 1.

   For `arctanh(z)':  This is defined by `(ln(1+z) - ln(1-z)) / 2'.
The branch cuts are on the real axis, less than -1 and greater than 1.

   The following tables for `arcsin', `arccos', and `arctan' assume the
current angular mode is Radians.  The hyperbolic functions operate
independently of the angular mode.

            z             arcsin(z)            arccos(z)
     -------------------------------------------------------
      (-1..1),  0      (-pi/2..pi/2), 0       (0..pi), 0
      (-1..1), +eps    (-pi/2..pi/2), +eps    (0..pi), -eps
      (-1..1), -eps    (-pi/2..pi/2), -eps    (0..pi), +eps
        <-1,    0          -pi/2,     +         pi,    -
        <-1,  +eps      -pi/2 + eps,  +      pi - eps, -
        <-1,  -eps      -pi/2 + eps,  -      pi - eps, +
         >1,    0           pi/2,     -          0,    +
         >1,  +eps       pi/2 - eps,  +        +eps,   -
         >1,  -eps       pi/2 - eps,  -        +eps,   +

            z            arccosh(z)         arctanh(z)
     -----------------------------------------------------
      (-1..1),  0        0,  (0..pi)       any,     0
      (-1..1), +eps    +eps, (0..pi)       any,    +eps
      (-1..1), -eps    +eps, (-pi..0)      any,    -eps
        <-1,    0        +,    pi           -,     pi/2
        <-1,  +eps       +,  pi - eps       -,  pi/2 - eps
        <-1,  -eps       +, -pi + eps       -, -pi/2 + eps
         >1,    0        +,     0           +,    -pi/2
         >1,  +eps       +,   +eps          +,  pi/2 - eps
         >1,  -eps       +,   -eps          +, -pi/2 + eps

            z           arcsinh(z)           arctan(z)
     -----------------------------------------------------
        0, (-1..1)    0, (-pi/2..pi/2)         0,     any
        0,   <-1      -,    -pi/2            -pi/2,    -
      +eps,  <-1      +, -pi/2 + eps       pi/2 - eps, -
      -eps,  <-1      -, -pi/2 + eps      -pi/2 + eps, -
        0,    >1      +,     pi/2             pi/2,    +
      +eps,   >1      +,  pi/2 - eps       pi/2 - eps, +
      -eps,   >1      -,  pi/2 - eps      -pi/2 + eps, +

   Finally, the following identities help to illustrate the relationship
between the complex trigonometric and hyperbolic functions.  They are
valid everywhere, including on the branch cuts.

     sin(i*z)  = i*sinh(z)       arcsin(i*z)  = i*arcsinh(z)
     cos(i*z)  =   cosh(z)       arcsinh(i*z) = i*arcsin(z)
     tan(i*z)  = i*tanh(z)       arctan(i*z)  = i*arctanh(z)
     sinh(i*z) = i*sin(z)        cosh(i*z)    =   cos(z)

   The "advanced math" functions (gamma, Bessel, etc.) are also defined
for general complex arguments, but their branch cuts and principal
values are not rigorously specified at present.

File: calc,  Node: Random Numbers,  Next: Combinatorial Functions,  Prev: Branch Cuts,  Up: Scientific Functions

10.5 Random Numbers
===================

The `k r' (`calc-random') [`random'] command produces random numbers of
various sorts.

   Given a positive numeric prefix argument `M', it produces a random
integer `N' in the range `0 <= N < M'.  Each possible value `N' appears
with equal probability.

   With no numeric prefix argument, the `k r' command takes its argument
from the stack instead.  Once again, if this is a positive integer `M'
the result is a random integer less than `M'.  However, note that while
numeric prefix arguments are limited to six digits or so, an `M' taken
from the stack can be arbitrarily large.  If `M' is negative, the
result is a random integer in the range `M < N <= 0'.

   If the value on the stack is a floating-point number `M', the result
is a random floating-point number `N' in the range `0 <= N < M' or `M <
N <= 0', according to the sign of `M'.

   If `M' is zero, the result is a Gaussian-distributed random real
number; the distribution has a mean of zero and a standard deviation of
one.  The algorithm used generates random numbers in pairs; thus, every
other call to this function will be especially fast.

   If `M' is an error form `m +/- s' where M and S are both real
numbers, the result uses a Gaussian distribution with mean M and
standard deviation S.

   If `M' is an interval form, the lower and upper bounds specify the
acceptable limits of the random numbers.  If both bounds are integers,
the result is a random integer in the specified range.  If either bound
is floating-point, the result is a random real number in the specified
range.  If the interval is open at either end, the result will be sure
not to equal that end value.  (This makes a big difference for integer
intervals, but for floating-point intervals it's relatively minor: with
a precision of 6, `random([1.0..2.0))' will return any of one million
numbers from 1.00000 to 1.99999; `random([1.0..2.0])' may additionally
return 2.00000, but the probability of this happening is extremely
small.)

   If `M' is a vector, the result is one element taken at random from
the vector.  All elements of the vector are given equal probabilities.

   The sequence of numbers produced by `k r' is completely random by
default, i.e., the sequence is seeded each time you start Calc using
the current time and other information.  You can get a reproducible
sequence by storing a particular "seed value" in the Calc variable
`RandSeed'.  Any integer will do for a seed; integers of from 1 to 12
digits are good.  If you later store a different integer into
`RandSeed', Calc will switch to a different pseudo-random sequence.  If
you "unstore" `RandSeed', Calc will re-seed itself from the current
time.  If you store the same integer that you used before back into
`RandSeed', you will get the exact same sequence of random numbers as
before.

   The `calc-rrandom' command (not on any key) produces a random real
number between zero and one.  It is equivalent to `random(1.0)'.

   The `k a' (`calc-random-again') command produces another random
number, re-using the most recent value of `M'.  With a numeric prefix
argument N, it produces N more random numbers using that value of `M'.

   The `k h' (`calc-shuffle') command produces a vector of several
random values with no duplicates.  The value on the top of the stack
specifies the set from which the random values are drawn, and may be any
of the `M' formats described above.  The numeric prefix argument gives
the length of the desired list.  (If you do not provide a numeric
prefix argument, the length of the list is taken from the top of the
stack, and `M' from second-to-top.)

   If `M' is a floating-point number, zero, or an error form (so that
the random values are being drawn from the set of real numbers) there
is little practical difference between using `k h' and using `k r'
several times.  But if the set of possible values consists of just a
few integers, or the elements of a vector, then there is a very real
chance that multiple `k r''s will produce the same number more than
once.  The `k h' command produces a vector whose elements are always
distinct.  (Actually, there is a slight exception: If `M' is a vector,
no given vector element will be drawn more than once, but if several
elements of `M' are equal, they may each make it into the result
vector.)

   One use of `k h' is to rearrange a list at random.  This happens if
the prefix argument is equal to the number of values in the list: `[1,
1.5, 2, 2.5, 3] 5 k h' might produce the permuted list `[2.5, 1, 1.5,
3, 2]'.  As a convenient feature, if the argument N is negative it is
replaced by the size of the set represented by `M'.  Naturally, this is
allowed only when `M' specifies a small discrete set of possibilities.

   To do the equivalent of `k h' but with duplications allowed, given
`M' on the stack and with N just entered as a numeric prefix, use `v b'
to build a vector of copies of `M', then use `V M k r' to "map" the
normal `k r' function over the elements of this vector.  *Note Matrix
Functions::.

* Menu:

* Random Number Generator::     (Complete description of Calc's algorithm)

File: calc,  Node: Random Number Generator,  Prev: Random Numbers,  Up: Random Numbers

10.5.1 Random Number Generator
------------------------------

Calc's random number generator uses several methods to ensure that the
numbers it produces are highly random.  Knuth's _Art of Computer
Programming_, Volume II, contains a thorough description of the theory
of random number generators and their measurement and characterization.

   If `RandSeed' has no stored value, Calc calls Emacs' built-in
`random' function to get a stream of random numbers, which it then
treats in various ways to avoid problems inherent in the simple random
number generators that many systems use to implement `random'.

   When Calc's random number generator is first invoked, it "seeds" the
low-level random sequence using the time of day, so that the random
number sequence will be different every time you use Calc.

   Since Emacs Lisp doesn't specify the range of values that will be
returned by its `random' function, Calc exercises the function several
times to estimate the range.  When Calc subsequently uses the `random'
function, it takes only 10 bits of the result near the most-significant
end.  (It avoids at least the bottom four bits, preferably more, and
also tries to avoid the top two bits.)  This strategy works well with
the linear congruential generators that are typically used to implement
`random'.

   If `RandSeed' contains an integer, Calc uses this integer to seed an
"additive congruential" method (Knuth's algorithm 3.2.2A, computing
`X_n-55 - X_n-24').  This method expands the seed value into a large
table which is maintained internally; the variable `RandSeed' is
changed from, e.g., 42 to the vector `[42]' to indicate that the seed
has been absorbed into this table.  When `RandSeed' contains a vector,
`k r' and related commands continue to use the same internal table as
last time.  There is no way to extract the complete state of the random
number generator so that you can restart it from any point; you can
only restart it from the same initial seed value.  A simple way to
restart from the same seed is to type `s r RandSeed' to get the seed
vector, `v u' to unpack it back into a number, then `s t RandSeed' to
reseed the generator with that number.

   Calc uses a "shuffling" method as described in algorithm 3.2.2B of
Knuth.  It fills a table with 13 random 10-bit numbers.  Then, to
generate a new random number, it uses the previous number to index into
the table, picks the value it finds there as the new random number,
then replaces that table entry with a new value obtained from a call to
the base random number generator (either the additive congruential
generator or the `random' function supplied by the system).  If there
are any flaws in the base generator, shuffling will tend to even them
out.  But if the system provides an excellent `random' function,
shuffling will not damage its randomness.

   To create a random integer of a certain number of digits, Calc
builds the integer three decimal digits at a time.  For each group of
three digits, Calc calls its 10-bit shuffling random number generator
(which returns a value from 0 to 1023); if the random value is 1000 or
more, Calc throws it out and tries again until it gets a suitable value.

   To create a random floating-point number with precision P, Calc
simply creates a random P-digit integer and multiplies by `10^-p'.  The
resulting random numbers should be very clean, but note that relatively
small numbers will have few significant random digits.  In other words,
with a precision of 12, you will occasionally get numbers on the order
of `10^-9' or `10^-10', but those numbers will only have two or three
random digits since they correspond to small integers times `10^-12'.

   To create a random integer in the interval `[0 .. M)', Calc counts
the digits in M, creates a random integer with three additional digits,
then reduces modulo M.  Unless M is a power of ten the resulting values
will be very slightly biased toward the lower numbers, but this bias
will be less than 0.1%.  (For example, if M is 42, Calc will reduce a
random integer less than 100000 modulo 42 to get a result less than 42.
It is easy to show that the numbers 40 and 41 will be only 2380/2381 as
likely to result from this modulo operation as numbers 39 and below.)
If M is a power of ten, however, the numbers should be completely
unbiased.

   The Gaussian random numbers generated by `random(0.0)' use the
"polar" method described in Knuth section 3.4.1C.  This method
generates a pair of Gaussian random numbers at a time, so only every
other call to `random(0.0)' will require significant calculations.

File: calc,  Node: Combinatorial Functions,  Next: Probability Distribution Functions,  Prev: Random Numbers,  Up: Scientific Functions

10.6 Combinatorial Functions
============================

Commands relating to combinatorics and number theory begin with the `k'
key prefix.

   The `k g' (`calc-gcd') [`gcd'] command computes the Greatest Common
Divisor of two integers.  It also accepts fractions; the GCD of two
fractions is defined by taking the GCD of the numerators, and the LCM
of the denominators.  This definition is consistent with the idea that
`a / gcd(a,x)' should yield an integer for any `a' and `x'.  For other
types of arguments, the operation is left in symbolic form.

   The `k l' (`calc-lcm') [`lcm'] command computes the Least Common
Multiple of two integers or fractions.  The product of the LCM and GCD
of two numbers is equal to the product of the numbers.

   The `k E' (`calc-extended-gcd') [`egcd'] command computes the GCD of
two integers `x' and `y' and returns a vector `[g, a, b]' where `g =
gcd(x,y) = a x + b y'.

   The `!' (`calc-factorial') [`fact'] command computes the factorial
of the number at the top of the stack.  If the number is an integer,
the result is an exact integer.  If the number is an integer-valued
float, the result is a floating-point approximation.  If the number is
a non-integral real number, the generalized factorial is used, as
defined by the Euler Gamma function.  Please note that computation of
large factorials can be slow; using floating-point format will help
since fewer digits must be maintained.  The same is true of many of the
commands in this section.

   The `k d' (`calc-double-factorial') [`dfact'] command computes the
"double factorial" of an integer.  For an even integer, this is the
product of even integers from 2 to `N'.  For an odd integer, this is
the product of odd integers from 3 to `N'.  If the argument is an
integer-valued float, the result is a floating-point approximation.
This function is undefined for negative even integers.  The notation
`N!!' is also recognized for double factorials.

   The `k c' (`calc-choose') [`choose'] command computes the binomial
coefficient `N'-choose-`M', where `M' is the number on the top of the
stack and `N' is second-to-top.  If both arguments are integers, the
result is an exact integer.  Otherwise, the result is a floating-point
approximation.  The binomial coefficient is defined for all real
numbers by `N! / M! (N-M)!'.

   The `H k c' (`calc-perm') [`perm'] command computes the
number-of-permutations function `N! / (N-M)!'.

   The `k b' (`calc-bernoulli-number') [`bern'] command computes a
given Bernoulli number.  The value at the top of the stack is a
nonnegative integer `n' that specifies which Bernoulli number is
desired.  The `H k b' command computes a Bernoulli polynomial, taking
`n' from the second-to-top position and `x' from the top of the stack.
If `x' is a variable or formula the result is a polynomial in `x'; if
`x' is a number the result is a number.

   The `k e' (`calc-euler-number') [`euler'] command similarly computes
an Euler number, and `H k e' computes an Euler polynomial.  Bernoulli
and Euler numbers occur in the Taylor expansions of several functions.

   The `k s' (`calc-stirling-number') [`stir1'] command computes a
Stirling number of the first kind, given two integers `n' and `m' on
the stack.  The `H k s' [`stir2'] command computes a Stirling number of
the second kind.  These are the number of `m'-cycle permutations of `n'
objects, and the number of ways to partition `n' objects into `m'
non-empty sets, respectively.

   The `k p' (`calc-prime-test') command checks if the integer on the
top of the stack is prime.  For integers less than eight million, the
answer is always exact and reasonably fast.  For larger integers, a
probabilistic method is used (see Knuth vol. II, section 4.5.4,
algorithm P).  The number is first checked against small prime factors
(up to 13).  Then, any number of iterations of the algorithm are
performed.  Each step either discovers that the number is non-prime, or
substantially increases the certainty that the number is prime.  After
a few steps, the chance that a number was mistakenly described as prime
will be less than one percent.  (Indeed, this is a worst-case estimate
of the probability; in practice even a single iteration is quite
reliable.)  After the `k p' command, the number will be reported as
definitely prime or non-prime if possible, or otherwise "probably"
prime with a certain probability of error.

   The normal `k p' command performs one iteration of the primality
test.  Pressing `k p' repeatedly for the same integer will perform
additional iterations.  Also, `k p' with a numeric prefix performs the
specified number of iterations.  There is also an algebraic function
`prime(n)' or `prime(n,iters)' which returns 1 if `n' is (probably)
prime and 0 if not.

   The `k f' (`calc-prime-factors') [`prfac'] command attempts to
decompose an integer into its prime factors.  For numbers up to 25
million, the answer is exact although it may take some time.  The
result is a vector of the prime factors in increasing order.  For larger
inputs, prime factors above 5000 may not be found, in which case the
last number in the vector will be an unfactored integer greater than 25
million (with a warning message).  For negative integers, the first
element of the list will be -1.  For inputs -1, 0, and 1, the result is
a list of the same number.

   The `k n' (`calc-next-prime') [`nextprime'] command finds the next
prime above a given number.  Essentially, it searches by calling
`calc-prime-test' on successive integers until it finds one that passes
the test.  This is quite fast for integers less than eight million, but
once the probabilistic test comes into play the search may be rather
slow.  Ordinarily this command stops for any prime that passes one
iteration of the primality test.  With a numeric prefix argument, a
number must pass the specified number of iterations before the search
stops.  (This only matters when searching above eight million.)  You
can always use additional `k p' commands to increase your certainty
that the number is indeed prime.

   The `I k n' (`calc-prev-prime') [`prevprime'] command analogously
finds the next prime less than a given number.

   The `k t' (`calc-totient') [`totient'] command computes the Euler
"totient" function, the number of integers less than `n' which are
relatively prime to `n'.

   The `k m' (`calc-moebius') [`moebius'] command computes the Moebius
"mu" function.  If the input number is a product of `k' distinct
factors, this is `(-1)^k'.  If the input number has any duplicate
factors (i.e., can be divided by the same prime more than once), the
result is zero.

File: calc,  Node: Probability Distribution Functions,  Prev: Combinatorial Functions,  Up: Scientific Functions

10.7 Probability Distribution Functions
=======================================

The functions in this section compute various probability distributions.
For continuous distributions, this is the integral of the probability
density function from `x' to infinity.  (These are the "upper tail"
distribution functions; there are also corresponding "lower tail"
functions which integrate from minus infinity to `x'.)  For discrete
distributions, the upper tail function gives the sum from `x' to
infinity; the lower tail function gives the sum from minus infinity up
to, but not including, `x'.

   To integrate from `x' to `y', just use the distribution function
twice and subtract.  For example, the probability that a Gaussian
random variable with mean 2 and standard deviation 1 will lie in the
range from 2.5 to 2.8 is `utpn(2.5,2,1) - utpn(2.8,2,1)' ("the
probability that it is greater than 2.5, but not greater than 2.8"), or
equivalently `ltpn(2.8,2,1) - ltpn(2.5,2,1)'.

   The `k B' (`calc-utpb') [`utpb'] function uses the binomial
distribution.  Push the parameters N, P, and then X onto the stack; the
result (`utpb(x,n,p)') is the probability that an event will occur X or
more times out of N trials, if its probability of occurring in any given
trial is P.  The `I k B' [`ltpb'] function is the probability that the
event will occur fewer than X times.

   The other probability distribution functions similarly take the form
`k X' (`calc-utpX') [`utpX'] and `I k X' [`ltpX'], for various letters
X.  The arguments to the algebraic functions are the value of the
random variable first, then whatever other parameters define the
distribution.  Note these are among the few Calc functions where the
order of the arguments in algebraic form differs from the order of
arguments as found on the stack.  (The random variable comes last on
the stack, so that you can type, e.g., `2 <RET> 1 <RET> 2.5 k N M-<RET>
<DEL> 2.8 k N -', using `M-<RET> <DEL>' to recover the original
arguments but substitute a new value for `x'.)

   The `utpc(x,v)' function uses the chi-square distribution with `v'
degrees of freedom.  It is the probability that a model is correct if
its chi-square statistic is `x'.

   The `utpf(F,v1,v2)' function uses the F distribution, used in
various statistical tests.  The parameters `v1' and `v2' are the
degrees of freedom in the numerator and denominator, respectively, used
in computing the statistic `F'.

   The `utpn(x,m,s)' function uses a normal (Gaussian) distribution
with mean `m' and standard deviation `s'.  It is the probability that
such a normal-distributed random variable would exceed `x'.

   The `utpp(n,x)' function uses a Poisson distribution with mean `x'.
It is the probability that `n' or more such Poisson random events will
occur.

   The `utpt(t,v)' function uses the Student's "t" distribution with `v'
degrees of freedom.  It is the probability that a t-distributed random
variable will be greater than `t'.  (Note:  This computes the
distribution function `A(t|v)' where `A(0|v) = 1' and `A(inf|v) -> 0'.
The `UTPT' operation on the HP-48 uses a different definition which
returns half of Calc's value:  `UTPT(t,v) = .5*utpt(t,v)'.)

   While Calc does not provide inverses of the probability distribution
functions, the `a R' command can be used to solve for the inverse.
Since the distribution functions are monotonic, `a R' is guaranteed to
be able to find a solution given any initial guess.  *Note Numerical
Solutions::.

File: calc,  Node: Matrix Functions,  Next: Algebra,  Prev: Scientific Functions,  Up: Top

11 Vector/Matrix Functions
**************************

Many of the commands described here begin with the `v' prefix.  (For
convenience, the shift-`V' prefix is equivalent to `v'.)  The commands
usually apply to both plain vectors and matrices; some apply only to
matrices or only to square matrices.  If the argument has the wrong
dimensions the operation is left in symbolic form.

   Vectors are entered and displayed using `[a,b,c]' notation.
Matrices are vectors of which all elements are vectors of equal length.
(Though none of the standard Calc commands use this concept, a
three-dimensional matrix or rank-3 tensor could be defined as a vector
of matrices, and so on.)

* Menu:

* Packing and Unpacking::
* Building Vectors::
* Extracting Elements::
* Manipulating Vectors::
* Vector and Matrix Arithmetic::
* Set Operations::
* Statistical Operations::
* Reducing and Mapping::
* Vector and Matrix Formats::

File: calc,  Node: Packing and Unpacking,  Next: Building Vectors,  Prev: Matrix Functions,  Up: Matrix Functions

11.1 Packing and Unpacking
==========================

Calc's "pack" and "unpack" commands collect stack entries to build
composite objects such as vectors and complex numbers.  They are
described in this chapter because they are most often used to build
vectors.

   The `v p' (`calc-pack') [`pack'] command collects several elements
from the stack into a matrix, complex number, HMS form, error form,
etc.  It uses a numeric prefix argument to specify the kind of object
to be built; this argument is referred to as the "packing mode."  If
the packing mode is a nonnegative integer, a vector of that length is
created.  For example, `C-u 5 v p' will pop the top five stack elements
and push back a single vector of those five elements.  (`C-u 0 v p'
simply creates an empty vector.)

   The same effect can be had by pressing `[' to push an incomplete
vector on the stack, using <TAB> (`calc-roll-down') to sneak the
incomplete object up past a certain number of elements, and then
pressing `]' to complete the vector.

   Negative packing modes create other kinds of composite objects:

`-1'
     Two values are collected to build a complex number.  For example,
     `5 <RET> 7 C-u -1 v p' creates the complex number `(5, 7)'.  The
     result is always a rectangular complex number.  The two input
     values must both be real numbers, i.e., integers, fractions, or
     floats.  If they are not, Calc will instead build a formula like
     `a + (0, 1) b'.  (The other packing modes also create a symbolic
     answer if the components are not suitable.)

`-2'
     Two values are collected to build a polar complex number.  The
     first is the magnitude; the second is the phase expressed in
     either degrees or radians according to the current angular mode.

`-3'
     Three values are collected into an HMS form.  The first two values
     (hours and minutes) must be integers or integer-valued floats.
     The third value may be any real number.

`-4'
     Two values are collected into an error form.  The inputs may be
     real numbers or formulas.

`-5'
     Two values are collected into a modulo form.  The inputs must be
     real numbers.

`-6'
     Two values are collected into the interval `[a .. b]'.  The inputs
     may be real numbers, HMS or date forms, or formulas.

`-7'
     Two values are collected into the interval `[a .. b)'.

`-8'
     Two values are collected into the interval `(a .. b]'.

`-9'
     Two values are collected into the interval `(a .. b)'.

`-10'
     Two integer values are collected into a fraction.

`-11'
     Two values are collected into a floating-point number.  The first
     is the mantissa; the second, which must be an integer, is the
     exponent.  The result is the mantissa times ten to the power of
     the exponent.

`-12'
     This is treated the same as -11 by the `v p' command.  When
     unpacking, -12 specifies that a floating-point mantissa is desired.

`-13'
     A real number is converted into a date form.

`-14'
     Three numbers (year, month, day) are packed into a pure date form.

`-15'
     Six numbers are packed into a date/time form.

   With any of the two-input negative packing modes, either or both of
the inputs may be vectors.  If both are vectors of the same length, the
result is another vector made by packing corresponding elements of the
input vectors.  If one input is a vector and the other is a plain
number, the number is packed along with each vector element to produce
a new vector.  For example, `C-u -4 v p' could be used to convert a
vector of numbers and a vector of errors into a single vector of error
forms; `C-u -5 v p' could convert a vector of numbers and a single
number M into a vector of numbers modulo M.

   If you don't give a prefix argument to `v p', it takes the packing
mode from the top of the stack.  The elements to be packed then begin
at stack level 2.  Thus `1 <RET> 2 <RET> 4 n v p' is another way to
enter the error form `1 +/- 2'.

   If the packing mode taken from the stack is a vector, the result is a
matrix with the dimensions specified by the elements of the vector,
which must each be integers.  For example, if the packing mode is `[2,
3]', then six numbers will be taken from the stack and returned in the
form `[[a, b, c], [d, e, f]]'.

   If any elements of the vector are negative, other kinds of packing
are done at that level as described above.  For example, `[2, 3, -4]'
takes 12 objects and creates a 2x3 matrix of error forms: `[[a +/- b, c
+/- d ... ]]'.  Also, `[-4, -10]' will convert four integers into an
error form consisting of two fractions:  `a:b +/- c:d'.

   There is an equivalent algebraic function, `pack(MODE, ITEMS)' where
MODE is a packing mode (an integer or a vector of integers) and ITEMS
is a vector of objects to be packed (re-packed, really) according to
that mode.  For example, `pack([3, -4], [a,b,c,d,e,f])' yields `[a +/-
b, c +/- d, e +/- f]'.  The function is left in symbolic form if the
packing mode is invalid, or if the number of data items does not match
the number of items required by the mode.

   The `v u' (`calc-unpack') command takes the vector, complex number,
HMS form, or other composite object on the top of the stack and
"unpacks" it, pushing each of its elements onto the stack as separate
objects.  Thus, it is the "inverse" of `v p'.  If the value at the top
of the stack is a formula, `v u' unpacks it by pushing each of the
arguments of the top-level operator onto the stack.

   You can optionally give a numeric prefix argument to `v u' to
specify an explicit (un)packing mode.  If the packing mode is negative
and the input is actually a vector or matrix, the result will be two or
more similar vectors or matrices of the elements.  For example, given
the vector `[a +/- b, c^2, d +/- 7]', the result of `C-u -4 v u' will
be the two vectors `[a, c^2, d]' and `[b, 0, 7]'.

   Note that the prefix argument can have an effect even when the input
is not a vector.  For example, if the input is the number -5, then `c-u
-1 v u' yields -5 and 0 (the components of -5 when viewed as a
rectangular complex number); `C-u -2 v u' yields 5 and 180 (assuming
Degrees mode); and `C-u -10 v u' yields -5 and 1 (the numerator and
denominator of -5, viewed as a rational number).  Plain `v u' with this
input would complain that the input is not a composite object.

   Unpacking mode -11 converts a float into an integer mantissa and an
integer exponent, where the mantissa is not divisible by 10 (except
that 0.0 is represented by a mantissa and exponent of 0).  Unpacking
mode -12 converts a float into a floating-point mantissa and integer
exponent, where the mantissa (for non-zero numbers) is guaranteed to
lie in the range [1 .. 10).  In both cases, the mantissa is shifted
left or right (and the exponent adjusted to compensate) in order to
satisfy these constraints.

   Positive unpacking modes are treated differently than for `v p'.  A
mode of 1 is much like plain `v u' with no prefix argument, except that
in addition to the components of the input object, a suitable packing
mode to re-pack the object is also pushed.  Thus, `C-u 1 v u' followed
by `v p' will re-build the original object.

   A mode of 2 unpacks two levels of the object; the resulting
re-packing mode will be a vector of length 2.  This might be used to
unpack a matrix, say, or a vector of error forms.  Higher unpacking
modes unpack the input even more deeply.

   There are two algebraic functions analogous to `v u'.  The
`unpack(MODE, ITEM)' function unpacks the ITEM using the given MODE,
returning the result as a vector of components.  Here the MODE must be
an integer, not a vector.  For example, `unpack(-4, a +/- b)' returns
`[a, b]', as does `unpack(1, a +/- b)'.

   The `unpackt' function is like `unpack' but instead of returning a
simple vector of items, it returns a vector of two things:  The mode,
and the vector of items.  For example, `unpackt(1, 2:3 +/- 1:4)'
returns `[-4, [2:3, 1:4]]', and `unpackt(2, 2:3 +/- 1:4)' returns
`[[-4, -10], [2, 3, 1, 4]]'.  The identity for re-building the original
object is `apply(pack, unpackt(N, X)) = X'.  (The `apply' function
builds a function call given the function name and a vector of
arguments.)

   Subscript notation is a useful way to extract a particular part of
an object.  For example, to get the numerator of a rational number, you
can use `unpack(-10, X)_1'.

File: calc,  Node: Building Vectors,  Next: Extracting Elements,  Prev: Packing and Unpacking,  Up: Matrix Functions

11.2 Building Vectors
=====================

Vectors and matrices can be added, subtracted, multiplied, and divided;
*note Basic Arithmetic::.

   The `|' (`calc-concat') [`vconcat'] command "concatenates" two
vectors into one.  For example, after `[ 1 , 2 ] [ 3 , 4 ] |', the stack
will contain the single vector `[1, 2, 3, 4]'.  If the arguments are
matrices, the rows of the first matrix are concatenated with the rows
of the second.  (In other words, two matrices are just two vectors of
row-vectors as far as `|' is concerned.)

   If either argument to `|' is a scalar (a non-vector), it is treated
like a one-element vector for purposes of concatenation:  `1 [ 2 , 3 ]
|' produces the vector `[1, 2, 3]'.  Likewise, if one argument is a
matrix and the other is a plain vector, the vector is treated as a
one-row matrix.

   The `H |' (`calc-append') [`append'] command concatenates two
vectors without any special cases.  Both inputs must be vectors.
Whether or not they are matrices is not taken into account.  If either
argument is a scalar, the `append' function is left in symbolic form.
See also `cons' and `rcons' below.

   The `I |' and `H I |' commands are similar, but they use their two
stack arguments in the opposite order.  Thus `I |' is equivalent to
`<TAB> |', but possibly more convenient and also a bit faster.

   The `v d' (`calc-diag') [`diag'] function builds a diagonal square
matrix.  The optional numeric prefix gives the number of rows and
columns in the matrix.  If the value at the top of the stack is a
vector, the elements of the vector are used as the diagonal elements;
the prefix, if specified, must match the size of the vector.  If the
value on the stack is a scalar, it is used for each element on the
diagonal, and the prefix argument is required.

   To build a constant square matrix, e.g., a 3x3 matrix filled with
ones, use `0 M-3 v d 1 +', i.e., build a zero matrix first and then add
a constant value to that matrix.  (Another alternative would be to use
`v b' and `v a'; see below.)

   The `v i' (`calc-ident') [`idn'] function builds an identity matrix
of the specified size.  It is a convenient form of `v d' where the
diagonal element is always one.  If no prefix argument is given, this
command prompts for one.

   In algebraic notation, `idn(a,n)' acts much like `diag(a,n)', except
that `a' is required to be a scalar (non-vector) quantity.  If `n' is
omitted, `idn(a)' represents `a' times an identity matrix of unknown
size.  Calc can operate algebraically on such generic identity
matrices, and if one is combined with a matrix whose size is known, it
is converted automatically to an identity matrix of a suitable matching
size.  The `v i' command with an argument of zero creates a generic
identity matrix, `idn(1)'.  Note that in dimensioned Matrix mode (*note
Matrix Mode::), generic identity matrices are immediately expanded to
the current default dimensions.

   The `v x' (`calc-index') [`index'] function builds a vector of
consecutive integers from 1 to N, where N is the numeric prefix
argument.  If you do not provide a prefix argument, you will be
prompted to enter a suitable number.  If N is negative, the result is a
vector of negative integers from N to -1.

   With a prefix argument of just `C-u', the `v x' command takes three
values from the stack: N, START, and INCR (with INCR at top-of-stack).
Counting starts at START and increases by INCR for successive vector
elements.  If START or N is in floating-point format, the resulting
vector elements will also be floats.  Note that START and INCR may in
fact be any kind of numbers or formulas.

   When START and INCR are specified, a negative N has a different
interpretation:  It causes a geometric instead of arithmetic sequence
to be generated.  For example, `index(-3, a, b)' produces `[a, a b, a
b^2]'.  If you omit INCR in the algebraic form, `index(N, START)', the
default value for INCR is one for positive N or two for negative N.

   The `v b' (`calc-build-vector') [`cvec'] function builds a vector of
N copies of the value on the top of the stack, where N is the numeric
prefix argument.  In algebraic formulas, `cvec(x,n,m)' can also be used
to build an N-by-M matrix of copies of X.  (Interactively, just use `v
b' twice: once to build a row, then again to build a matrix of copies
of that row.)

   The `v h' (`calc-head') [`head'] function returns the first element
of a vector.  The `I v h' (`calc-tail') [`tail'] function returns the
vector with its first element removed.  In both cases, the argument
must be a non-empty vector.

   The `v k' (`calc-cons') [`cons'] function takes a value H and a
vector T from the stack, and produces the vector whose head is H and
whose tail is T.  This is similar to `|', except if H is itself a
vector, `|' will concatenate the two vectors whereas `cons' will insert
H at the front of the vector T.

   Each of these three functions also accepts the Hyperbolic flag
[`rhead', `rtail', `rcons'] in which case T instead represents the
_last_ single element of the vector, with H representing the remainder
of the vector.  Thus the vector `[a, b, c, d] = cons(a, [b, c, d]) =
rcons([a, b, c], d)'.  Also, `head([a, b, c, d]) = a', `tail([a, b, c,
d]) = [b, c, d]', `rhead([a, b, c, d]) = [a, b, c]', and `rtail([a, b,
c, d]) = d'.

File: calc,  Node: Extracting Elements,  Next: Manipulating Vectors,  Prev: Building Vectors,  Up: Matrix Functions

11.3 Extracting Vector Elements
===============================

The `v r' (`calc-mrow') [`mrow'] command extracts one row of the matrix
on the top of the stack, or one element of the plain vector on the top
of the stack.  The row or element is specified by the numeric prefix
argument; the default is to prompt for the row or element number.  The
matrix or vector is replaced by the specified row or element in the
form of a vector or scalar, respectively.

   With a prefix argument of `C-u' only, `v r' takes the index of the
element or row from the top of the stack, and the vector or matrix from
the second-to-top position.  If the index is itself a vector of
integers, the result is a vector of the corresponding elements of the
input vector, or a matrix of the corresponding rows of the input matrix.
This command can be used to obtain any permutation of a vector.

   With `C-u', if the index is an interval form with integer components,
it is interpreted as a range of indices and the corresponding subvector
or submatrix is returned.

   Subscript notation in algebraic formulas (`a_b') stands for the Calc
function `subscr', which is synonymous with `mrow'.  Thus, `[x, y,
z]_k' produces `x', `y', or `z' if `k' is one, two, or three,
respectively.  A double subscript (`M_i_j', equivalent to
`subscr(subscr(M, i), j)') will access the element at row `i', column
`j' of a matrix.  The `a _' (`calc-subscript') command creates a
subscript formula `a_b' out of two stack entries.  (It is on the `a'
"algebra" prefix because subscripted variables are often used purely as
an algebraic notation.)

   Given a negative prefix argument, `v r' instead deletes one row or
element from the matrix or vector on the top of the stack.  Thus `C-u 2
v r' replaces a matrix with its second row, but `C-u -2 v r' replaces
the matrix with the same matrix with its second row removed.  In
algebraic form this function is called `mrrow'.

   Given a prefix argument of zero, `v r' extracts the diagonal elements
of a square matrix in the form of a vector.  In algebraic form this
function is called `getdiag'.

   The `v c' (`calc-mcol') [`mcol' or `mrcol'] command is the analogous
operation on columns of a matrix.  Given a plain vector it extracts (or
removes) one element, just like `v r'.  If the index in `C-u v c' is an
interval or vector and the argument is a matrix, the result is a
submatrix with only the specified columns retained (and possibly
permuted in the case of a vector index).

   To extract a matrix element at a given row and column, use `v r' to
extract the row as a vector, then `v c' to extract the column element
from that vector.  In algebraic formulas, it is often more convenient to
use subscript notation:  `m_i_j' gives row `i', column `j' of matrix
`m'.

   The `v s' (`calc-subvector') [`subvec'] command extracts a subvector
of a vector.  The arguments are the vector, the starting index, and the
ending index, with the ending index in the top-of-stack position.  The
starting index indicates the first element of the vector to take.  The
ending index indicates the first element _past_ the range to be taken.
Thus, `subvec([a, b, c, d, e], 2, 4)' produces the subvector `[b, c]'.
You could get the same result using `mrow([a, b, c, d, e], [2 .. 4))'.

   If either the start or the end index is zero or negative, it is
interpreted as relative to the end of the vector.  Thus `subvec([a, b,
c, d, e], 2, -2)' also produces `[b, c]'.  In the algebraic form, the
end index can be omitted in which case it is taken as zero, i.e.,
elements from the starting element to the end of the vector are used.
The infinity symbol, `inf', also has this effect when used as the
ending index.

   With the Inverse flag, `I v s' [`rsubvec'] removes a subvector from
a vector.  The arguments are interpreted the same as for the normal `v
s' command.  Thus, `rsubvec([a, b, c, d, e], 2, 4)' produces `[a, d,
e]'.  It is always true that `subvec' and `rsubvec' return
complementary parts of the input vector.

   *Note Selecting Subformulas::, for an alternative way to operate on
vectors one element at a time.

File: calc,  Node: Manipulating Vectors,  Next: Vector and Matrix Arithmetic,  Prev: Extracting Elements,  Up: Matrix Functions

11.4 Manipulating Vectors
=========================

The `v l' (`calc-vlength') [`vlen'] command computes the length of a
vector.  The length of a non-vector is considered to be zero.  Note
that matrices are just vectors of vectors for the purposes of this
command.

   With the Hyperbolic flag, `H v l' [`mdims'] computes a vector of the
dimensions of a vector, matrix, or higher-order object.  For example,
`mdims([[a,b,c],[d,e,f]])' returns `[2, 3]' since its argument is a 2x3
matrix.

   The `v f' (`calc-vector-find') [`find'] command searches along a
vector for the first element equal to a given target.  The target is on
the top of the stack; the vector is in the second-to-top position.  If
a match is found, the result is the index of the matching element.
Otherwise, the result is zero.  The numeric prefix argument, if given,
allows you to select any starting index for the search.

   The `v a' (`calc-arrange-vector') [`arrange'] command rearranges a
vector to have a certain number of columns and rows.  The numeric
prefix argument specifies the number of columns; if you do not provide
an argument, you will be prompted for the number of columns.  The
vector or matrix on the top of the stack is "flattened" into a plain
vector.  If the number of columns is nonzero, this vector is then
formed into a matrix by taking successive groups of N elements.  If the
number of columns does not evenly divide the number of elements in the
vector, the last row will be short and the result will not be suitable
for use as a matrix.  For example, with the matrix `[[1, 2], [3, 4]]'
on the stack, `v a 4' produces `[[1, 2, 3, 4]]' (a 1x4 matrix), `v a 1'
produces `[[1], [2], [3], [4]]' (a 4x1 matrix), `v a 2' produces `[[1,
2], [3, 4]]' (the original 2x2 matrix), `v a 3' produces `[[1, 2, 3],
[4]]' (not a matrix), and `v a 0' produces the flattened list `[1, 2,
3, 4]'.

   The `V S' (`calc-sort') [`sort'] command sorts the elements of a
vector into increasing order.  Real numbers, real infinities, and
constant interval forms come first in this ordering; next come other
kinds of numbers, then variables (in alphabetical order), then finally
come formulas and other kinds of objects; these are sorted according to
a kind of lexicographic ordering with the useful property that one
vector is less or greater than another if the first corresponding
unequal elements are less or greater, respectively.  Since quoted
strings are stored by Calc internally as vectors of ASCII character
codes (*note Strings::), this means vectors of strings are also sorted
into alphabetical order by this command.

   The `I V S' [`rsort'] command sorts a vector into decreasing order.

   The `V G' (`calc-grade') [`grade', `rgrade'] command produces an
index table or permutation vector which, if applied to the input vector
(as the index of `C-u v r', say), would sort the vector.  A permutation
vector is just a vector of integers from 1 to N, where each integer
occurs exactly once.  One application of this is to sort a matrix of
data rows using one column as the sort key; extract that column, grade
it with `V G', then use the result to reorder the original matrix with
`C-u v r'.  Another interesting property of the `V G' command is that,
if the input is itself a permutation vector, the result will be the
inverse of the permutation.  The inverse of an index table is a rank
table, whose Kth element says where the Kth original vector element
will rest when the vector is sorted.  To get a rank table, just use `V
G V G'.

   With the Inverse flag, `I V G' produces an index table that would
sort the input into decreasing order.  Note that `V S' and `V G' use a
"stable" sorting algorithm, i.e., any two elements which are equal will
not be moved out of their original order.  Generally there is no way to
tell with `V S', since two elements which are equal look the same, but
with `V G' this can be an important issue.  In the matrix-of-rows
example, suppose you have names and telephone numbers as two columns and
you wish to sort by phone number primarily, and by name when the numbers
are equal.  You can sort the data matrix by names first, and then again
by phone numbers.  Because the sort is stable, any two rows with equal
phone numbers will remain sorted by name even after the second sort.

   The `V H' (`calc-histogram') [`histogram'] command builds a
histogram of a vector of numbers.  Vector elements are assumed to be
integers or real numbers in the range [0..N) for some "number of bins"
N, which is the numeric prefix argument given to the command.  The
result is a vector of N counts of how many times each value appeared in
the original vector.  Non-integers in the input are rounded down to
integers.  Any vector elements outside the specified range are ignored.
(You can tell if elements have been ignored by noting that the counts
in the result vector don't add up to the length of the input vector.)

   With the Hyperbolic flag, `H V H' pulls two vectors from the stack.
The second-to-top vector is the list of numbers as before.  The top
vector is an equal-sized list of "weights" to attach to the elements of
the data vector.  For example, if the first data element is 4.2 and the
first weight is 10, then 10 will be added to bin 4 of the result
vector.  Without the hyperbolic flag, every element has a weight of one.

   The `v t' (`calc-transpose') [`trn'] command computes the transpose
of the matrix at the top of the stack.  If the argument is a plain
vector, it is treated as a row vector and transposed into a one-column
matrix.

   The `v v' (`calc-reverse-vector') [`rev'] command reverses a vector
end-for-end.  Given a matrix, it reverses the order of the rows.  (To
reverse the columns instead, just use `v t v v v t'.  The same
principle can be used to apply other vector commands to the columns of
a matrix.)

   The `v m' (`calc-mask-vector') [`vmask'] command uses one vector as
a mask to extract elements of another vector.  The mask is in the
second-to-top position; the target vector is on the top of the stack.
These vectors must have the same length.  The result is the same as the
target vector, but with all elements which correspond to zeros in the
mask vector deleted.  Thus, for example, `vmask([1, 0, 1, 0, 1], [a, b,
c, d, e])' produces `[a, c, e]'.  *Note Logical Operations::.

   The `v e' (`calc-expand-vector') [`vexp'] command expands a vector
according to another mask vector.  The result is a vector the same
length as the mask, but with nonzero elements replaced by successive
elements from the target vector.  The length of the target vector is
normally the number of nonzero elements in the mask.  If the target
vector is longer, its last few elements are lost.  If the target vector
is shorter, the last few nonzero mask elements are left unreplaced in
the result.  Thus `vexp([2, 0, 3, 0, 7], [a, b])' produces `[a, 0, b,
0, 7]'.

   With the Hyperbolic flag, `H v e' takes a filler value from the top
of the stack; the mask and target vectors come from the third and
second elements of the stack.  This filler is used where the mask is
zero:  `vexp([2, 0, 3, 0, 7], [a, b], z)' produces `[a, z, c, z, 7]'.
If the filler value is itself a vector, then successive values are
taken from it, so that the effect is to interleave two vectors
according to the mask: `vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])'
produces `[a, x, b, 7, y, 0]'.

   Another variation on the masking idea is to combine `[a, b, c, d, e]'
with the mask `[1, 0, 1, 0, 1]' to produce `[a, 0, c, 0, e]'.  You can
accomplish this with `V M a &', mapping the logical "and" operation
across the two vectors.  *Note Logical Operations::.  Note that the `?
:' operation also discussed there allows other types of masking using
vectors.

File: calc,  Node: Vector and Matrix Arithmetic,  Next: Set Operations,  Prev: Manipulating Vectors,  Up: Matrix Functions

11.5 Vector and Matrix Arithmetic
=================================

Basic arithmetic operations like addition and multiplication are defined
for vectors and matrices as well as for numbers.  Division of matrices,
in the sense of multiplying by the inverse, is supported.  (Division by
a matrix actually uses LU-decomposition for greater accuracy and speed.)
*Note Basic Arithmetic::.

   The following functions are applied element-wise if their arguments
are vectors or matrices: `change-sign', `conj', `arg', `re', `im',
`polar', `rect', `clean', `float', `frac'.  *Note Function Index::.

   The `V J' (`calc-conj-transpose') [`ctrn'] command computes the
conjugate transpose of its argument, i.e., `conj(trn(x))'.

   The `A' (`calc-abs') [`abs'] command computes the Frobenius norm of
a vector or matrix argument.  This is the square root of the sum of the
squares of the absolute values of the elements of the vector or matrix.
If the vector is interpreted as a point in two- or three-dimensional
space, this is the distance from that point to the origin.

   The `v n' (`calc-rnorm') [`rnorm'] command computes the
infinity-norm of a vector, or the row norm of a matrix.  For a plain
vector, this is the maximum of the absolute values of the elements.  For
a matrix, this is the maximum of the row-absolute-value-sums, i.e., of
the sums of the absolute values of the elements along the various rows.

   The `V N' (`calc-cnorm') [`cnorm'] command computes the one-norm of
a vector, or column norm of a matrix.  For a plain vector, this is the
sum of the absolute values of the elements.  For a matrix, this is the
maximum of the column-absolute-value-sums.  General `k'-norms for `k'
other than one or infinity are not provided.  However, the 2-norm (or
Frobenius norm) is provided for vectors by the `A' (`calc-abs') command.

   The `V C' (`calc-cross') [`cross'] command computes the right-handed
cross product of two vectors, each of which must have exactly three
elements.

   The `&' (`calc-inv') [`inv'] command computes the inverse of a
square matrix.  If the matrix is singular, the inverse operation is
left in symbolic form.  Matrix inverses are recorded so that once an
inverse (or determinant) of a particular matrix has been computed, the
inverse and determinant of the matrix can be recomputed quickly in the
future.

   If the argument to `&' is a plain number `x', this command simply
computes `1/x'.  This is okay, because the `/' operator also does a
matrix inversion when dividing one by a matrix.

   The `V D' (`calc-mdet') [`det'] command computes the determinant of
a square matrix.

   The `V L' (`calc-mlud') [`lud'] command computes the LU
decomposition of a matrix.  The result is a list of three matrices
which, when multiplied together left-to-right, form the original matrix.
The first is a permutation matrix that arises from pivoting in the
algorithm, the second is lower-triangular with ones on the diagonal,
and the third is upper-triangular.

   The `V T' (`calc-mtrace') [`tr'] command computes the trace of a
square matrix.  This is defined as the sum of the diagonal elements of
the matrix.

   The `V K' (`calc-kron') [`kron'] command computes the Kronecker
product of two matrices.

File: calc,  Node: Set Operations,  Next: Statistical Operations,  Prev: Vector and Matrix Arithmetic,  Up: Matrix Functions

11.6 Set Operations using Vectors
=================================

Calc includes several commands which interpret vectors as "sets" of
objects.  A set is a collection of objects; any given object can appear
only once in the set.  Calc stores sets as vectors of objects in sorted
order.  Objects in a Calc set can be any of the usual things, such as
numbers, variables, or formulas.  Two set elements are considered equal
if they are identical, except that numerically equal numbers like the
integer 4 and the float 4.0 are considered equal even though they are
not "identical."  Variables are treated like plain symbols without
attached values by the set operations; subtracting the set `[b]' from
`[a, b]' always yields the set `[a]' even though if the variables `a'
and `b' both equaled 17, you might expect the answer `[]'.

   If a set contains interval forms, then it is assumed to be a set of
real numbers.  In this case, all set operations require the elements of
the set to be only things that are allowed in intervals:  Real numbers,
plus and minus infinity, HMS forms, and date forms.  If there are
variables or other non-real objects present in a real set, all set
operations on it will be left in unevaluated form.

   If the input to a set operation is a plain number or interval form
A, it is treated like the one-element vector `[A]'.  The result is
always a vector, except that if the set consists of a single interval,
the interval itself is returned instead.

   *Note Logical Operations::, for the `in' function which tests if a
certain value is a member of a given set.  To test if the set `A' is a
subset of the set `B', use `vdiff(A, B) = []'.

   The `V +' (`calc-remove-duplicates') [`rdup'] command converts an
arbitrary vector into set notation.  It works by sorting the vector as
if by `V S', then removing duplicates.  (For example, `[a, 5, 4, a,
4.0]' is sorted to `[4, 4.0, 5, a, a]' and then reduced to `[4, 5,
a]').  Overlapping intervals are merged as necessary.  You rarely need
to use `V +' explicitly, since all the other set-based commands apply
`V +' to their inputs before using them.

   The `V V' (`calc-set-union') [`vunion'] command computes the union
of two sets.  An object is in the union of two sets if and only if it
is in either (or both) of the input sets.  (You could accomplish the
same thing by concatenating the sets with `|', then using `V +'.)

   The `V ^' (`calc-set-intersect') [`vint'] command computes the
intersection of two sets.  An object is in the intersection if and only
if it is in both of the input sets.  Thus if the input sets are
disjoint, i.e., if they share no common elements, the result will be
the empty vector `[]'.  Note that the characters `V' and `^' were
chosen to be close to the conventional mathematical notation for set
union and intersection.

   The `V -' (`calc-set-difference') [`vdiff'] command computes the
difference between two sets.  An object is in the difference `A - B' if
and only if it is in `A' but not in `B'.  Thus subtracting `[y,z]' from
a set will remove the elements `y' and `z' if they are present.  You
can also think of this as a general "set complement" operator; if `A'
is the set of all possible values, then `A - B' is the "complement" of
`B'.  Obviously this is only practical if the set of all possible
values in your problem is small enough to list in a Calc vector (or
simple enough to express in a few intervals).

   The `V X' (`calc-set-xor') [`vxor'] command computes the
"exclusive-or," or "symmetric difference" of two sets.  An object is in
the symmetric difference of two sets if and only if it is in one, but
_not_ both, of the sets.  Objects that occur in both sets "cancel out."

   The `V ~' (`calc-set-complement') [`vcompl'] command computes the
complement of a set with respect to the real numbers.  Thus `vcompl(x)'
is equivalent to `vdiff([-inf .. inf], x)'.  For example, `vcompl([2,
(3 .. 4]])' evaluates to `[[-inf .. 2), (2 .. 3], (4 .. inf]]'.

   The `V F' (`calc-set-floor') [`vfloor'] command reinterprets a set
as a set of integers.  Any non-integer values, and intervals that do
not enclose any integers, are removed.  Open intervals are converted to
equivalent closed intervals.  Successive integers are converted into
intervals of integers.  For example, the complement of the set `[2, 6,
7, 8]' is messy, but if you wanted the complement with respect to the
set of integers you could type `V ~ V F' to get `[[-inf .. 1], [3 ..
5], [9 .. inf]]'.

   The `V E' (`calc-set-enumerate') [`venum'] command converts a set of
integers into an explicit vector.  Intervals in the set are expanded
out to lists of all integers encompassed by the intervals.  This only
works for finite sets (i.e., sets which do not involve `-inf' or `inf').

   The `V :' (`calc-set-span') [`vspan'] command converts any set of
reals into an interval form that encompasses all its elements.  The
lower limit will be the smallest element in the set; the upper limit
will be the largest element.  For an empty set, `vspan([])' returns the
empty interval `[0 .. 0)'.

   The `V #' (`calc-set-cardinality') [`vcard'] command counts the
number of integers in a set.  The result is the length of the vector
that would be produced by `V E', although the computation is much more
efficient than actually producing that vector.

   Another representation for sets that may be more appropriate in some
cases is binary numbers.  If you are dealing with sets of integers in
the range 0 to 49, you can use a 50-bit binary number where a
particular bit is 1 if the corresponding element is in the set.  *Note
Binary Functions::, for a list of commands that operate on binary
numbers.  Note that many of the above set operations have direct
equivalents in binary arithmetic:  `b o' (`calc-or'), `b a'
(`calc-and'), `b d' (`calc-diff'), `b x' (`calc-xor'), and `b n'
(`calc-not'), respectively.  You can use whatever representation for
sets is most convenient to you.

   The `b u' (`calc-unpack-bits') [`vunpack'] command converts an
integer that represents a set in binary into a set in vector/interval
notation.  For example, `vunpack(67)' returns `[[0 .. 1], 6]'.  If the
input is negative, the set it represents is semi-infinite: `vunpack(-4)
= [2 .. inf)'.  Use `V E' afterwards to expand intervals to individual
values if you wish.  Note that this command uses the `b' (binary)
prefix key.

   The `b p' (`calc-pack-bits') [`vpack'] command converts the other
way, from a vector or interval representing a set of nonnegative
integers into a binary integer describing the same set.  The set may
include positive infinity, but must not include any negative numbers.
The input is interpreted as a set of integers in the sense of `V F'
(`vfloor').  Beware that a simple input like `[100]' can result in a
huge integer representation (`2^100', a 31-digit integer, in this case).

File: calc,  Node: Statistical Operations,  Next: Reducing and Mapping,  Prev: Set Operations,  Up: Matrix Functions

11.7 Statistical Operations on Vectors
======================================

The commands in this section take vectors as arguments and compute
various statistical measures on the data stored in the vectors.  The
references used in the definitions of these functions are Bevington's
_Data Reduction and Error Analysis for the Physical Sciences_, and
_Numerical Recipes_ by Press, Flannery, Teukolsky and Vetterling.

   The statistical commands use the `u' prefix key followed by a
shifted letter or other character.

   *Note Manipulating Vectors::, for a description of `V H'
(`calc-histogram').

   *Note Curve Fitting::, for the `a F' command for doing least-squares
fits to statistical data.

   *Note Probability Distribution Functions::, for several common
probability distribution functions.

* Menu:

* Single-Variable Statistics::
* Paired-Sample Statistics::

File: calc,  Node: Single-Variable Statistics,  Next: Paired-Sample Statistics,  Prev: Statistical Operations,  Up: Statistical Operations

11.7.1 Single-Variable Statistics
---------------------------------

These functions do various statistical computations on single vectors.
Given a numeric prefix argument, they actually pop N objects from the
stack and combine them into a data vector.  Each object may be either a
number or a vector; if a vector, any sub-vectors inside it are
"flattened" as if by `v a 0'; *note Manipulating Vectors::.  By default
one object is popped, which (in order to be useful) is usually a vector.

   If an argument is a variable name, and the value stored in that
variable is a vector, then the stored vector is used.  This method has
the advantage that if your data vector is large, you can avoid the slow
process of manipulating it directly on the stack.

   These functions are left in symbolic form if any of their arguments
are not numbers or vectors, e.g., if an argument is a formula, or a
non-vector variable.  However, formulas embedded within vector
arguments are accepted; the result is a symbolic representation of the
computation, based on the assumption that the formula does not itself
represent a vector.  All varieties of numbers such as error forms and
interval forms are acceptable.

   Some of the functions in this section also accept a single error form
or interval as an argument.  They then describe a property of the
normal or uniform (respectively) statistical distribution described by
the argument.  The arguments are interpreted in the same way as the M
argument of the random number function `k r'.  In particular, an
interval with integer limits is considered an integer distribution, so
that `[2 .. 6)' is the same as `[2 .. 5]'.  An interval with at least
one floating-point limit is a continuous distribution:  `[2.0 .. 6.0)'
is _not_ the same as `[2.0 .. 5.0]'!

   The `u #' (`calc-vector-count') [`vcount'] command computes the
number of data values represented by the inputs.  For example,
`vcount(1, [2, 3], [[4, 5], [], x, y])' returns 7.  If the argument is
a single vector with no sub-vectors, this simply computes the length of
the vector.

   The `u +' (`calc-vector-sum') [`vsum'] command computes the sum of
the data values.  The `u *' (`calc-vector-prod') [`vprod'] command
computes the product of the data values.  If the input is a single flat
vector, these are the same as `V R +' and `V R *' (*note Reducing and
Mapping::).

   The `u X' (`calc-vector-max') [`vmax'] command computes the maximum
of the data values, and the `u N' (`calc-vector-min') [`vmin'] command
computes the minimum.  If the argument is an interval, this finds the
minimum or maximum value in the interval.  (Note that `vmax([2..6)) =
5' as described above.)  If the argument is an error form, this returns
plus or minus infinity.

   The `u M' (`calc-vector-mean') [`vmean'] command computes the
average (arithmetic mean) of the data values.  If the inputs are error
forms `x +/- s', this is the weighted mean of the `x' values with
weights `1 / s^2'.  If the inputs are not error forms, this is simply
the sum of the values divided by the count of the values.

   Note that a plain number can be considered an error form with error
`s = 0'.  If the input to `u M' is a mixture of plain numbers and error
forms, the result is the mean of the plain numbers, ignoring all values
with non-zero errors.  (By the above definitions it's clear that a
plain number effectively has an infinite weight, next to which an error
form with a finite weight is completely negligible.)

   This function also works for distributions (error forms or
intervals).  The mean of an error form `A +/- B' is simply `a'.  The
mean of an interval is the mean of the minimum and maximum values of
the interval.

   The `I u M' (`calc-vector-mean-error') [`vmeane'] command computes
the mean of the data points expressed as an error form.  This includes
the estimated error associated with the mean.  If the inputs are error
forms, the error is the square root of the reciprocal of the sum of the
reciprocals of the squares of the input errors.  (I.e., the variance is
the reciprocal of the sum of the reciprocals of the variances.)  If the
inputs are plain numbers, the error is equal to the standard deviation
of the values divided by the square root of the number of values.
(This works out to be equivalent to calculating the standard deviation
and then assuming each value's error is equal to this standard
deviation.)

   The `H u M' (`calc-vector-median') [`vmedian'] command computes the
median of the data values.  The values are first sorted into numerical
order; the median is the middle value after sorting.  (If the number of
data values is even, the median is taken to be the average of the two
middle values.)  The median function is different from the other
functions in this section in that the arguments must all be real
numbers; variables are not accepted even when nested inside vectors.
(Otherwise it is not possible to sort the data values.)  If any of the
input values are error forms, their error parts are ignored.

   The median function also accepts distributions.  For both normal
(error form) and uniform (interval) distributions, the median is the
same as the mean.

   The `H I u M' (`calc-vector-harmonic-mean') [`vhmean'] command
computes the harmonic mean of the data values.  This is defined as the
reciprocal of the arithmetic mean of the reciprocals of the values.

   The `u G' (`calc-vector-geometric-mean') [`vgmean'] command computes
the geometric mean of the data values.  This is the Nth root of the
product of the values.  This is also equal to the `exp' of the
arithmetic mean of the logarithms of the data values.

   The `H u G' [`agmean'] command computes the "arithmetic-geometric
mean" of two numbers taken from the stack.  This is computed by
replacing the two numbers with their arithmetic mean and geometric
mean, then repeating until the two values converge.

   Another commonly used mean, the RMS (root-mean-square), can be
computed for a vector of numbers simply by using the `A' command.

   The `u S' (`calc-vector-sdev') [`vsdev'] command computes the
standard deviation of the data values.  If the values are error forms,
the errors are used as weights just as for `u M'.  This is the _sample_
standard deviation, whose value is the square root of the sum of the
squares of the differences between the values and the mean of the `N'
values, divided by `N-1'.

   This function also applies to distributions.  The standard deviation
of a single error form is simply the error part.  The standard deviation
of a continuous interval happens to equal the difference between the
limits, divided by `sqrt(12)'.  The standard deviation of an integer
interval is the same as the standard deviation of a vector of those
integers.

   The `I u S' (`calc-vector-pop-sdev') [`vpsdev'] command computes the
_population_ standard deviation.  It is defined by the same formula as
above but dividing by `N' instead of by `N-1'.  The population standard
deviation is used when the input represents the entire set of data
values in the distribution; the sample standard deviation is used when
the input represents a sample of the set of all data values, so that
the mean computed from the input is itself only an estimate of the true
mean.

   For error forms and continuous intervals, `vpsdev' works exactly
like `vsdev'.  For integer intervals, it computes the population
standard deviation of the equivalent vector of integers.

   The `H u S' (`calc-vector-variance') [`vvar'] and `H I u S'
(`calc-vector-pop-variance') [`vpvar'] commands compute the variance of
the data values.  The variance is the square of the standard deviation,
i.e., the sum of the squares of the deviations of the data values from
the mean.  (This definition also applies when the argument is a
distribution.)

   The `vflat' algebraic function returns a vector of its arguments,
interpreted in the same way as the other functions in this section.
For example, `vflat(1, [2, [3, 4]], 5)' returns `[1, 2, 3, 4, 5]'.

File: calc,  Node: Paired-Sample Statistics,  Prev: Single-Variable Statistics,  Up: Statistical Operations

11.7.2 Paired-Sample Statistics
-------------------------------

The functions in this section take two arguments, which must be vectors
of equal size.  The vectors are each flattened in the same way as by
the single-variable statistical functions.  Given a numeric prefix
argument of 1, these functions instead take one object from the stack,
which must be an Nx2 matrix of data values.  Once again, variable names
can be used in place of actual vectors and matrices.

   The `u C' (`calc-vector-covariance') [`vcov'] command computes the
sample covariance of two vectors.  The covariance of vectors X and Y is
the sum of the products of the differences between the elements of X
and the mean of X times the differences between the corresponding
elements of Y and the mean of Y, all divided by `N-1'.  Note that the
variance of a vector is just the covariance of the vector with itself.
Once again, if the inputs are error forms the errors are used as weight
factors.  If both X and Y are composed of error forms, the error for a
given data point is taken as the square root of the sum of the squares
of the two input errors.

   The `I u C' (`calc-vector-pop-covariance') [`vpcov'] command
computes the population covariance, which is the same as the sample
covariance computed by `u C' except dividing by `N' instead of `N-1'.

   The `H u C' (`calc-vector-correlation') [`vcorr'] command computes
the linear correlation coefficient of two vectors.  This is defined by
the covariance of the vectors divided by the product of their standard
deviations.  (There is no difference between sample or population
statistics here.)

File: calc,  Node: Reducing and Mapping,  Next: Vector and Matrix Formats,  Prev: Statistical Operations,  Up: Matrix Functions

11.8 Reducing and Mapping Vectors
=================================

The commands in this section allow for more general operations on the
elements of vectors.

   The simplest of these operations is `V A' (`calc-apply') [`apply'],
which applies a given operator to the elements of a vector.  For
example, applying the hypothetical function `f' to the vector
`[1, 2, 3]' would produce the function call `f(1, 2, 3)'.  Applying the
`+' function to the vector `[a, b]' gives `a + b'.  Applying `+' to the
vector `[a, b, c]' is an error, since the `+' function expects exactly
two arguments.

   While `V A' is useful in some cases, you will usually find that
either `V R' or `V M', described below, is closer to what you want.

* Menu:

* Specifying Operators::
* Mapping::
* Reducing::
* Nesting and Fixed Points::
* Generalized Products::

File: calc,  Node: Specifying Operators,  Next: Mapping,  Prev: Reducing and Mapping,  Up: Reducing and Mapping

11.8.1 Specifying Operators
---------------------------

Commands in this section (like `V A') prompt you to press the key
corresponding to the desired operator.  Press `?' for a partial list of
the available operators.  Generally, an operator is any key or sequence
of keys that would normally take one or more arguments from the stack
and replace them with a result.  For example, `V A H C' uses the
hyperbolic cosine operator, `cosh'.  (Since `cosh' expects one
argument, `V A H C' requires a vector with a single element as its
argument.)

   You can press `x' at the operator prompt to select any algebraic
function by name to use as the operator.  This includes functions you
have defined yourself using the `Z F' command.  (*Note Algebraic
Definitions::.)  If you give a name for which no function has been
defined, the result is left in symbolic form, as in `f(1, 2, 3)'.  Calc
will prompt for the number of arguments the function takes if it can't
figure it out on its own (say, because you named a function that is
currently undefined).  It is also possible to type a digit key before
the function name to specify the number of arguments, e.g., `V M 3 x f
<RET>' calls `f' with three arguments even if it looks like it ought to
have only two.  This technique may be necessary if the function allows
a variable number of arguments.  For example, the `v e' [`vexp']
function accepts two or three arguments; if you want to map with the
three-argument version, you will have to type `V M 3 v e'.

   It is also possible to apply any formula to a vector by treating that
formula as a function.  When prompted for the operator to use, press
`'' (the apostrophe) and type your formula as an algebraic entry.  You
will then be prompted for the argument list, which defaults to a list
of all variables that appear in the formula, sorted into alphabetic
order.  For example, suppose you enter the formula `x + 2y^x'.  The
default argument list would be `(x y)', which means that if this
function is applied to the arguments `[3, 10]' the result will be `3 +
2*10^3'.  (If you plan to use a certain formula in this way often, you
might consider defining it as a function with `Z F'.)

   Another way to specify the arguments to the formula you enter is with
`$', `$$', and so on.  For example, `V A ' $$ + 2$^$$' has the same
effect as the previous example.  The argument list is automatically
taken to be `($$ $)'.  (The order of the arguments may seem backwards,
but it is analogous to the way normal algebraic entry interacts with
the stack.)

   If you press `$' at the operator prompt, the effect is similar to
the apostrophe except that the relevant formula is taken from
top-of-stack instead.  The actual vector arguments of the `V A $' or
related command then start at the second-to-top stack position.  You
will still be prompted for an argument list.

   A function can be written without a name using the notation `<#1 -
#2>', which means "a function of two arguments that computes the first
argument minus the second argument."  The symbols `#1' and `#2' are
placeholders for the arguments.  You can use any names for these
placeholders if you wish, by including an argument list followed by a
colon:  `<x, y : x - y>'.  When you type `V A ' $$ + 2$^$$ <RET>', Calc
builds the nameless function `<#1 + 2 #2^#1>' as the function to map
across the vectors.  When you type `V A ' x + 2y^x <RET> <RET>', Calc
builds the nameless function `<x, y : x + 2 y^x>'.  In both cases, Calc
also writes the nameless function to the Trail so that you can get it
back later if you wish.

   If there is only one argument, you can write `#' in place of `#1'.
(Note that `< >' notation is also used for date forms.  Calc tells that
`<STUFF>' is a nameless function by the presence of `#' signs inside
STUFF, or by the fact that STUFF begins with a list of variables
followed by a colon.)

   You can type a nameless function directly to `V A '', or put one on
the stack and use it with `V A $'.  Calc will not prompt for an
argument list in this case, since the nameless function specifies the
argument list as well as the function itself.  In `V A '', you can omit
the `< >' marks if you use `#' notation for the arguments, so that `V A
' #1+#2 <RET>' is the same as `V A ' <#1+#2> <RET>', which in turn is
the same as `V A ' $$+$ <RET>'.

   The internal format for `<x, y : x + y>' is `lambda(x, y, x + y)'.
(The word `lambda' derives from Lisp notation and the theory of
functions.)  The internal format for `<#1 + #2>' is `lambda(ArgA, ArgB,
ArgA + ArgB)'.  Note that there is no actual Calc function called
`lambda'; the whole point is that the `lambda' expression is used in
its symbolic form, not evaluated for an answer until it is applied to
specific arguments by a command like `V A' or `V M'.

   (Actually, `lambda' does have one special property:  Its arguments
are never evaluated; for example, putting `<(2/3) #>' on the stack will
not simplify the `2/3' until the nameless function is actually called.)

   As usual, commands like `V A' have algebraic function name
equivalents.  For example, `V A k g' with an argument of `v' is
equivalent to `apply(gcd, v)'.  The first argument specifies the
operator name, and is either a variable whose name is the same as the
function name, or a nameless function like `<#^3+1>'.  Operators that
are normally written as algebraic symbols have the names `add', `sub',
`mul', `div', `pow', `neg', `mod', and `vconcat'.

   The `call' function builds a function call out of several arguments:
`call(gcd, x, y)' is the same as `apply(gcd, [x, y])', which in turn is
the same as `gcd(x, y)'.  The first argument of `call', like the other
functions described here, may be either a variable naming a function,
or a nameless function (`call(<#1+2#2>, x, y)' is the same as `x + 2y').

   (Experts will notice that it's not quite proper to use a variable to
name a function, since the name `gcd' corresponds to the Lisp variable
`var-gcd' but to the Lisp function `calcFunc-gcd'.  Calc automatically
makes this translation, so you don't have to worry about it.)

File: calc,  Node: Mapping,  Next: Reducing,  Prev: Specifying Operators,  Up: Reducing and Mapping

11.8.2 Mapping
--------------

The `V M' (`calc-map') [`map'] command applies a given operator
elementwise to one or more vectors.  For example, mapping `A' [`abs']
produces a vector of the absolute values of the elements in the input
vector.  Mapping `+' pops two vectors from the stack, which must be of
equal length, and produces a vector of the pairwise sums of the
elements.  If either argument is a non-vector, it is duplicated for
each element of the other vector.  For example, `[1,2,3] 2 V M ^'
squares the elements of the specified vector.  With the 2 listed first,
it would have computed a vector of powers of two.  Mapping a
user-defined function pops as many arguments from the stack as the
function requires.  If you give an undefined name, you will be prompted
for the number of arguments to use.

   If any argument to `V M' is a matrix, the operator is normally mapped
across all elements of the matrix.  For example, given the matrix `[[1,
-2, 3], [-4, 5, -6]]', `V M A' takes six absolute values to produce
another 3x2 matrix, `[[1, 2, 3], [4, 5, 6]]'.

   The command `V M _' [`mapr'] (i.e., type an underscore at the
operator prompt) maps by rows instead.  For example, `V M _ A' views
the above matrix as a vector of two 3-element row vectors.  It produces
a new vector which contains the absolute values of those row vectors,
namely `[3.74, 8.77]'.  (Recall, the absolute value of a vector is
defined as the square root of the sum of the squares of the elements.)
Some operators accept vectors and return new vectors; for example, `v
v' reverses a vector, so `V M _ v v' would reverse each row of the
matrix to get a new matrix, `[[3, -2, 1], [-6, 5, -4]]'.

   Sometimes a vector of vectors (representing, say, strings, sets, or
lists) happens to look like a matrix.  If so, remember to use `V M _'
if you want to map a function across the whole strings or sets rather
than across their individual elements.

   The command `V M :' [`mapc'] maps by columns.  Basically, it
transposes the input matrix, maps by rows, and then, if the result is a
matrix, transposes again.  For example, `V M : A' takes the absolute
values of the three columns of the matrix, treating each as a 2-vector,
and `V M : v v' reverses the columns to get the matrix `[[-4, 5, -6],
[1, -2, 3]]'.

   (The symbols `_' and `:' were chosen because they had row-like and
column-like appearances, and were not already taken by useful
operators.  Also, they appear shifted on most keyboards so they are easy
to type after `V M'.)

   The `_' and `:' modifiers have no effect on arguments that are not
matrices (so if none of the arguments are matrices, they have no effect
at all).  If some of the arguments are matrices and others are plain
numbers, the plain numbers are held constant for all rows of the matrix
(so that `2 V M _ ^' squares every row of a matrix; squaring a vector
takes a dot product of the vector with itself).

   If some of the arguments are vectors with the same lengths as the
rows (for `V M _') or columns (for `V M :') of the matrix arguments,
those vectors are also held constant for every row or column.

   Sometimes it is useful to specify another mapping command as the
operator to use with `V M'.  For example, `V M _ V A +' applies `V A +'
to each row of the input matrix, which in turn adds the two values on
that row.  If you give another vector-operator command as the operator
for `V M', it automatically uses map-by-rows mode if you don't specify
otherwise; thus `V M V A +' is equivalent to `V M _ V A +'.  (If you
really want to map-by-elements another mapping command, you can use a
triple-nested mapping command:  `V M V M V A +' means to map `V M V A
+' over the rows of the matrix; in turn, `V A +' is mapped over the
elements of each row.)

   Previous versions of Calc had "map across" and "map down" modes that
are now considered obsolete; the old "map across" is now simply `V M V
A', and "map down" is now `V M : V A'.  The algebraic functions `mapa'
and `mapd' are still supported, though.  Note also that, while the old
mapping modes were persistent (once you set the mode, it would apply to
later mapping commands until you reset it), the new `:' and `_'
modifiers apply only to the current mapping command.  The default `V M'
always means map-by-elements.

   *Note Algebraic Manipulation::, for the `a M' command, which is like
`V M' but for equations and inequalities instead of vectors.  *Note
Storing Variables::, for the `s m' command which modifies a variable's
stored value using a `V M'-like operator.

File: calc,  Node: Reducing,  Next: Nesting and Fixed Points,  Prev: Mapping,  Up: Reducing and Mapping

11.8.3 Reducing
---------------

The `V R' (`calc-reduce') [`reduce'] command applies a given binary
operator across all the elements of a vector.  A binary operator is a
function such as `+' or `max' which takes two arguments.  For example,
reducing `+' over a vector computes the sum of the elements of the
vector.  Reducing `-' computes the first element minus each of the
remaining elements.  Reducing `max' computes the maximum element and so
on.  In general, reducing `f' over the vector `[a, b, c, d]' produces
`f(f(f(a, b), c), d)'.

   The `I V R' [`rreduce'] command is similar to `V R' except that
works from right to left through the vector.  For example, plain `V R
-' on the vector `[a, b, c, d]' produces `a - b - c - d' but `I V R -'
on the same vector produces `a - (b - (c - d))', or `a - b + c - d'.
This "alternating sum" occurs frequently in power series expansions.

   The `V U' (`calc-accumulate') [`accum'] command does an accumulation
operation.  Here Calc does the corresponding reduction operation, but
instead of producing only the final result, it produces a vector of all
the intermediate results.  Accumulating `+' over the vector `[a, b, c,
d]' produces the vector `[a, a + b, a + b + c, a + b + c + d]'.

   The `I V U' [`raccum'] command does a right-to-left accumulation.
For example, `I V U -' on the vector `[a, b, c, d]' produces the vector
`[a - b + c - d, b - c + d, c - d, d]'.

   As for `V M', `V R' normally reduces a matrix elementwise.  For
example, given the matrix `[[a, b, c], [d, e, f]]', `V R +' will
compute `a + b + c + d + e + f'.  You can type `V R _' or `V R :' to
modify this behavior.  The `V R _' [`reducea'] command reduces "across"
the matrix; it reduces each row of the matrix as a vector, then
collects the results.  Thus `V R _ +' of this matrix would produce `[a
+ b + c, d + e + f]'.  Similarly, `V R :' [`reduced'] reduces down; `V
R : +' would produce `[a + d, b + e, c + f]'.

   There is a third "by rows" mode for reduction that is occasionally
useful; `V R =' [`reducer'] simply reduces the operator over the rows
of the matrix themselves.  Thus `V R = +' on the above matrix would get
the same result as `V R : +', since adding two row vectors is
equivalent to adding their elements.  But `V R = *' would multiply the
two rows (to get a single number, their dot product), while `V R : *'
would produce a vector of the products of the columns.

   These three matrix reduction modes work with `V R' and `I V R', but
they are not currently supported with `V U' or `I V U'.

   The obsolete reduce-by-columns function, `reducec', is still
supported but there is no way to get it through the `V R' command.

   The commands `C-x * :' and `C-x * _' are equivalent to typing `C-x *
r' to grab a rectangle of data into Calc, and then typing `V R : +' or
`V R _ +', respectively, to sum the columns or rows of the matrix.
*Note Grabbing From Buffers::.

File: calc,  Node: Nesting and Fixed Points,  Next: Generalized Products,  Prev: Reducing,  Up: Reducing and Mapping

11.8.4 Nesting and Fixed Points
-------------------------------

The `H V R' [`nest'] command applies a function to a given argument
repeatedly.  It takes two values, `a' and `n', from the stack, where
`n' must be an integer.  It then applies the function nested `n' times;
if the function is `f' and `n' is 3, the result is `f(f(f(a)))'.  The
number `n' may be negative if Calc knows an inverse for the function
`f'; for example, `nest(sin, a, -2)' returns `arcsin(arcsin(a))'.

   The `H V U' [`anest'] command is an accumulating version of `nest':
It returns a vector of `n+1' values, e.g., `[a, f(a), f(f(a)),
f(f(f(a)))]'.  If `n' is negative and `F' is the inverse of `f', then
the result is of the form `[a, F(a), F(F(a)), F(F(F(a)))]'.

   The `H I V R' [`fixp'] command is like `H V R', except that it takes
only an `a' value from the stack; the function is applied until it
reaches a "fixed point," i.e., until the result no longer changes.

   The `H I V U' [`afixp'] command is an accumulating `fixp'.  The
first element of the return vector will be the initial value `a'; the
last element will be the final result that would have been returned by
`fixp'.

   For example, 0.739085 is a fixed point of the cosine function (in
radians): `cos(0.739085) = 0.739085'.  You can find this value by
putting, say, 1.0 on the stack and typing `H I V U C'.  (We use the
accumulating version so we can see the intermediate results:  `[1,
0.540302, 0.857553, 0.65329, ...]'.  With a precision of six, this
command will take 36 steps to converge to 0.739085.)

   Newton's method for finding roots is a classic example of iteration
to a fixed point.  To find the square root of five starting with an
initial guess, Newton's method would look for a fixed point of the
function `(x + 5/x) / 2'.  Putting a guess of 1 on the stack and typing
`H I V R ' ($ + 5/$)/2 <RET>' quickly yields the result 2.23607.  This
is equivalent to using the `a R' (`calc-find-root') command to find a
root of the equation `x^2 = 5'.

   These examples used numbers for `a' values.  Calc keeps applying the
function until two successive results are equal to within the current
precision.  For complex numbers, both the real parts and the imaginary
parts must be equal to within the current precision.  If `a' is a
formula (say, a variable name), then the function is applied until two
successive results are exactly the same formula.  It is up to you to
ensure that the function will eventually converge; if it doesn't, you
may have to press `C-g' to stop the Calculator.

   The algebraic `fixp' function takes two optional arguments, `n' and
`tol'.  The first is the maximum number of steps to be allowed, and
must be either an integer or the symbol `inf' (infinity, the default).
The second is a convergence tolerance.  If a tolerance is specified,
all results during the calculation must be numbers, not formulas, and
the iteration stops when the magnitude of the difference between two
successive results is less than or equal to the tolerance.  (This
implies that a tolerance of zero iterates until the results are exactly
equal.)

   Putting it all together, `fixp(<(# + A/#)/2>, B, 20, 1e-10)'
computes the square root of `A' given the initial guess `B', stopping
when the result is correct within the specified tolerance, or when 20
steps have been taken, whichever is sooner.

File: calc,  Node: Generalized Products,  Prev: Nesting and Fixed Points,  Up: Reducing and Mapping

11.8.5 Generalized Products
---------------------------

The `V O' (`calc-outer-product') [`outer'] command applies a given
binary operator to all possible pairs of elements from two vectors, to
produce a matrix.  For example, `V O *' with `[a, b]' and `[x, y, z]'
on the stack produces a multiplication table: `[[a x, a y, a z], [b x,
b y, b z]]'.  Element R,C of the result matrix is obtained by applying
the operator to element R of the lefthand vector and element C of the
righthand vector.

   The `V I' (`calc-inner-product') [`inner'] command computes the
generalized inner product of two vectors or matrices, given a
"multiplicative" operator and an "additive" operator.  These can each
actually be any binary operators; if they are `*' and `+',
respectively, the result is a standard matrix multiplication.  Element
R,C of the result matrix is obtained by mapping the multiplicative
operator across row R of the lefthand matrix and column C of the
righthand matrix, and then reducing with the additive operator.  Just
as for the standard `*' command, this can also do a vector-matrix or
matrix-vector inner product, or a vector-vector generalized dot product.

   Since `V I' requires two operators, it prompts twice.  In each case,
you can use any of the usual methods for entering the operator.  If you
use `$' twice to take both operator formulas from the stack, the first
(multiplicative) operator is taken from the top of the stack and the
second (additive) operator is taken from second-to-top.

File: calc,  Node: Vector and Matrix Formats,  Prev: Reducing and Mapping,  Up: Matrix Functions

11.9 Vector and Matrix Display Formats
======================================

Commands for controlling vector and matrix display use the `v' prefix
instead of the usual `d' prefix.  But they are display modes; in
particular, they are influenced by the `I' and `H' prefix keys in the
same way (*note Display Modes::).  Matrix display is also influenced by
the `d O' (`calc-flat-language') mode; *note Normal Language Modes::.

   The commands `v <' (`calc-matrix-left-justify'), `v >'
(`calc-matrix-right-justify'), and `v =' (`calc-matrix-center-justify')
control whether matrix elements are justified to the left, right, or
center of their columns.

   The `v [' (`calc-vector-brackets') command turns the square brackets
that surround vectors and matrices displayed in the stack on and off.
The `v {' (`calc-vector-braces') and `v (' (`calc-vector-parens')
commands use curly braces or parentheses, respectively, instead of
square brackets.  For example, `v {' might be used in preparation for
yanking a matrix into a buffer running Mathematica.  (In fact, the
Mathematica language mode uses this mode; *note Mathematica Language
Mode::.)  Note that, regardless of the display mode, either brackets or
braces may be used to enter vectors, and parentheses may never be used
for this purpose.

   The `v ]' (`calc-matrix-brackets') command controls the "big" style
display of matrices.  It prompts for a string of code letters;
currently implemented letters are `R', which enables brackets on each
row of the matrix; `O', which enables outer brackets in opposite
corners of the matrix; and `C', which enables commas or semicolons at
the ends of all rows but the last.  The default format is `RO'.
(Before Calc 2.00, the format was fixed at `ROC'.)  Here are some
example matrices:

     [ [ 123,  0,   0  ]       [ [ 123,  0,   0  ],
       [  0,  123,  0  ]         [  0,  123,  0  ],
       [  0,   0,  123 ] ]       [  0,   0,  123 ] ]

              RO                        ROC

       [ 123,  0,   0            [ 123,  0,   0 ;
          0,  123,  0               0,  123,  0 ;
          0,   0,  123 ]            0,   0,  123 ]

               O                        OC

       [ 123,  0,   0  ]           123,  0,   0
       [  0,  123,  0  ]            0,  123,  0
       [  0,   0,  123 ]            0,   0,  123

               R                       blank

Note that of the formats shown here, `RO', `ROC', and `OC' are all
recognized as matrices during reading, while the others are useful for
display only.

   The `v ,' (`calc-vector-commas') command turns commas on and off in
vector and matrix display.

   In vectors of length one, and in all vectors when commas have been
turned off, Calc adds extra parentheses around formulas that might
otherwise be ambiguous.  For example, `[a b]' could be a vector of the
one formula `a b', or it could be a vector of two variables with commas
turned off.  Calc will display the former case as `[(a b)]'.  You can
disable these extra parentheses (to make the output less cluttered at
the expense of allowing some ambiguity) by adding the letter `P' to the
control string you give to `v ]' (as described above).

   The `v .' (`calc-full-vectors') command turns abbreviated display of
long vectors on and off.  In this mode, vectors of six or more
elements, or matrices of six or more rows or columns, will be displayed
in an abbreviated form that displays only the first three elements and
the last element:  `[a, b, c, ..., z]'.  When very large vectors are
involved this will substantially improve Calc's display speed.

   The `t .' (`calc-full-trail-vectors') command controls a similar
mode for recording vectors in the Trail.  If you turn on this mode,
vectors of six or more elements and matrices of six or more rows or
columns will be abbreviated when they are put in the Trail.  The `t y'
(`calc-trail-yank') command will be unable to recover those vectors.
If you are working with very large vectors, this mode will improve the
speed of all operations that involve the trail.

   The `v /' (`calc-break-vectors') command turns multi-line vector
display on and off.  Normally, matrices are displayed with one row per
line but all other types of vectors are displayed in a single line.
This mode causes all vectors, whether matrices or not, to be displayed
with a single element per line.  Sub-vectors within the vectors will
still use the normal linear form.

File: calc,  Node: Algebra,  Next: Units,  Prev: Matrix Functions,  Up: Top

12 Algebra
**********

This section covers the Calc features that help you work with algebraic
formulas.  First, the general sub-formula selection mechanism is
described; this works in conjunction with any Calc commands.  Then,
commands for specific algebraic operations are described.  Finally, the
flexible "rewrite rule" mechanism is discussed.

   The algebraic commands use the `a' key prefix; selection commands
use the `j' (for "just a letter that wasn't used for anything else")
prefix.

   *Note Editing Stack Entries::, to see how to manipulate formulas
using regular Emacs editing commands.

   When doing algebraic work, you may find several of the Calculator's
modes to be helpful, including Algebraic Simplification mode (`m A') or
No-Simplification mode (`m O'), Algebraic entry mode (`m a'), Fraction
mode (`m f'), and Symbolic mode (`m s').  *Note Mode Settings::, for
discussions of these modes.  You may also wish to select Big display
mode (`d B').  *Note Normal Language Modes::.

* Menu:

* Selecting Subformulas::
* Algebraic Manipulation::
* Simplifying Formulas::
* Polynomials::
* Calculus::
* Solving Equations::
* Numerical Solutions::
* Curve Fitting::
* Summations::
* Logical Operations::
* Rewrite Rules::

File: calc,  Node: Selecting Subformulas,  Next: Algebraic Manipulation,  Prev: Algebra,  Up: Algebra

12.1 Selecting Sub-Formulas
===========================

When working with an algebraic formula it is often necessary to
manipulate a portion of the formula rather than the formula as a whole.
Calc allows you to "select" a portion of any formula on the stack.
Commands which would normally operate on that stack entry will now
operate only on the sub-formula, leaving the surrounding part of the
stack entry alone.

   One common non-algebraic use for selection involves vectors.  To work
on one element of a vector in-place, simply select that element as a
"sub-formula" of the vector.

* Menu:

* Making Selections::
* Changing Selections::
* Displaying Selections::
* Operating on Selections::
* Rearranging with Selections::

File: calc,  Node: Making Selections,  Next: Changing Selections,  Prev: Selecting Subformulas,  Up: Selecting Subformulas

12.1.1 Making Selections
------------------------

To select a sub-formula, move the Emacs cursor to any character in that
sub-formula, and press `j s' (`calc-select-here').  Calc will highlight
the smallest portion of the formula that contains that character.  By
default the sub-formula is highlighted by blanking out all of the rest
of the formula with dots.  Selection works in any display mode but is
perhaps easiest in Big mode (`d B').  Suppose you enter the following
formula:

                3    ___
         (a + b)  + V c
     1:  ---------------
             2 x + 1

(by typing `' ((a+b)^3 + sqrt(c)) / (2x+1)').  If you move the cursor
to the letter `b' and press `j s', the display changes to

                .    ...
         .. . b.  . . .
     1*  ...............
             . . . .

Every character not part of the sub-formula `b' has been changed to a
dot.  The `*' next to the line number is to remind you that the formula
has a portion of it selected.  (In this case, it's very obvious, but it
might not always be.  If Embedded mode is enabled, the word `Sel' also
appears in the mode line because the stack may not be visible.  *note
Embedded Mode::.)

   If you had instead placed the cursor on the parenthesis immediately
to the right of the `b', the selection would have been:

                .    ...
         (a + b)  . . .
     1*  ...............
             . . . .

The portion selected is always large enough to be considered a complete
formula all by itself, so selecting the parenthesis selects the whole
formula that it encloses.  Putting the cursor on the `+' sign would
have had the same effect.

   (Strictly speaking, the Emacs cursor is really the manifestation of
the Emacs "point," which is a position _between_ two characters in the
buffer.  So purists would say that Calc selects the smallest
sub-formula which contains the character to the right of "point.")

   If you supply a numeric prefix argument N, the selection is expanded
to the Nth enclosing sub-formula.  Thus, positioning the cursor on the
`b' and typing `C-u 1 j s' will select `a + b'; typing `C-u 2 j s' will
select `(a + b)^3', and so on.

   If the cursor is not on any part of the formula, or if you give a
numeric prefix that is too large, the entire formula is selected.

   If the cursor is on the `.' line that marks the top of the stack
(i.e., its normal "rest position"), this command selects the entire
formula at stack level 1.  Most selection commands similarly operate on
the formula at the top of the stack if you haven't positioned the
cursor on any stack entry.

   The `j a' (`calc-select-additional') command enlarges the current
selection to encompass the cursor.  To select the smallest sub-formula
defined by two different points, move to the first and press `j s',
then move to the other and press `j a'.  This is roughly analogous to
using `C-@' (`set-mark-command') to select the two ends of a region of
text during normal Emacs editing.

   The `j o' (`calc-select-once') command selects a formula in exactly
the same way as `j s', except that the selection will last only as long
as the next command that uses it.  For example, `j o 1 +' is a handy
way to add one to the sub-formula indicated by the cursor.

   (A somewhat more precise definition: The `j o' command sets a flag
such that the next command involving selected stack entries will clear
the selections on those stack entries afterwards.  All other selection
commands except `j a' and `j O' clear this flag.)

   The `j S' (`calc-select-here-maybe') and `j O'
(`calc-select-once-maybe') commands are equivalent to `j s' and `j o',
respectively, except that if the formula already has a selection they
have no effect.  This is analogous to the behavior of some commands
such as `j r' (`calc-rewrite-selection'; *note Selections with Rewrite
Rules::) and is mainly intended to be used in keyboard macros that
implement your own selection-oriented commands.

   Selection of sub-formulas normally treats associative terms like `a
+ b - c + d' and `x * y * z' as single levels of the formula.  If you
place the cursor anywhere inside `a + b - c + d' except on one of the
variable names and use `j s', you will select the entire four-term sum.

   The `j b' (`calc-break-selections') command controls a mode in which
the "deep structure" of these associative formulas shows through.  Calc
actually stores the above formulas as `((a + b) - c) + d' and `x * (y *
z)'.  (Note that for certain obscure reasons, by default Calc treats
multiplication as right-associative.)  Once you have enabled `j b'
mode, selecting with the cursor on the `-' sign would only select the
`a + b - c' portion, which makes sense when the deep structure of the
sum is considered.  There is no way to select the `b - c + d' portion;
although this might initially look like just as legitimate a sub-formula
as `a + b - c', the deep structure shows that it isn't.  The `d U'
command can be used to view the deep structure of any formula (*note
Normal Language Modes::).

   When `j b' mode has not been enabled, the deep structure is
generally hidden by the selection commands--what you see is what you
get.

   The `j u' (`calc-unselect') command unselects the formula that the
cursor is on.  If there was no selection in the formula, this command
has no effect.  With a numeric prefix argument, it unselects the Nth
stack element rather than using the cursor position.

   The `j c' (`calc-clear-selections') command unselects all stack
elements.

File: calc,  Node: Changing Selections,  Next: Displaying Selections,  Prev: Making Selections,  Up: Selecting Subformulas

12.1.2 Changing Selections
--------------------------

Once you have selected a sub-formula, you can expand it using the `j m'
(`calc-select-more') command.  If `a + b' is selected, pressing `j m'
repeatedly works as follows:

                3    ...                3    ___                3    ___
         (a + b)  . . .          (a + b)  + V c          (a + b)  + V c
     1*  ...............     1*  ...............     1*  ---------------
             . . . .                 . . . .                 2 x + 1

In the last example, the entire formula is selected.  This is roughly
the same as having no selection at all, but because there are subtle
differences the `*' character is still there on the line number.

   With a numeric prefix argument N, `j m' expands N times (or until
the entire formula is selected).  Note that `j s' with argument N is
equivalent to plain `j s' followed by `j m' with argument N.  If `j m'
is used when there is no current selection, it is equivalent to `j s'.

   Even though `j m' does not explicitly use the location of the cursor
within the formula, it nevertheless uses the cursor to determine which
stack element to operate on.  As usual, `j m' when the cursor is not on
any stack element operates on the top stack element.

   The `j l' (`calc-select-less') command reduces the current selection
around the cursor position.  That is, it selects the immediate
sub-formula of the current selection which contains the cursor, the
opposite of `j m'.  If the cursor is not inside the current selection,
the command de-selects the formula.

   The `j 1' through `j 9' (`calc-select-part') commands select the Nth
sub-formula of the current selection.  They are like `j l'
(`calc-select-less') except they use counting rather than the cursor
position to decide which sub-formula to select.  For example, if the
current selection is `a + b + c' or `f(a, b, c)' or `[a, b, c]', then
`j 1' selects `a', `j 2' selects `b', and `j 3' selects `c'; in each of
these cases, `j 4' through `j 9' would be errors.

   If there is no current selection, `j 1' through `j 9' select the Nth
top-level sub-formula.  (In other words, they act as if the entire
stack entry were selected first.)  To select the Nth sub-formula where
N is greater than nine, you must instead invoke `j 1' with N as a
numeric prefix argument.

   The `j n' (`calc-select-next') and `j p' (`calc-select-previous')
commands change the current selection to the next or previous
sub-formula at the same level.  For example, if `b' is selected in
`2 + a*b*c + x', then `j n' selects `c'.  Further `j n' commands would
be in error because, even though there is something to the right of `c'
(namely, `x'), it is not at the same level; in this case, it is not a
term of the same product as `b' and `c'.  However, `j m' (to select the
whole product `a*b*c' as a term of the sum) followed by `j n' would
successfully select the `x'.

   Similarly, `j p' moves the selection from the `b' in this sample
formula to the `a'.  Both commands accept numeric prefix arguments to
move several steps at a time.

   It is interesting to compare Calc's selection commands with the
Emacs Info system's commands for navigating through hierarchically
organized documentation.  Calc's `j n' command is completely analogous
to Info's `n' command.  Likewise, `j p' maps to `p', `j 2' maps to `2',
and Info's `u' is like `j m'.  (Note that `j u' stands for
`calc-unselect', not "up".)  The Info `m' command is somewhat similar
to Calc's `j s' and `j l'; in each case, you can jump directly to a
sub-component of the hierarchy simply by pointing to it with the cursor.

File: calc,  Node: Displaying Selections,  Next: Operating on Selections,  Prev: Changing Selections,  Up: Selecting Subformulas

12.1.3 Displaying Selections
----------------------------

The `j d' (`calc-show-selections') command controls how selected
sub-formulas are displayed.  One of the alternatives is illustrated in
the above examples; if we press `j d' we switch to the other style in
which the selected portion itself is obscured by `#' signs:

                3    ...                  #    ___
         (a + b)  . . .            ## # ##  + V c
     1*  ...............       1*  ---------------
             . . . .                   2 x + 1

File: calc,  Node: Operating on Selections,  Next: Rearranging with Selections,  Prev: Displaying Selections,  Up: Selecting Subformulas

12.1.4 Operating on Selections
------------------------------

Once a selection is made, all Calc commands that manipulate items on
the stack will operate on the selected portions of the items instead.
(Note that several stack elements may have selections at once, though
there can be only one selection at a time in any given stack element.)

   The `j e' (`calc-enable-selections') command disables the effect
that selections have on Calc commands.  The current selections still
exist, but Calc commands operate on whole stack elements anyway.  This
mode can be identified by the fact that the `*' markers on the line
numbers are gone, even though selections are visible.  To reactivate
the selections, press `j e' again.

   To extract a sub-formula as a new formula, simply select the
sub-formula and press <RET>.  This normally duplicates the top stack
element; here it duplicates only the selected portion of that element.

   To replace a sub-formula with something different, you can enter the
new value onto the stack and press <TAB>.  This normally exchanges the
top two stack elements; here it swaps the value you entered into the
selected portion of the formula, returning the old selected portion to
the top of the stack.

                3    ...                    ...                    ___
         (a + b)  . . .           17 x y . . .           17 x y + V c
     2*  ...............      2*  .............      2:  -------------
             . . . .                 . . . .                2 x + 1

                                         3                      3
     1:  17 x y               1:  (a + b)            1:  (a + b)

   In this example we select a sub-formula of our original example,
enter a new formula, <TAB> it into place, then deselect to see the
complete, edited formula.

   If you want to swap whole formulas around even though they contain
selections, just use `j e' before and after.

   The `j '' (`calc-enter-selection') command is another way to replace
a selected sub-formula.  This command does an algebraic entry just like
the regular `'' key.  When you press <RET>, the formula you type
replaces the original selection.  You can use the `$' symbol in the
formula to refer to the original selection.  If there is no selection
in the formula under the cursor, the cursor is used to make a temporary
selection for the purposes of the command.  Thus, to change a term of a
formula, all you have to do is move the Emacs cursor to that term and
press `j ''.

   The `j `' (`calc-edit-selection') command is a similar analogue of
the ``' (`calc-edit') command.  It edits the selected sub-formula in a
separate buffer.  If there is no selection, it edits the sub-formula
indicated by the cursor.

   To delete a sub-formula, press <DEL>.  This generally replaces the
sub-formula with the constant zero, but in a few suitable contexts it
uses the constant one instead.  The <DEL> key automatically deselects
and re-simplifies the entire formula afterwards.  Thus:

                   ###
         17 x y + # #          17 x y         17 # y          17 y
     1*  -------------     1:  -------    1*  -------    1:  -------
            2 x + 1            2 x + 1        2 x + 1        2 x + 1

   In this example, we first delete the `sqrt(c)' term; Calc
accomplishes this by replacing `sqrt(c)' with zero and resimplifying.
We then delete the `x' in the numerator; since this is part of a
product, Calc replaces it with `1' and resimplifies.

   If you select an element of a vector and press <DEL>, that element
is deleted from the vector.  If you delete one side of an equation or
inequality, only the opposite side remains.

   The `j <DEL>' (`calc-del-selection') command is like <DEL> but with
the auto-selecting behavior of `j '' and `j `'.  It deletes the
selected portion of the formula indicated by the cursor, or, in the
absence of a selection, it deletes the sub-formula indicated by the
cursor position.

   (There is also an auto-selecting `j <RET>' (`calc-copy-selection')
command.)

   Normal arithmetic operations also apply to sub-formulas.  Here we
select the denominator, press `5 -' to subtract five from the
denominator, press `n' to negate the denominator, then press `Q' to
take the square root.

          .. .           .. .           .. .             .. .
     1*  .......    1*  .......    1*  .......    1*  ..........
         2 x + 1        2 x - 4        4 - 2 x         _________
                                                      V 4 - 2 x

   Certain types of operations on selections are not allowed.  For
example, for an arithmetic function like `-' no more than one of the
arguments may be a selected sub-formula.  (As the above example shows,
the result of the subtraction is spliced back into the argument which
had the selection; if there were more than one selection involved, this
would not be well-defined.)  If you try to subtract two selections, the
command will abort with an error message.

   Operations on sub-formulas sometimes leave the formula as a whole in
an "un-natural" state.  Consider negating the `2 x' term of our sample
formula by selecting it and pressing `n' (`calc-change-sign').

            .. .                .. .
     1*  ..........      1*  ...........
          .........           ..........
         . . . 2 x           . . . -2 x

   Unselecting the sub-formula reveals that the minus sign, which would
normally have cancelled out with the subtraction automatically, has not
been able to do so because the subtraction was not part of the selected
portion.  Pressing `=' (`calc-evaluate') or doing any other
mathematical operation on the whole formula will cause it to be
simplified.

            17 y                17 y
     1:  -----------     1:  ----------
          __________          _________
         V 4 - -2 x          V 4 + 2 x

File: calc,  Node: Rearranging with Selections,  Prev: Operating on Selections,  Up: Selecting Subformulas

12.1.5 Rearranging Formulas using Selections
--------------------------------------------

The `j R' (`calc-commute-right') command moves the selected sub-formula
to the right in its surrounding formula.  Generally the selection is
one term of a sum or product; the sum or product is rearranged
according to the commutative laws of algebra.

   As with `j '' and `j <DEL>', the term under the cursor is used if
there is no selection in the current formula.  All commands described
in this section share this property.  In this example, we place the
cursor on the `a' and type `j R', then repeat.

     1:  a + b - c          1:  b + a - c          1:  b - c + a

Note that in the final step above, the `a' is switched with the `c' but
the signs are adjusted accordingly.  When moving terms of sums and
products, `j R' will never change the mathematical meaning of the
formula.

   The selected term may also be an element of a vector or an argument
of a function.  The term is exchanged with the one to its right.  In
this case, the "meaning" of the vector or function may of course be
drastically changed.

     1:  [a, b, c]          1:  [b, a, c]          1:  [b, c, a]

     1:  f(a, b, c)         1:  f(b, a, c)         1:  f(b, c, a)

   The `j L' (`calc-commute-left') command is like `j R' except that it
swaps the selected term with the one to its left.

   With numeric prefix arguments, these commands move the selected term
several steps at a time.  It is an error to try to move a term left or
right past the end of its enclosing formula.  With numeric prefix
arguments of zero, these commands move the selected term as far as
possible in the given direction.

   The `j D' (`calc-sel-distribute') command mixes the selected sum or
product into the surrounding formula using the distributive law.  For
example, in `a * (b - c)' with the `b - c' selected, the result is `a b
- a c'.  This also distributes products or quotients into surrounding
powers, and can also do transformations like `exp(a + b)' to `exp(a)
exp(b)', where `a + b' is the selected term, and `ln(a ^ b)' to `ln(a)
b', where `a ^ b' is the selected term.

   For multiple-term sums or products, `j D' takes off one term at a
time:  `a * (b + c - d)' goes to `a * (c - d) + a b' with the `c - d'
selected so that you can type `j D' repeatedly to expand completely.
The `j D' command allows a numeric prefix argument which specifies the
maximum number of times to expand at once; the default is one time only.

   The `j D' command is implemented using rewrite rules.  *Note
Selections with Rewrite Rules::.  The rules are stored in the Calc
variable `DistribRules'.  A convenient way to view these rules is to
use `s e' (`calc-edit-variable') which displays and edits the stored
value of a variable.  Press `C-c C-c' to return from editing mode; be
careful not to make any actual changes or else you will affect the
behavior of future `j D' commands!

   To extend `j D' to handle new cases, just edit `DistribRules' as
described above.  You can then use the `s p' command to save this
variable's value permanently for future Calc sessions.  *Note
Operations on Variables::.

   The `j M' (`calc-sel-merge') command is the complement of `j D';
given `a b - a c' with either `a b' or `a c' selected, the result is `a
* (b - c)'.  Once again, `j M' can also merge calls to functions like
`exp' and `ln'; examine the variable `MergeRules' to see all the
relevant rules.

   The `j C' (`calc-sel-commute') command swaps the arguments of the
selected sum, product, or equation.  It always behaves as if `j b' mode
were in effect, i.e., the sum `a + b + c' is treated as the nested sums
`(a + b) + c' by this command.  If you put the cursor on the first `+',
the result is `(b + a) + c'; if you put the cursor on the second `+',
the result is `c + (a + b)' (which the default simplifications will
rearrange to `(c + a) + b').  The relevant rules are stored in the
variable `CommuteRules'.

   You may need to turn default simplifications off (with the `m O'
command) in order to get the full benefit of `j C'.  For example,
commuting `a - b' produces `-b + a', but the default simplifications
will "simplify" this right back to `a - b' if you don't turn them off.
The same is true of some of the other manipulations described in this
section.

   The `j N' (`calc-sel-negate') command replaces the selected term
with the negative of that term, then adjusts the surrounding formula in
order to preserve the meaning.  For example, given `exp(a - b)' where
`a - b' is selected, the result is `1 / exp(b - a)'.  By contrast,
selecting a term and using the regular `n' (`calc-change-sign') command
negates the term without adjusting the surroundings, thus changing the
meaning of the formula as a whole.  The rules variable is `NegateRules'.

   The `j &' (`calc-sel-invert') command is similar to `j N' except it
takes the reciprocal of the selected term.  For example, given `a -
ln(b)' with `b' selected, the result is `a + ln(1/b)'.  The rules
variable is `InvertRules'.

   The `j E' (`calc-sel-jump-equals') command moves the selected term
from one side of an equation to the other.  Given `a + b = c + d' with
`c' selected, the result is `a + b - c = d'.  This command also works
if the selected term is part of a `*', `/', or `^' formula.  The
relevant rules variable is `JumpRules'.

   The `j I' (`calc-sel-isolate') command isolates the selected term on
its side of an equation.  It uses the `a S' (`calc-solve-for') command
to solve the equation, and the Hyperbolic flag affects it in the same
way.  *Note Solving Equations::.  When it applies, `j I' is often
easier to use than `j E'.  It understands more rules of algebra, and
works for inequalities as well as equations.

   The `j *' (`calc-sel-mult-both-sides') command prompts for a formula
using algebraic entry, then multiplies both sides of the selected
quotient or equation by that formula.  It simplifies each side with `a
s' (`calc-simplify') before re-forming the quotient or equation.  You
can suppress this simplification by providing a prefix argument: `C-u j
*'.  There is also a `j /' (`calc-sel-div-both-sides') which is similar
to `j *' but dividing instead of multiplying by the factor you enter.

   If the selection is a quotient with numerator 1, then Calc's default
simplifications would normally cancel the new factors.  To prevent
this, when the `j *' command is used on a selection whose numerator is
1 or -1, the denominator is expanded at the top level using the
distributive law (as if using the `C-u 1 a x' command).  Suppose the
formula on the stack is `1 / (a + 1)' and you wish to multiplying the
top and bottom by `a - 1'.  Calc's default simplifications would
normally change the result `(a - 1) /(a + 1) (a - 1)' back to the
original form by cancellation; when `j *' is used, Calc expands the
denominator to  `a (a - 1) + a - 1' to prevent this.

   If you wish the `j *' command to completely expand the denominator
of a quotient you can call it with a zero prefix: `C-u 0 j *'.  For
example, if the formula on the stack is `1 / (sqrt(a) + 1)', you may
wish to eliminate the square root in the denominator by multiplying the
top and bottom by `sqrt(a) - 1'.  If you did this simply by using a
simple `j *' command, you would get `(sqrt(a)-1)/ (sqrt(a) (sqrt(a) -
1) + sqrt(a) - 1)'.  Instead, you would probably want to use `C-u 0 j
*', which would expand the bottom and give you the desired result
`(sqrt(a)-1)/(a-1)'.  More generally, if `j *' is called with an
argument of a positive integer N, then the denominator of the
expression will be expanded N times (as if with the `C-u N a x'
command).

   If the selection is an inequality, `j *' and `j /' will accept any
factor, but will warn unless they can prove the factor is either
positive or negative.  (In the latter case the direction of the
inequality will be switched appropriately.)  *Note Declarations::, for
ways to inform Calc that a given variable is positive or negative.  If
Calc can't tell for sure what the sign of the factor will be, it will
assume it is positive and display a warning message.

   For selections that are not quotients, equations, or inequalities,
these commands pull out a multiplicative factor:  They divide (or
multiply) by the entered formula, simplify, then multiply (or divide)
back by the formula.

   The `j +' (`calc-sel-add-both-sides') and `j -'
(`calc-sel-sub-both-sides') commands analogously add to or subtract
from both sides of an equation or inequality.  For other types of
selections, they extract an additive factor.  A numeric prefix argument
suppresses simplification of the intermediate results.

   The `j U' (`calc-sel-unpack') command replaces the selected function
call with its argument.  For example, given `a + sin(x^2)' with
`sin(x^2)' selected, the result is `a + x^2'.  (The `x^2' will remain
selected; if you wanted to change the `sin' to `cos', just press `C'
now to take the cosine of the selected part.)

   The `j v' (`calc-sel-evaluate') command performs the normal default
simplifications on the selected sub-formula.  These are the
simplifications that are normally done automatically on all results,
but which may have been partially inhibited by previous
selection-related operations, or turned off altogether by the `m O'
command.  This command is just an auto-selecting version of the `a v'
command (*note Algebraic Manipulation::).

   With a numeric prefix argument of 2, `C-u 2 j v' applies the `a s'
(`calc-simplify') command to the selected sub-formula.  With a prefix
argument of 3 or more, e.g., `C-u j v' applies the `a e'
(`calc-simplify-extended') command.  *Note Simplifying Formulas::.
With a negative prefix argument it simplifies at the top level only,
just as with `a v'.  Here the "top" level refers to the top level of
the selected sub-formula.

   The `j "' (`calc-sel-expand-formula') command is to `a "' (*note
Algebraic Manipulation::) what `j v' is to `a v'.

   You can use the `j r' (`calc-rewrite-selection') command to define
other algebraic operations on sub-formulas.  *Note Rewrite Rules::.

File: calc,  Node: Algebraic Manipulation,  Next: Simplifying Formulas,  Prev: Selecting Subformulas,  Up: Algebra

12.2 Algebraic Manipulation
===========================

The commands in this section perform general-purpose algebraic
manipulations.  They work on the whole formula at the top of the stack
(unless, of course, you have made a selection in that formula).

   Many algebra commands prompt for a variable name or formula.  If you
answer the prompt with a blank line, the variable or formula is taken
from top-of-stack, and the normal argument for the command is taken
from the second-to-top stack level.

   The `a v' (`calc-alg-evaluate') command performs the normal default
simplifications on a formula; for example, `a - -b' is changed to `a +
b'.  These simplifications are normally done automatically on all Calc
results, so this command is useful only if you have turned default
simplifications off with an `m O' command.  *Note Simplification
Modes::.

   It is often more convenient to type `=', which is like `a v' but
which also substitutes stored values for variables in the formula.  Use
`a v' if you want the variables to ignore their stored values.

   If you give a numeric prefix argument of 2 to `a v', it simplifies
as if in Algebraic Simplification mode.  This is equivalent to typing
`a s'; *note Simplifying Formulas::.  If you give a numeric prefix of 3
or more, it uses Extended Simplification mode (`a e').

   If you give a negative prefix argument -1, -2, or -3, it simplifies
in the corresponding mode but only works on the top-level function call
of the formula.  For example, `(2 + 3) * (2 + 3)' will simplify to `(2
+ 3)^2', without simplifying the sub-formulas `2 + 3'.  As another
example, typing `V R +' to sum the vector `[1, 2, 3, 4]' produces the
formula `reduce(add, [1, 2, 3, 4])' in No-Simplify mode.  Using `a v'
will evaluate this all the way to 10; using `C-u - a v' will evaluate
it only to `1 + 2 + 3 + 4'.  (*Note Reducing and Mapping::.)

   The `=' command corresponds to the `evalv' function, and the related
`N' command, which is like `=' but temporarily disables Symbolic mode
(`m s') during the evaluation, corresponds to the `evalvn' function.
(These commands interpret their prefix arguments differently than `a
v'; `=' treats the prefix as the number of stack elements to evaluate
at once, and `N' treats it as a temporary different working precision.)

   The `evalvn' function can take an alternate working precision as an
optional second argument.  This argument can be either an integer, to
set the precision absolutely, or a vector containing a single integer,
to adjust the precision relative to the current precision.  Note that
`evalvn' with a larger than current precision will do the calculation
at this higher precision, but the result will as usual be rounded back
down to the current precision afterward.  For example, `evalvn(pi -
3.1415)' at a precision of 12 will return `9.265359e-5'; `evalvn(pi -
3.1415, 30)' will return `9.26535897932e-5' (computing a 25-digit
result which is then rounded down to 12); and `evalvn(pi - 3.1415,
[-2])' will return `9.2654e-5'.

   The `a "' (`calc-expand-formula') command expands functions into
their defining formulas wherever possible.  For example, `deg(x^2)' is
changed to `180 x^2 / pi'.  Most functions, like `sin' and `gcd', are
not defined by simple formulas and so are unaffected by this command.
One important class of functions which _can_ be expanded is the
user-defined functions created by the `Z F' command.  *Note Algebraic
Definitions::.  Other functions which `a "' can expand include the
probability distribution functions, most of the financial functions,
and the hyperbolic and inverse hyperbolic functions.  A numeric prefix
argument affects `a "' in the same way as it does `a v':  A positive
argument expands all functions in the formula and then simplifies in
various ways; a negative argument expands and simplifies only the
top-level function call.

   The `a M' (`calc-map-equation') [`mapeq'] command applies a given
function or operator to one or more equations.  It is analogous to `V
M', which operates on vectors instead of equations.  *note Reducing and
Mapping::.  For example, `a M S' changes `x = y+1' to `sin(x) =
sin(y+1)', and `a M +' with `x = y+1' and `6' on the stack produces
`x+6 = y+7'.  With two equations on the stack, `a M +' would add the
lefthand sides together and the righthand sides together to get the two
respective sides of a new equation.

   Mapping also works on inequalities.  Mapping two similar inequalities
produces another inequality of the same type.  Mapping an inequality
with an equation produces an inequality of the same type.  Mapping a
`<=' with a `<' or `!=' (not-equal) produces a `<'.  If inequalities
with opposite direction (e.g., `<' and `>') are mapped, the direction
of the second inequality is reversed to match the first:  Using `a M +'
on `a < b' and `a > 2' reverses the latter to get `2 < a', which then
allows the combination `a + 2 < b + a', which the `a s' command can
then simplify to get `2 < b'.

   Using `a M *', `a M /', `a M n', or `a M &' to negate or invert an
inequality will reverse the direction of the inequality.  Other
adjustments to inequalities are _not_ done automatically; `a M S' will
change `x < y' to `sin(x) < sin(y)' even though this is not true for
all values of the variables.

   With the Hyperbolic flag, `H a M' [`mapeqp'] does a plain mapping
operation without reversing the direction of any inequalities.  Thus,
`H a M &' would change `x > 2' to `1/x > 0.5'.  (This change is
mathematically incorrect, but perhaps you were fixing an inequality
which was already incorrect.)

   With the Inverse flag, `I a M' [`mapeqr'] always reverses the
direction of the inequality.  You might use `I a M C' to change `x < y'
to `cos(x) > cos(y)' if you know you are working with small positive
angles.

   The `a b' (`calc-substitute') [`subst'] command substitutes all
occurrences of some variable or sub-expression of an expression with a
new sub-expression.  For example, substituting `sin(x)' with `cos(y)'
in `2 sin(x)^2 + x sin(x) + sin(2 x)' produces `2 cos(y)^2 + x cos(y) +
sin(2 x)'.  Note that this is a purely structural substitution; the
lone `x' and the `sin(2 x)' stayed the same because they did not look
like `sin(x)'.  *Note Rewrite Rules::, for a more general method for
doing substitutions.

   The `a b' command normally prompts for two formulas, the old one and
the new one.  If you enter a blank line for the first prompt, all three
arguments are taken from the stack (new, then old, then target
expression).  If you type an old formula but then enter a blank line
for the new one, the new formula is taken from top-of-stack and the
target from second-to-top.  If you answer both prompts, the target is
taken from top-of-stack as usual.

   Note that `a b' has no understanding of commutativity or
associativity.  The pattern `x+y' will not match the formula `y+x'.
Also, `y+z' will not match inside the formula `x+y+z' because the `+'
operator is left-associative, so the "deep structure" of that formula
is `(x+y) + z'.  Use `d U' (`calc-unformatted-language') mode to see
the true structure of a formula.  The rewrite rule mechanism, discussed
later, does not have these limitations.

   As an algebraic function, `subst' takes three arguments: Target
expression, old, new.  Note that `subst' is always evaluated
immediately, even if its arguments are variables, so if you wish to put
a call to `subst' onto the stack you must turn the default
simplifications off first (with `m O').

File: calc,  Node: Simplifying Formulas,  Next: Polynomials,  Prev: Algebraic Manipulation,  Up: Algebra

12.3 Simplifying Formulas
=========================

The `a s' (`calc-simplify') [`simplify'] command applies various
algebraic rules to simplify a formula.  This includes rules which are
not part of the default simplifications because they may be too slow to
apply all the time, or may not be desirable all of the time.  For
example, non-adjacent terms of sums are combined, as in `a + b + 2 a'
to `b + 3 a', and some formulas like `sin(arcsin(x))' are simplified to
`x'.

   The sections below describe all the various kinds of algebraic
simplifications Calc provides in full detail.  None of Calc's
simplification commands are designed to pull rabbits out of hats; they
simply apply certain specific rules to put formulas into less redundant
or more pleasing forms.  Serious algebra in Calc must be done manually,
usually with a combination of selections and rewrite rules.  *Note
Rearranging with Selections::.  *Note Rewrite Rules::.

   *Note Simplification Modes::, for commands to control what level of
simplification occurs automatically.  Normally only the "default
simplifications" occur.

* Menu:

* Default Simplifications::
* Algebraic Simplifications::
* Unsafe Simplifications::
* Simplification of Units::

File: calc,  Node: Default Simplifications,  Next: Algebraic Simplifications,  Prev: Simplifying Formulas,  Up: Simplifying Formulas

12.3.1 Default Simplifications
------------------------------

This section describes the "default simplifications," those which are
normally applied to all results.  For example, if you enter the variable
`x' on the stack twice and push `+', Calc's default simplifications
automatically change `x + x' to `2 x'.

   The `m O' command turns off the default simplifications, so that `x
+ x' will remain in this form unless you give an explicit "simplify"
command like `=' or `a v'.  *Note Algebraic Manipulation::.  The `m D'
command turns the default simplifications back on.

   The most basic default simplification is the evaluation of functions.
For example, `2 + 3' is evaluated to `5', and `sqrt(9)' is evaluated to
`3'.  Evaluation does not occur if the arguments to a function are
somehow of the wrong type `tan([2,3,4])'), range (`tan(90)'), or number
(`tan(3,5)'), or if the function name is not recognized (`f(5)'), or if
Symbolic mode (*note Symbolic Mode::) prevents evaluation (`sqrt(2)').

   Calc simplifies (evaluates) the arguments to a function before it
simplifies the function itself.  Thus `sqrt(5+4)' is simplified to
`sqrt(9)' before the `sqrt' function itself is applied.  There are very
few exceptions to this rule: `quote', `lambda', and `condition' (the
`::' operator) do not evaluate their arguments, `if' (the `? :'
operator) does not evaluate all of its arguments, and `evalto' does not
evaluate its lefthand argument.

   Most commands apply the default simplifications to all arguments they
take from the stack, perform a particular operation, then simplify the
result before pushing it back on the stack.  In the common special case
of regular arithmetic commands like `+' and `Q' [`sqrt'], the arguments
are simply popped from the stack and collected into a suitable function
call, which is then simplified (the arguments being simplified first as
part of the process, as described above).

   The default simplifications are too numerous to describe completely
here, but this section will describe the ones that apply to the major
arithmetic operators.  This list will be rather technical in nature,
and will probably be interesting to you only if you are a serious user
of Calc's algebra facilities.

   As well as the simplifications described here, if you have stored
any rewrite rules in the variable `EvalRules' then these rules will
also be applied before any built-in default simplifications.  *Note
Automatic Rewrites::, for details.

   And now, on with the default simplifications:

   Arithmetic operators like `+' and `*' always take two arguments in
Calc's internal form.  Sums and products of three or more terms are
arranged by the associative law of algebra into a left-associative form
for sums, `((a + b) + c) + d', and (by default) a right-associative
form for products, `a * (b * (c * d))'.  Formulas like `(a + b) + (c +
d)' are rearranged to left-associative form, though this rarely matters
since Calc's algebra commands are designed to hide the inner structure
of sums and products as much as possible.  Sums and products in their
proper associative form will be written without parentheses in the
examples below.

   Sums and products are _not_ rearranged according to the commutative
law (`a + b' to `b + a') except in a few special cases described below.
Some algebra programs always rearrange terms into a canonical order,
which enables them to see that `a b + b a' can be simplified to `2 a b'.
Calc assumes you have put the terms into the order you want and
generally leaves that order alone, with the consequence that formulas
like the above will only be simplified if you explicitly give the `a s'
command.  *Note Algebraic Simplifications::.

   Differences `a - b' are treated like sums `a + (-b)' for purposes of
simplification; one of the default simplifications is to rewrite `a +
(-b)' or `(-b) + a', where `-b' represents a "negative-looking" term,
into `a - b' form.  "Negative-looking" means negative numbers, negated
formulas like `-x', and products or quotients in which either term is
negative-looking.

   Other simplifications involving negation are `-(-x)' to `x'; `-(a
b)' or `-(a/b)' where either `a' or `b' is negative-looking, simplified
by negating that term, or else where `a' or `b' is any number, by
negating that number; `-(a + b)' to `-a - b', and `-(b - a)' to `a - b'.
(This, and rewriting `(-b) + a' to `a - b', are the only cases where
the order of terms in a sum is changed by the default simplifications.)

   The distributive law is used to simplify sums in some cases: `a x +
b x' to `(a + b) x', where `a' represents a number or an implicit 1 or
-1 (as in `x' or `-x') and similarly for `b'.  Use the `a c', `a f', or
`j M' commands to merge sums with non-numeric coefficients using the
distributive law.

   The distributive law is only used for sums of two terms, or for
adjacent terms in a larger sum.  Thus `a + b + b + c' is simplified to
`a + 2 b + c', but `a + b + c + b' is not simplified.  The reason is
that comparing all terms of a sum with one another would require time
proportional to the square of the number of terms; Calc relegates
potentially slow operations like this to commands that have to be
invoked explicitly, like `a s'.

   Finally, `a + 0' and `0 + a' are simplified to `a'.  A consequence
of the above rules is that `0 - a' is simplified to `-a'.

   The products `1 a' and `a 1' are simplified to `a'; `(-1) a' and `a
(-1)' are simplified to `-a'; `0 a' and `a 0' are simplified to `0',
except that in Matrix mode where `a' is not provably scalar the result
is the generic zero matrix `idn(0)', and that if `a' is infinite the
result is `nan'.

   Also, `(-a) b' and `a (-b)' are simplified to `-(a b)', where this
occurs for negated formulas but not for regular negative numbers.

   Products are commuted only to move numbers to the front: `a b 2' is
commuted to `2 a b'.

   The product `a (b + c)' is distributed over the sum only if `a' and
at least one of `b' and `c' are numbers: `2 (x + 3)' goes to `2 x + 6'.
The formula `(-a) (b - c)', where `-a' is a negative number, is
rewritten to `a (c - b)'.

   The distributive law of products and powers is used for adjacent
terms of the product: `x^a x^b' goes to `x^(a+b)' where `a' is a
number, or an implicit 1 (as in `x'), or the implicit one-half of
`sqrt(x)', and similarly for `b'.  The result is written using `sqrt'
or `1/sqrt' if the sum of the powers is `1/2' or `-1/2', respectively.
If the sum of the powers is zero, the product is simplified to `1' or
to `idn(1)' if Matrix mode is enabled.

   The product of a negative power times anything but another negative
power is changed to use division: `x^(-2) y' goes to `y / x^2' unless
Matrix mode is in effect and neither `x' nor `y' are scalar (in which
case it is considered unsafe to rearrange the order of the terms).

   Finally, `a (b/c)' is rewritten to `(a b)/c', and also `(a/b) c' is
changed to `(a c)/b' unless in Matrix mode.

   Simplifications for quotients are analogous to those for products.
The quotient `0 / x' is simplified to `0', with the same exceptions
that were noted for `0 x'.  Likewise, `x / 1' and `x / (-1)' are
simplified to `x' and `-x', respectively.

   The quotient `x / 0' is left unsimplified or changed to an infinite
quantity, as directed by the current infinite mode.  *Note Infinite
Mode::.

   The expression `a / b^(-c)' is changed to `a b^c', where `-c' is any
negative-looking power.  Also, `1 / b^c' is changed to `b^(-c)' for any
power `c'.

   Also, `(-a) / b' and `a / (-b)' go to `-(a/b)'; `(a/b) / c' goes to
`a / (b c)'; and `a / (b/c)' goes to `(a c) / b' unless Matrix mode
prevents this rearrangement.  Similarly, `a / (b:c)' is simplified to
`(c:b) a' for any fraction `b:c'.

   The distributive law is applied to `(a + b) / c' only if `c' and at
least one of `a' and `b' are numbers.  Quotients of powers and square
roots are distributed just as described for multiplication.

   Quotients of products cancel only in the leading terms of the
numerator and denominator.  In other words, `a x b / a y b' is
cancelled to `x b / y b' but not to `x / y'.  Once again this is
because full cancellation can be slow; use `a s' to cancel all terms of
the quotient.

   Quotients of negative-looking values are simplified according to
`(-a) / (-b)' to `a / b', `(-a) / (b - c)' to `a / (c - b)', and `(a -
b) / (-c)' to `(b - a) / c'.

   The formula `x^0' is simplified to `1', or to `idn(1)' in Matrix
mode.  The formula `0^x' is simplified to `0' unless `x' is a negative
number, complex number or zero.  If `x' is negative, complex or `0.0',
`0^x' is an infinity or an unsimplified formula according to the
current infinite mode.  The expression `0^0' is simplified to `1'.

   Powers of products or quotients `(a b)^c', `(a/b)^c' are distributed
to `a^c b^c', `a^c / b^c' only if `c' is an integer, or if either `a'
or `b' are nonnegative real numbers.  Powers of powers `(a^b)^c' are
simplified to `a^(b c)' only when `c' is an integer and `b c' also
evaluates to an integer.  Without these restrictions these
simplifications would not be safe because of problems with principal
values.  (In other words, `((-3)^1:2)^2' is safe to simplify, but
`((-3)^2)^1:2' is not.)  *Note Declarations::, for ways to inform Calc
that your variables satisfy these requirements.

   As a special case of this rule, `sqrt(x)^n' is simplified to
`x^(n/2)' only for even integers `n'.

   If `a' is known to be real, `b' is an even integer, and `c' is a
half- or quarter-integer, then `(a^b)^c' is simplified to `abs(a^(b
c))'.

   Also, `(-a)^b' is simplified to `a^b' if `b' is an even integer, or
to `-(a^b)' if `b' is an odd integer, for any negative-looking
expression `-a'.

   Square roots `sqrt(x)' generally act like one-half powers `x^1:2'
for the purposes of the above-listed simplifications.

   Also, note that `1 / x^1:2' is changed to `x^(-1:2)', but `1 /
sqrt(x)' is left alone.

   Generic identity matrices (*note Matrix Mode::) are simplified by the
following rules:  `idn(a) + b' to `a + b' if `b' is provably scalar, or
expanded out if `b' is a matrix; `idn(a) + idn(b)' to `idn(a + b)';
`-idn(a)' to `idn(-a)'; `a idn(b)' to `idn(a b)' if `a' is provably
scalar, or to `a b' if `a' is provably non-scalar;  `idn(a) idn(b)' to
`idn(a b)'; analogous simplifications for quotients involving `idn';
and `idn(a)^n' to `idn(a^n)' where `n' is an integer.

   The `floor' function and other integer truncation functions vanish
if the argument is provably integer-valued, so that `floor(round(x))'
simplifies to `round(x)'.  Also, combinations of `float', `floor' and
its friends, and `ffloor' and its friends, are simplified in appropriate
ways.  *Note Integer Truncation::.

   The expression `abs(-x)' changes to `abs(x)'.  The expression
`abs(abs(x))' changes to `abs(x)';  in fact, `abs(x)' changes to `x' or
`-x' if `x' is provably nonnegative or nonpositive (*note
Declarations::).

   While most functions do not recognize the variable `i' as an
imaginary number, the `arg' function does handle the two cases `arg(i)'
and `arg(-i)' just for convenience.

   The expression `conj(conj(x))' simplifies to `x'.  Various other
expressions involving `conj', `re', and `im' are simplified, especially
if some of the arguments are provably real or involve the constant `i'.
For example, `conj(a + b i)' is changed to `conj(a) - conj(b) i',  or
to `a - b i' if `a' and `b' are known to be real.

   Functions like `sin' and `arctan' generally don't have any default
simplifications beyond simply evaluating the functions for suitable
numeric arguments and infinity.  The `a s' command described in the
next section does provide some simplifications for these functions,
though.

   One important simplification that does occur is that `ln(e)' is
simplified to 1, and `ln(e^x)' is simplified to `x' for any `x'.  This
occurs even if you have stored a different value in the Calc variable
`e'; but this would be a bad idea in any case if you were also using
natural logarithms!

   Among the logical functions, !(A <= B) changes to A > B and so on.
Equations and inequalities where both sides are either negative-looking
or zero are simplified by negating both sides and reversing the
inequality.  While it might seem reasonable to simplify `!!x' to `x',
this would not be valid in general because `!!2' is 1, not 2.

   Most other Calc functions have few if any default simplifications
defined, aside of course from evaluation when the arguments are
suitable numbers.

File: calc,  Node: Algebraic Simplifications,  Next: Unsafe Simplifications,  Prev: Default Simplifications,  Up: Simplifying Formulas

12.3.2 Algebraic Simplifications
--------------------------------

The `a s' command makes simplifications that may be too slow to do all
the time, or that may not be desirable all of the time.  If you find
these simplifications are worthwhile, you can type `m A' to have Calc
apply them automatically.

   This section describes all simplifications that are performed by the
`a s' command.  Note that these occur in addition to the default
simplifications; even if the default simplifications have been turned
off by an `m O' command, `a s' will turn them back on temporarily while
it simplifies the formula.

   There is a variable, `AlgSimpRules', in which you can put rewrites
to be applied by `a s'.  Its use is analogous to `EvalRules', but
without the special restrictions.  Basically, the simplifier does `a r
AlgSimpRules' with an infinite repeat count on the whole expression
being simplified, then it traverses the expression applying the
built-in rules described below.  If the result is different from the
original expression, the process repeats with the default
simplifications (including `EvalRules'), then `AlgSimpRules', then the
built-in simplifications, and so on.

   Sums are simplified in two ways.  Constant terms are commuted to the
end of the sum, so that `a + 2 + b' changes to `a + b + 2'.  The only
exception is that a constant will not be commuted away from the first
position of a difference, i.e., `2 - x' is not commuted to `-x + 2'.

   Also, terms of sums are combined by the distributive law, as in `x +
y + 2 x' to `y + 3 x'.  This always occurs for adjacent terms, but `a
s' compares all pairs of terms including non-adjacent ones.

   Products are sorted into a canonical order using the commutative
law.  For example, `b c a' is commuted to `a b c'.  This allows easier
comparison of products; for example, the default simplifications will
not change `x y + y x' to `2 x y', but `a s' will; it first rewrites
the sum to `x y + x y', and then the default simplifications are able
to recognize a sum of identical terms.

   The canonical ordering used to sort terms of products has the
property that real-valued numbers, interval forms and infinities come
first, and are sorted into increasing order.  The `V S' command uses
the same ordering when sorting a vector.

   Sorting of terms of products is inhibited when Matrix mode is turned
on; in this case, Calc will never exchange the order of two terms
unless it knows at least one of the terms is a scalar.

   Products of powers are distributed by comparing all pairs of terms,
using the same method that the default simplifications use for adjacent
terms of products.

   Even though sums are not sorted, the commutative law is still taken
into account when terms of a product are being compared.  Thus `(x + y)
(y + x)' will be simplified to `(x + y)^2'.  A subtle point is that `(x
- y) (y - x)' will _not_ be simplified to `-(x - y)^2'; Calc does not
notice that one term can be written as a constant times the other, even
if that constant is -1.

   A fraction times any expression, `(a:b) x', is changed to a quotient
involving integers:  `a x / b'.  This is not done for floating-point
numbers like `0.5', however.  This is one reason why you may find it
convenient to turn Fraction mode on while doing algebra; *note Fraction
Mode::.

   Quotients are simplified by comparing all terms in the numerator
with all terms in the denominator for possible cancellation using the
distributive law.  For example, `a x^2 b / c x^3 d' will cancel `x^2'
from the top and bottom to get `a b / c x d'.  (The terms in the
denominator will then be rearranged to `c d x' as described above.)  If
there is any common integer or fractional factor in the numerator and
denominator, it is cancelled out; for example, `(4 x + 6) / 8 x'
simplifies to `(2 x + 3) / 4 x'.

   Non-constant common factors are not found even by `a s'.  To cancel
the factor `a' in `(a x + a) / a^2' you could first use `j M' on the
product `a x' to Merge the numerator to `a (1+x)', which can then be
simplified successfully.

   Integer powers of the variable `i' are simplified according to the
identity `i^2 = -1'.  If you store a new value other than the complex
number `(0,1)' in `i', this simplification will no longer occur.  This
is done by `a s' instead of by default in case someone (unwisely) uses
the name `i' for a variable unrelated to complex numbers; it would be
unfortunate if Calc quietly and automatically changed this formula for
reasons the user might not have been thinking of.

   Square roots of integer or rational arguments are simplified in
several ways.  (Note that these will be left unevaluated only in
Symbolic mode.)  First, square integer or rational factors are pulled
out so that `sqrt(8)' is rewritten as `2 sqrt(2)'.  Conceptually
speaking this implies factoring the argument into primes and moving
pairs of primes out of the square root, but for reasons of efficiency
Calc only looks for primes up to 29.

   Square roots in the denominator of a quotient are moved to the
numerator:  `1 / sqrt(3)' changes to `sqrt(3) / 3'.  The same effect
occurs for the square root of a fraction: `sqrt(2:3)' changes to
`sqrt(6) / 3'.

   The `%' (modulo) operator is simplified in several ways when the
modulus `M' is a positive real number.  First, if the argument is of
the form `x + n' for some real number `n', then `n' is itself reduced
modulo `M'.  For example, `(x - 23) % 10' is simplified to `(x + 7) %
10'.

   If the argument is multiplied by a constant, and this constant has a
common integer divisor with the modulus, then this factor is cancelled
out.  For example, `12 x % 15' is changed to `3 (4 x % 5)' by factoring
out 3.  Also, `(12 x + 1) % 15' is changed to `3 ((4 x + 1:3) % 5)'.
While these forms may not seem "simpler," they allow Calc to discover
useful information about modulo forms in the presence of declarations.

   If the modulus is 1, then Calc can use `int' declarations to
evaluate the expression.  For example, the idiom `x % 2' is often used
to check whether a number is odd or even.  As described above,
`2 n % 2' and `(2 n + 1) % 2' are simplified to `2 (n % 1)' and `2 ((n
+ 1:2) % 1)', respectively; Calc can simplify these to 0 and 1
(respectively) if `n' has been declared to be an integer.

   Trigonometric functions are simplified in several ways.  Whenever a
products of two trigonometric functions can be replaced by a single
function, the replacement is made; for example, `tan(x) cos(x)' is
simplified to `sin(x)'.  Reciprocals of trigonometric functions are
replaced by their reciprocal function; for example, `1/sec(x)' is
simplified to `cos(x)'.  The corresponding simplifications for the
hyperbolic functions are also handled.

   Trigonometric functions of their inverse functions are simplified.
The expression `sin(arcsin(x))' is simplified to `x', and similarly for
`cos' and `tan'.  Trigonometric functions of inverses of different
trigonometric functions can also be simplified, as in `sin(arccos(x))'
to `sqrt(1 - x^2)'.

   If the argument to `sin' is negative-looking, it is simplified to
`-sin(x)', and similarly for `cos' and `tan'.  Finally, certain special
values of the argument are recognized; *note Trigonometric and
Hyperbolic Functions::.

   Hyperbolic functions of their inverses and of negative-looking
arguments are also handled, as are exponentials of inverse hyperbolic
functions.

   No simplifications for inverse trigonometric and hyperbolic
functions are known, except for negative arguments of `arcsin',
`arctan', `arcsinh', and `arctanh'.  Note that `arcsin(sin(x))' can
_not_ safely change to `x', since this only correct within an integer
multiple of `2 pi' radians or 360 degrees.  However, `arcsinh(sinh(x))'
is simplified to `x' if `x' is known to be real.

   Several simplifications that apply to logarithms and exponentials
are that `exp(ln(x))', `e^ln(x)', and `10^log10(x)' all reduce to `x'.
Also, `ln(exp(x))', etc., can reduce to `x' if `x' is provably real.
The form `exp(x)^y' is simplified to `exp(x y)'.  If `x' is a suitable
multiple of `pi i' (as described above for the trigonometric
functions), then `exp(x)' or `e^x' will be expanded.  Finally, `ln(x)'
is simplified to a form involving `pi' and `i' where `x' is provably
negative, positive imaginary, or negative imaginary.

   The error functions `erf' and `erfc' are simplified when their
arguments are negative-looking or are calls to the `conj' function.

   Equations and inequalities are simplified by cancelling factors of
products, quotients, or sums on both sides.  Inequalities change sign
if a negative multiplicative factor is cancelled.  Non-constant
multiplicative factors as in `a b = a c' are cancelled from equations
only if they are provably nonzero (generally because they were declared
so; *note Declarations::).  Factors are cancelled from inequalities
only if they are nonzero and their sign is known.

   Simplification also replaces an equation or inequality with 1 or 0
("true" or "false") if it can through the use of declarations.  If `x'
is declared to be an integer greater than 5, then `x < 3', `x = 3', and
`x = 7.5' are all simplified to 0, but `x > 3' is simplified to 1.  By
a similar analysis, `abs(x) >= 0' is simplified to 1, as is `x^2 >= 0'
if `x' is known to be real.

File: calc,  Node: Unsafe Simplifications,  Next: Simplification of Units,  Prev: Algebraic Simplifications,  Up: Simplifying Formulas

12.3.3 "Unsafe" Simplifications
-------------------------------

The `a e' (`calc-simplify-extended') [`esimplify'] command is like `a s'
except that it applies some additional simplifications which are not
"safe" in all cases.  Use this only if you know the values in your
formula lie in the restricted ranges for which these simplifications
are valid.  The symbolic integrator uses `a e'; one effect of this is
that the integrator's results must be used with caution.  Where an
integral table will often attach conditions like "for positive `a'
only," Calc (like most other symbolic integration programs) will simply
produce an unqualified result.

   Because `a e''s simplifications are unsafe, it is sometimes better
to type `C-u -3 a v', which does extended simplification only on the
top level of the formula without affecting the sub-formulas.  In fact,
`C-u -3 j v' allows you to target extended simplification to any
specific part of a formula.

   The variable `ExtSimpRules' contains rewrites to be applied by the
`a e' command.  These are applied in addition to `EvalRules' and
`AlgSimpRules'.  (The `a r AlgSimpRules' step described above is simply
followed by an `a r ExtSimpRules' step.)

   Following is a complete list of "unsafe" simplifications performed
by `a e'.

   Inverse trigonometric or hyperbolic functions, called with their
corresponding non-inverse functions as arguments, are simplified by `a
e'.  For example, `arcsin(sin(x))' changes to `x'.  Also,
`arcsin(cos(x))' and `arccos(sin(x))' both change to `pi/2 - x'.  These
simplifications are unsafe because they are valid only for values of
`x' in a certain range; outside that range, values are folded down to
the 360-degree range that the inverse trigonometric functions always
produce.

   Powers of powers `(x^a)^b' are simplified to `x^(a b)' for all `a'
and `b'.  These results will be valid only in a restricted range of
`x'; for example, in `(x^2)^1:2' the powers cancel to get `x', which is
valid for positive values of `x' but not for negative or complex values.

   Similarly, `sqrt(x^a)' and `sqrt(x)^a' are both simplified (possibly
unsafely) to `x^(a/2)'.

   Forms like `sqrt(1 - sin(x)^2)' are simplified to, e.g., `cos(x)'.
Calc has identities of this sort for `sin', `cos', `tan', `sinh', and
`cosh'.

   Arguments of square roots are partially factored to look for squared
terms that can be extracted.  For example, `sqrt(a^2 b^3 + a^3 b^2)'
simplifies to `a b sqrt(a+b)'.

   The simplifications of `ln(exp(x))', `ln(e^x)', and `log10(10^x)' to
`x' are also unsafe because of problems with principal values (although
these simplifications are safe if `x' is known to be real).

   Common factors are cancelled from products on both sides of an
equation, even if those factors may be zero:  `a x / b x' to `a / b'.
Such factors are never cancelled from inequalities:  Even `a e' is not
bold enough to reduce `a x < b x' to `a < b' (or `a > b', depending on
whether you believe `x' is positive or negative).  The `a M /' command
can be used to divide a factor out of both sides of an inequality.

File: calc,  Node: Simplification of Units,  Prev: Unsafe Simplifications,  Up: Simplifying Formulas

12.3.4 Simplification of Units
------------------------------

The simplifications described in this section are applied by the `u s'
(`calc-simplify-units') command.  These are in addition to the regular
`a s' (but not `a e') simplifications described earlier.  *Note Basic
Operations on Units::.

   The variable `UnitSimpRules' contains rewrites to be applied by the
`u s' command.  These are applied in addition to `EvalRules' and
`AlgSimpRules'.

   Scalar mode is automatically put into effect when simplifying units.
*Note Matrix Mode::.

   Sums `a + b' involving units are simplified by extracting the units
of `a' as if by the `u x' command (call the result `u_a'), then
simplifying the expression `b / u_a' using `u b' and `u s'.  If the
result has units then the sum is inconsistent and is left alone.
Otherwise, it is rewritten in terms of the units `u_a'.

   If units auto-ranging mode is enabled, products or quotients in
which the first argument is a number which is out of range for the
leading unit are modified accordingly.

   When cancelling and combining units in products and quotients, Calc
accounts for unit names that differ only in the prefix letter.  For
example, `2 km m' is simplified to `2000 m^2'.  However, compatible but
different units like `ft' and `in' are not combined in this way.

   Quotients `a / b' are simplified in three additional ways.  First,
if `b' is a number or a product beginning with a number, Calc computes
the reciprocal of this number and moves it to the numerator.

   Second, for each pair of unit names from the numerator and
denominator of a quotient, if the units are compatible (e.g., they are
both units of area) then they are replaced by the ratio between those
units.  For example, in `3 s in N / kg cm' the units `in / cm' will be
replaced by `2.54'.

   Third, if the units in the quotient exactly cancel out, so that a `u
b' command on the quotient would produce a dimensionless number for an
answer, then the quotient simplifies to that number.

   For powers and square roots, the "unsafe" simplifications `(a b)^c'
to `a^c b^c', `(a/b)^c' to `a^c / b^c', and `(a^b)^c' to `a^(b c)' are
done if the powers are real numbers.  (These are safe in the context of
units because all numbers involved can reasonably be assumed to be
real.)

   Also, if a unit name is raised to a fractional power, and the base
units in that unit name all occur to powers which are a multiple of the
denominator of the power, then the unit name is expanded out into its
base units, which can then be simplified according to the previous
paragraph.  For example, `acre^1.5' is simplified by noting that `1.5 =
3:2', that `acre' is defined in terms of `m^2', and that the 2 in the
power of `m' is a multiple of 2 in `3:2'.  Thus, `acre^1.5' is replaced
by approximately `(4046 m^2)^1.5', which is then changed to `4046^1.5
(m^2)^1.5', then to `257440 m^3'.

   The functions `float', `frac', `clean', `abs', as well as `floor'
and the other integer truncation functions, applied to unit names or
products or quotients involving units, are simplified.  For example,
`round(1.6 in)' is changed to `round(1.6) round(in)'; the lefthand term
evaluates to 2, and the righthand term simplifies to `in'.

   The functions `sin', `cos', and `tan' with arguments that have
angular units like `rad' or `arcmin' are simplified by converting to
base units (radians), then evaluating with the angular mode temporarily
set to radians.

File: calc,  Node: Polynomials,  Next: Calculus,  Prev: Simplifying Formulas,  Up: Algebra

12.4 Polynomials
================

A "polynomial" is a sum of terms which are coefficients times various
powers of a "base" variable.  For example, `2 x^2 + 3 x - 4' is a
polynomial in `x'.  Some formulas can be considered polynomials in
several different variables:  `1 + 2 x + 3 y + 4 x y^2' is a polynomial
in both `x' and `y'.  Polynomial coefficients are often numbers, but
they may in general be any formulas not involving the base variable.

   The `a f' (`calc-factor') [`factor'] command factors a polynomial
into a product of terms.  For example, the polynomial `x^3 + 2 x^2 + x'
is factored into `x*(x+1)^2'.  As another example, `a c + b d + b c + a
d' is factored into the product `(a + b) (c + d)'.

   Calc currently has three algorithms for factoring.  Formulas which
are linear in several variables, such as the second example above, are
merged according to the distributive law.  Formulas which are
polynomials in a single variable, with constant integer or fractional
coefficients, are factored into irreducible linear and/or quadratic
terms.  The first example above factors into three linear terms (`x',
`x+1', and `x+1' again).  Finally, formulas which do not fit the above
criteria are handled by the algebraic rewrite mechanism.

   Calc's polynomial factorization algorithm works by using the general
root-finding command (`a P') to solve for the roots of the polynomial.
It then looks for roots which are rational numbers or complex-conjugate
pairs, and converts these into linear and quadratic terms,
respectively.  Because it uses floating-point arithmetic, it may be
unable to find terms that involve large integers (whose number of
digits approaches the current precision).  Also, irreducible factors of
degree higher than quadratic are not found, and polynomials in more
than one variable are not treated.  (A more robust factorization
algorithm may be included in a future version of Calc.)

   The rewrite-based factorization method uses rules stored in the
variable `FactorRules'.  *Note Rewrite Rules::, for a discussion of the
operation of rewrite rules.  The default `FactorRules' are able to
factor quadratic forms symbolically into two linear terms, `(a x + b)
(c x + d)'.  You can edit these rules to include other cases if you
wish.  To use the rules, Calc builds the formula `thecoefs(x, [a, b, c,
...])' where `x' is the polynomial base variable and `a', `b', etc.,
are polynomial coefficients (which may be numbers or formulas).  The
constant term is written first, i.e., in the `a' position.  When the
rules complete, they should have changed the formula into the form
`thefactors(x, [f1, f2, f3, ...])' where each `fi' should be a factored
term, e.g., `x - ai'.  Calc then multiplies these terms together to get
the complete factored form of the polynomial.  If the rules do not
change the `thecoefs' call to a `thefactors' call, `a f' leaves the
polynomial alone on the assumption that it is unfactorable.  (Note that
the function names `thecoefs' and `thefactors' are used only as
placeholders; there are no actual Calc functions by those names.)

   The `H a f' [`factors'] command also factors a polynomial, but it
returns a list of factors instead of an expression which is the product
of the factors.  Each factor is represented by a sub-vector of the
factor, and the power with which it appears.  For example, `x^5 + x^4 -
33 x^3 + 63 x^2' factors to `(x + 7) x^2 (x - 3)^2' in `a f', or to `[
[x, 2], [x+7, 1], [x-3, 2] ]' in `H a f'.  If there is an overall
numeric factor, it always comes first in the list.  The functions
`factor' and `factors' allow a second argument when written in
algebraic form; `factor(x,v)' factors `x' with respect to the specific
variable `v'.  The default is to factor with respect to all the
variables that appear in `x'.

   The `a c' (`calc-collect') [`collect'] command rearranges a formula
as a polynomial in a given variable, ordered in decreasing powers of
that variable.  For example, given `1 + 2 x + 3 y + 4 x y^2' on the
stack, `a c x' would produce `(2 + 4 y^2) x + (1 + 3 y)', and `a c y'
would produce `(4 x) y^2 + 3 y + (1 + 2 x)'.  The polynomial will be
expanded out using the distributive law as necessary:  Collecting `x'
in `(x - 1)^3' produces `x^3 - 3 x^2 + 3 x - 1'.  Terms not involving
`x' will not be expanded.

   The "variable" you specify at the prompt can actually be any
expression: `a c ln(x+1)' will collect together all terms multiplied by
`ln(x+1)' or integer powers thereof.  If `x' also appears in the
formula in a context other than `ln(x+1)', `a c' will treat those
occurrences as unrelated to `ln(x+1)', i.e., as constants.

   The `a x' (`calc-expand') [`expand'] command expands an expression
by applying the distributive law everywhere.  It applies to products,
quotients, and powers involving sums.  By default, it fully distributes
all parts of the expression.  With a numeric prefix argument, the
distributive law is applied only the specified number of times, then
the partially expanded expression is left on the stack.

   The `a x' and `j D' commands are somewhat redundant.  Use `a x' if
you want to expand all products of sums in your formula.  Use `j D' if
you want to expand a particular specified term of the formula.  There
is an exactly analogous correspondence between `a f' and `j M'.  (The
`j D' and `j M' commands also know many other kinds of expansions, such
as `exp(a + b) = exp(a) exp(b)', which `a x' and `a f' do not do.)

   Calc's automatic simplifications will sometimes reverse a partial
expansion.  For example, the first step in expanding `(x+1)^3' is to
write `(x+1) (x+1)^2'.  If `a x' stops there and tries to put this
formula onto the stack, though, Calc will automatically simplify it
back to `(x+1)^3' form.  The solution is to turn simplification off
first (*note Simplification Modes::), or to run `a x' without a numeric
prefix argument so that it expands all the way in one step.

   The `a a' (`calc-apart') [`apart'] command expands a rational
function by partial fractions.  A rational function is the quotient of
two polynomials; `apart' pulls this apart into a sum of rational
functions with simple denominators.  In algebraic notation, the `apart'
function allows a second argument that specifies which variable to use
as the "base"; by default, Calc chooses the base variable automatically.

   The `a n' (`calc-normalize-rat') [`nrat'] command attempts to
arrange a formula into a quotient of two polynomials.  For example,
given `1 + (a + b/c) / d', the result would be `(b + a c + c d) / c d'.
The quotient is reduced, so that `a n' will simplify `(x^2 + 2x + 1) /
(x^2 - 1)' by dividing out the common factor `x + 1', yielding `(x + 1)
/ (x - 1)'.

   The `a \' (`calc-poly-div') [`pdiv'] command divides two polynomials
`u' and `v', yielding a new polynomial `q'.  If several variables occur
in the inputs, the inputs are considered multivariate polynomials.
(Calc divides by the variable with the largest power in `u' first, or,
in the case of equal powers, chooses the variables in alphabetical
order.)  For example, dividing `x^2 + 3 x + 2' by `x + 2' yields `x +
1'.  The remainder from the division, if any, is reported at the bottom
of the screen and is also placed in the Trail along with the quotient.

   Using `pdiv' in algebraic notation, you can specify the particular
variable to be used as the base: `pdiv(A,B,X)'.  If `pdiv' is given
only two arguments (as is always the case with the `a \' command), then
it does a multivariate division as outlined above.

   The `a %' (`calc-poly-rem') [`prem'] command divides two polynomials
and keeps the remainder `r'.  The quotient `q' is discarded.  For any
formulas `a' and `b', the results of `a \' and `a %' satisfy `a = q b +
r'.  (This is analogous to plain `\' and `%', which compute the integer
quotient and remainder from dividing two numbers.)

   The `a /' (`calc-poly-div-rem') [`pdivrem'] command divides two
polynomials and reports both the quotient and the remainder as a vector
`[q, r]'.  The `H a /' [`pdivide'] command divides two polynomials and
constructs the formula `q + r/b' on the stack.  (Naturally if the
remainder is zero, this will immediately simplify to `q'.)

   The `a g' (`calc-poly-gcd') [`pgcd'] command computes the greatest
common divisor of two polynomials.  (The GCD actually is unique only to
within a constant multiplier; Calc attempts to choose a GCD which will
be unsurprising.)  For example, the `a n' command uses `a g' to take
the GCD of the numerator and denominator of a quotient, then divides
each by the result using `a \'.  (The definition of GCD ensures that
this division can take place without leaving a remainder.)

   While the polynomials used in operations like `a /' and `a g' often
have integer coefficients, this is not required.  Calc can also deal
with polynomials over the rationals or floating-point reals.
Polynomials with modulo-form coefficients are also useful in many
applications; if you enter `(x^2 + 3 x - 1) mod 5', Calc automatically
transforms this into a polynomial over the field of integers mod 5:
`(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)'.

   Congratulations and thanks go to Ove Ewerlid
(`ewerlidATmizar.SE'), who contributed many of the polynomial
routines used in the above commands.

   *Note Decomposing Polynomials::, for several useful functions for
extracting the individual coefficients of a polynomial.

File: calc,  Node: Calculus,  Next: Solving Equations,  Prev: Polynomials,  Up: Algebra

12.5 Calculus
=============

The following calculus commands do not automatically simplify their
inputs or outputs using `calc-simplify'.  You may find it helps to do
this by hand by typing `a s' or `a e'.  It may also help to use `a x'
and/or `a c' to arrange a result in the most readable way.

* Menu:

* Differentiation::
* Integration::
* Customizing the Integrator::
* Numerical Integration::
* Taylor Series::

File: calc,  Node: Differentiation,  Next: Integration,  Prev: Calculus,  Up: Calculus

12.5.1 Differentiation
----------------------

The `a d' (`calc-derivative') [`deriv'] command computes the derivative
of the expression on the top of the stack with respect to some
variable, which it will prompt you to enter.  Normally, variables in
the formula other than the specified differentiation variable are
considered constant, i.e., `deriv(y,x)' is reduced to zero.  With the
Hyperbolic flag, the `tderiv' (total derivative) operation is used
instead, in which derivatives of variables are not reduced to zero
unless those variables are known to be "constant," i.e., independent of
any other variables.  (The built-in special variables like `pi' are
considered constant, as are variables that have been declared `const';
*note Declarations::.)

   With a numeric prefix argument N, this command computes the Nth
derivative.

   When working with trigonometric functions, it is best to switch to
Radians mode first (with `m r').  The derivative of `sin(x)' in degrees
is `(pi/180) cos(x)', probably not the expected answer!

   If you use the `deriv' function directly in an algebraic formula,
you can write `deriv(f,x,x0)' which represents the derivative of `f'
with respect to `x', evaluated at the point `x=x0'.

   If the formula being differentiated contains functions which Calc
does not know, the derivatives of those functions are produced by adding
primes (apostrophe characters).  For example, `deriv(f(2x), x)'
produces `2 f'(2 x)', where the function `f'' represents the derivative
of `f'.

   For functions you have defined with the `Z F' command, Calc expands
the functions according to their defining formulas unless you have also
defined `f'' suitably.  For example, suppose we define `sinc(x) =
sin(x)/x' using `Z F'.  If we then differentiate the formula `sinc(2
x)', the formula will be expanded to `sin(2 x) / (2 x)' and
differentiated.  However, if we also define `sinc'(x) = dsinc(x)', say,
then Calc will write the result as `2 dsinc(2 x)'.  *Note Algebraic
Definitions::.

   For multi-argument functions `f(x,y,z)', the derivative with respect
to the first argument is written `f'(x,y,z)'; derivatives with respect
to the other arguments are `f'2(x,y,z)' and `f'3(x,y,z)'.  Various
higher-order derivatives can be formed in the obvious way, e.g.,
`f''(x)' (the second derivative of `f') or `f''2'3(x,y,z)' (`f'
differentiated with respect to each argument once).

File: calc,  Node: Integration,  Next: Customizing the Integrator,  Prev: Differentiation,  Up: Calculus

12.5.2 Integration
------------------

The `a i' (`calc-integral') [`integ'] command computes the indefinite
integral of the expression on the top of the stack with respect to a
prompted-for variable.  The integrator is not guaranteed to work for
all integrable functions, but it is able to integrate several large
classes of formulas.  In particular, any polynomial or rational
function (a polynomial divided by a polynomial) is acceptable.
(Rational functions don't have to be in explicit quotient form, however;
`x/(1+x^-2)' is not strictly a quotient of polynomials, but it is
equivalent to `x^3/(x^2+1)', which is.)  Also, square roots of terms
involving `x' and `x^2' may appear in rational functions being
integrated.  Finally, rational functions involving trigonometric or
hyperbolic functions can be integrated.

   With an argument (`C-u a i'), this command will compute the definite
integral of the expression on top of the stack.  In this case, the
command will again prompt for an integration variable, then prompt for a
lower limit and an upper limit.

   If you use the `integ' function directly in an algebraic formula,
you can also write `integ(f,x,v)' which expresses the resulting
indefinite integral in terms of variable `v' instead of `x'.  With four
arguments, `integ(f(x),x,a,b)' represents a definite integral from `a'
to `b'.

   Please note that the current implementation of Calc's integrator
sometimes produces results that are significantly more complex than
they need to be.  For example, the integral Calc finds for
`1/(x+sqrt(x^2+1))' is several times more complicated than the answer
Mathematica returns for the same input, although the two forms are
numerically equivalent.  Also, any indefinite integral should be
considered to have an arbitrary constant of integration added to it,
although Calc does not write an explicit constant of integration in its
result.  For example, Calc's solution for `1/(1+tan(x))' differs from
the solution given in the _CRC Math Tables_ by a constant factor of `pi
i / 2', due to a different choice of constant of integration.

   The Calculator remembers all the integrals it has done.  If
conditions change in a way that would invalidate the old integrals,
say, a switch from Degrees to Radians mode, then they will be thrown
out.  If you suspect this is not happening when it should, use the
`calc-flush-caches' command; *note Caches::.

   Calc normally will pursue integration by substitution or integration
by parts up to 3 nested times before abandoning an approach as
fruitless.  If the integrator is taking too long, you can lower this
limit by storing a number (like 2) in the variable `IntegLimit'.  (The
`s I' command is a convenient way to edit `IntegLimit'.)  If this
variable has no stored value or does not contain a nonnegative integer,
a limit of 3 is used.  The lower this limit is, the greater the chance
that Calc will be unable to integrate a function it could otherwise
handle.  Raising this limit allows the Calculator to solve more
integrals, though the time it takes may grow exponentially.  You can
monitor the integrator's actions by creating an Emacs buffer called
`*Trace*'.  If such a buffer exists, the `a i' command will write a log
of its actions there.

   If you want to manipulate integrals in a purely symbolic way, you can
set the integration nesting limit to 0 to prevent all but fast
table-lookup solutions of integrals.  You might then wish to define
rewrite rules for integration by parts, various kinds of substitutions,
and so on.  *Note Rewrite Rules::.

File: calc,  Node: Customizing the Integrator,  Next: Numerical Integration,  Prev: Integration,  Up: Calculus

12.5.3 Customizing the Integrator
---------------------------------

Calc has two built-in rewrite rules called `IntegRules' and
`IntegAfterRules' which you can edit to define new integration methods.
*Note Rewrite Rules::.  At each step of the integration process, Calc
wraps the current integrand in a call to the fictitious function
`integtry(EXPR,VAR)', where EXPR is the integrand and VAR is the
integration variable.  If your rules rewrite this to be a plain formula
(not a call to `integtry'), then Calc will use this formula as the
integral of EXPR.  For example, the rule `integtry(mysin(x),x) :=
-mycos(x)' would define a rule to integrate a function `mysin' that
acts like the sine function.  Then, putting `4 mysin(2y+1)' on the
stack and typing `a i y' will produce the integral `-2 mycos(2y+1)'.
Note that Calc has automatically made various transformations on the
integral to allow it to use your rule; integral tables generally give
rules for `mysin(a x + b)', but you don't need to use this much
generality in your `IntegRules'.

   As a more serious example, the expression `exp(x)/x' cannot be
integrated in terms of the standard functions, so the "exponential
integral" function `Ei(x)' was invented to describe it.  We can get
Calc to do this integral in terms of a made-up `Ei' function by adding
the rule `[integtry(exp(x)/x, x) := Ei(x)]' to `IntegRules'.  Now
entering `exp(2x)/x' on the stack and typing `a i x' yields `Ei(2 x)'.
This new rule will work with Calc's various built-in integration
methods (such as integration by substitution) to solve a variety of
other problems involving `Ei':  For example, now Calc will also be able
to integrate `exp(exp(x))' and `ln(ln(x))' (to get `Ei(exp(x))' and `x
ln(ln(x)) - Ei(ln(x))', respectively).

   Your rule may do further integration by calling `integ'.  For
example, `integtry(twice(u),x) := twice(integ(u))' allows Calc to
integrate `twice(sin(x))' to get `twice(-cos(x))'.  Note that `integ'
was called with only one argument.  This notation is allowed only
within `IntegRules'; it means "integrate this with respect to the same
integration variable."  If Calc is unable to integrate `u', the
integration that invoked `IntegRules' also fails.  Thus integrating
`twice(f(x))' fails, returning the unevaluated integral
`integ(twice(f(x)), x)'.  It is still valid to call `integ' with two or
more arguments, however; in this case, if `u' is not integrable,
`twice' itself will still be integrated:  If the above rule is changed
to `... := twice(integ(u,x))', then integrating `twice(f(x))' will
yield `twice(integ(f(x),x))'.

   If a rule instead produces the formula `integsubst(SEXPR, SVAR)',
either replacing the top-level `integtry' call or nested anywhere
inside the expression, then Calc will apply the substitution `U =
SEXPR(SVAR)' to try to integrate the original EXPR.  For example, the
rule `sqrt(a) := integsubst(sqrt(x),x)' says that if Calc ever finds a
square root in the integrand, it should attempt the substitution `u =
sqrt(x)'.  (This particular rule is unnecessary because Calc always
tries "obvious" substitutions where SEXPR actually appears in the
integrand.)  The variable SVAR may be the same as the VAR that appeared
in the call to `integtry', but it need not be.

   When integrating according to an `integsubst', Calc uses the
equation solver to find the inverse of SEXPR (if the integrand refers
to VAR anywhere except in subexpressions that exactly match SEXPR).  It
uses the differentiator to find the derivative of SEXPR and/or its
inverse (it has two methods that use one derivative or the other).  You
can also specify these items by adding extra arguments to the
`integsubst' your rules construct; the general form is
`integsubst(SEXPR, SVAR, SINV, SPRIME)', where SINV is the inverse of
SEXPR (still written as a function of SVAR), and SPRIME is the
derivative of SEXPR with respect to SVAR.  If you don't specify these
things, and Calc is not able to work them out on its own with the
information it knows, then your substitution rule will work only in
very specific, simple cases.

   Calc applies `IntegRules' as if by `C-u 1 a r IntegRules'; in other
words, Calc stops rewriting as soon as any rule in your rule set
succeeds.  (If it weren't for this, the `integsubst(sqrt(x),x)' example
above would keep on adding layers of `integsubst' calls forever!)

   Another set of rules, stored in `IntegSimpRules', are applied every
time the integrator uses `a s' to simplify an intermediate result.  For
example, putting the rule `twice(x) := 2 x' into `IntegSimpRules' would
tell Calc to convert the `twice' function into a form it knows whenever
integration is attempted.

   One more way to influence the integrator is to define a function with
the `Z F' command (*note Algebraic Definitions::).  Calc's integrator
automatically expands such functions according to their defining
formulas, even if you originally asked for the function to be left
unevaluated for symbolic arguments.  (Certain other Calc systems, such
as the differentiator and the equation solver, also do this.)

   Sometimes Calc is able to find a solution to your integral, but it
expresses the result in a way that is unnecessarily complicated.  If
this happens, you can either use `integsubst' as described above to try
to hint at a more direct path to the desired result, or you can use
`IntegAfterRules'.  This is an extra rule set that runs after the main
integrator returns its result; basically, Calc does an `a r
IntegAfterRules' on the result before showing it to you.  (It also does
an `a s', without `IntegSimpRules', after that to further simplify the
result.)  For example, Calc's integrator sometimes produces expressions
of the form `ln(1+x) - ln(1-x)'; the default `IntegAfterRules' rewrite
this into the more readable form `2 arctanh(x)'.  Note that, unlike
`IntegRules', `IntegSimpRules' and `IntegAfterRules' are applied any
number of times until no further changes are possible.  Rewriting by
`IntegAfterRules' occurs only after the main integrator has finished,
not at every step as for `IntegRules' and `IntegSimpRules'.

File: calc,  Node: Numerical Integration,  Next: Taylor Series,  Prev: Customizing the Integrator,  Up: Calculus

12.5.4 Numerical Integration
----------------------------

If you want a purely numerical answer to an integration problem, you can
use the `a I' (`calc-num-integral') [`ninteg'] command.  This command
prompts for an integration variable, a lower limit, and an upper limit.
Except for the integration variable, all other variables that appear in
the integrand formula must have stored values.  (A stored value, if
any, for the integration variable itself is ignored.)

   Numerical integration works by evaluating your formula at many
points in the specified interval.  Calc uses an "open Romberg" method;
this means that it does not evaluate the formula actually at the
endpoints (so that it is safe to integrate `sin(x)/x' from zero, for
example).  Also, the Romberg method works especially well when the
function being integrated is fairly smooth.  If the function is not
smooth, Calc will have to evaluate it at quite a few points before it
can accurately determine the value of the integral.

   Integration is much faster when the current precision is small.  It
is best to set the precision to the smallest acceptable number of digits
before you use `a I'.  If Calc appears to be taking too long, press
`C-g' to halt it and try a lower precision.  If Calc still appears to
need hundreds of evaluations, check to make sure your function is
well-behaved in the specified interval.

   It is possible for the lower integration limit to be `-inf' (minus
infinity).  Likewise, the upper limit may be plus infinity.  Calc
internally transforms the integral into an equivalent one with finite
limits.  However, integration to or across singularities is not
supported: The integral of `1/sqrt(x)' from 0 to 1 exists (it can be
found by Calc's symbolic integrator, for example), but `a I' will fail
because the integrand goes to infinity at one of the endpoints.

File: calc,  Node: Taylor Series,  Prev: Numerical Integration,  Up: Calculus

12.5.5 Taylor Series
--------------------

The `a t' (`calc-taylor') [`taylor'] command computes a power series
expansion or Taylor series of a function.  You specify the variable and
the desired number of terms.  You may give an expression of the form
`VAR = A' or `VAR - A' instead of just a variable to produce a Taylor
expansion about the point A.  You may specify the number of terms with
a numeric prefix argument; otherwise the command will prompt you for
the number of terms.  Note that many series expansions have
coefficients of zero for some terms, so you may appear to get fewer
terms than you asked for.

   If the `a i' command is unable to find a symbolic integral for a
function, you can get an approximation by integrating the function's
Taylor series.

File: calc,  Node: Solving Equations,  Next: Numerical Solutions,  Prev: Calculus,  Up: Algebra

12.6 Solving Equations
======================

The `a S' (`calc-solve-for') [`solve'] command rearranges an equation
to solve for a specific variable.  An equation is an expression of the
form `L = R'.  For example, the command `a S x' will rearrange `y = 3x
+ 6' to the form, `x = y/3 - 2'.  If the input is not an equation, it
is treated like an equation of the form `X = 0'.

   This command also works for inequalities, as in `y < 3x + 6'.  Some
inequalities cannot be solved where the analogous equation could be;
for example, solving `a < b c' for `b' is impossible without knowing
the sign of `c'.  In this case, `a S' will produce the result `b != a/c'
(using the not-equal-to operator) to signify that the direction of the
inequality is now unknown.  The inequality `a <= b c' is not even
partially solved.  *Note Declarations::, for a way to tell Calc that
the signs of the variables in a formula are in fact known.

   Two useful commands for working with the result of `a S' are `a .'
(*note Logical Operations::), which converts `x = y/3 - 2' to `y/3 -
2', and `s l' (*note Let Command::) which evaluates another formula
with `x' set equal to `y/3 - 2'.

* Menu:

* Multiple Solutions::
* Solving Systems of Equations::
* Decomposing Polynomials::

File: calc,  Node: Multiple Solutions,  Next: Solving Systems of Equations,  Prev: Solving Equations,  Up: Solving Equations

12.6.1 Multiple Solutions
-------------------------

Some equations have more than one solution.  The Hyperbolic flag (`H a
S') [`fsolve'] tells the solver to report the fully general family of
solutions.  It will invent variables `n1', `n2', ..., which represent
independent arbitrary integers, and `s1', `s2', ..., which represent
independent arbitrary signs (either +1 or -1).  If you don't use the
Hyperbolic flag, Calc will use zero in place of all arbitrary integers,
and plus one in place of all arbitrary signs.  Note that variables like
`n1' and `s1' are not given any special interpretation in Calc except by
the equation solver itself.  As usual, you can use the `s l'
(`calc-let') command to obtain solutions for various actual values of
these variables.

   For example, `' x^2 = y <RET> H a S x <RET>' solves to get `x = s1
sqrt(y)', indicating that the two solutions to the equation are
`sqrt(y)' and `-sqrt(y)'.  Another way to think about it is that the
square-root operation is really a two-valued function; since every Calc
function must return a single result, `sqrt' chooses to return the
positive result.  Then `H a S' doctors this result using `s1' to
indicate the full set of possible values of the mathematical
square-root.

   There is a similar phenomenon going the other direction:  Suppose we
solve `sqrt(y) = x' for `y'.  Calc squares both sides to get `y = x^2'.
This is correct, except that it introduces some dubious solutions.
Consider solving `sqrt(y) = -3': Calc will report `y = 9' as a valid
solution, which is true in the mathematical sense of square-root, but
false (there is no solution) for the actual Calc positive-valued
`sqrt'.  This happens for both `a S' and `H a S'.

   If you store a positive integer in the Calc variable `GenCount',
then Calc will generate formulas of the form `as(N)' for arbitrary
signs, and `an(N)' for arbitrary integers, where N represents
successive values taken by incrementing `GenCount' by one.  While the
normal arbitrary sign and integer symbols start over at `s1' and `n1'
with each new Calc command, the `GenCount' approach will give each
arbitrary value a name that is unique throughout the entire Calc
session.  Also, the arbitrary values are function calls instead of
variables, which is advantageous in some cases.  For example, you can
make a rewrite rule that recognizes all arbitrary signs using a pattern
like `as(n)'.  The `s l' command only works on variables, but you can
use the `a b' (`calc-substitute') command to substitute actual values
for function calls like `as(3)'.

   The `s G' (`calc-edit-GenCount') command is a convenient way to
create or edit this variable.  Press `C-c C-c' to finish.

   If you have not stored a value in `GenCount', or if the value in
that variable is not a positive integer, the regular `s1'/`n1' notation
is used.

   With the Inverse flag, `I a S' [`finv'] treats the expression on top
of the stack as a function of the specified variable and solves to find
the inverse function, written in terms of the same variable.  For
example, `I a S x' inverts `2x + 6' to `x/2 - 3'.  You can use both
Inverse and Hyperbolic [`ffinv'] to obtain a fully general inverse, as
described above.

   Some equations, specifically polynomials, have a known, finite number
of solutions.  The `a P' (`calc-poly-roots') [`roots'] command uses `H
a S' to solve an equation in general form, then, for all arbitrary-sign
variables like `s1', and all arbitrary-integer variables like `n1' for
which `n1' only usefully varies over a finite range, it expands these
variables out to all their possible values.  The results are collected
into a vector, which is returned.  For example, `roots(x^4 = 1, x)'
returns the four solutions `[1, -1, (0, 1), (0, -1)]'.  Generally an
Nth degree polynomial will always have N roots on the complex plane.
(If you have given a `real' declaration for the solution variable, then
only the real-valued solutions, if any, will be reported; *note
Declarations::.)

   Note that because `a P' uses `H a S', it is able to deliver symbolic
solutions if the polynomial has symbolic coefficients.  Also note that
Calc's solver is not able to get exact symbolic solutions to all
polynomials.  Polynomials containing powers up to `x^4' can always be
solved exactly; polynomials of higher degree sometimes can be:  `x^6 +
x^3 + 1' is converted to `(x^3)^2 + (x^3) + 1', which can be solved for
`x^3' using the quadratic equation, and then for `x' by taking cube
roots.  But in many cases, like `x^6 + x + 1', Calc does not know how
to rewrite the polynomial into a form it can solve.  The `a P' command
can still deliver a list of numerical roots, however, provided that
Symbolic mode (`m s') is not turned on.  (If you work with Symbolic
mode on, recall that the `N' (`calc-eval-num') key is a handy way to
reevaluate the formula on the stack with Symbolic mode temporarily
off.)  Naturally, `a P' can only provide numerical roots if the
polynomial coefficients are all numbers (real or complex).

File: calc,  Node: Solving Systems of Equations,  Next: Decomposing Polynomials,  Prev: Multiple Solutions,  Up: Solving Equations

12.6.2 Solving Systems of Equations
-----------------------------------

You can also use the commands described above to solve systems of
simultaneous equations.  Just create a vector of equations, then
specify a vector of variables for which to solve.  (You can omit the
surrounding brackets when entering the vector of variables at the
prompt.)

   For example, putting `[x + y = a, x - y = b]' on the stack and
typing `a S x,y <RET>' produces the vector of solutions `[x = a -
(a-b)/2, y = (a-b)/2]'.  The result vector will have the same length as
the variables vector, and the variables will be listed in the same
order there.  Note that the solutions are not always simplified as far
as possible; the solution for `x' here could be improved by an
application of the `a n' command.

   Calc's algorithm works by trying to eliminate one variable at a time
by solving one of the equations for that variable and then substituting
into the other equations.  Calc will try all the possibilities, but you
can speed things up by noting that Calc first tries to eliminate the
first variable with the first equation, then the second variable with
the second equation, and so on.  It also helps to put the simpler
(e.g., more linear) equations toward the front of the list.  Calc's
algorithm will solve any system of linear equations, and also many
kinds of nonlinear systems.

   Normally there will be as many variables as equations.  If you give
fewer variables than equations (an "over-determined" system of
equations), Calc will find a partial solution.  For example, typing `a
S y <RET>' with the above system of equations would produce `[y = a -
x]'.  There are now several ways to express this solution in terms of
the original variables; Calc uses the first one that it finds.  You can
control the choice by adding variable specifiers of the form `elim(V)'
to the variables list.  This says that V should be eliminated from the
equations; the variable will not appear at all in the solution.  For
example, typing `a S y,elim(x)' would yield `[y = a - (b+a)/2]'.

   If the variables list contains only `elim' specifiers, Calc simply
eliminates those variables from the equations and then returns the
resulting set of equations.  For example, `a S elim(x)' produces `[a -
2 y = b]'.  Every variable eliminated will reduce the number of
equations in the system by one.

   Again, `a S' gives you one solution to the system of equations.  If
there are several solutions, you can use `H a S' to get a general
family of solutions, or, if there is a finite number of solutions, you
can use `a P' to get a list.  (In the latter case, the result will take
the form of a matrix where the rows are different solutions and the
columns correspond to the variables you requested.)

   Another way to deal with certain kinds of overdetermined systems of
equations is the `a F' command, which does least-squares fitting to
satisfy the equations.  *Note Curve Fitting::.

File: calc,  Node: Decomposing Polynomials,  Prev: Solving Systems of Equations,  Up: Solving Equations

12.6.3 Decomposing Polynomials
------------------------------

The `poly' function takes a polynomial and a variable as arguments, and
returns a vector of polynomial coefficients (constant coefficient
first).  For example, `poly(x^3 + 2 x, x)' returns `[0, 2, 0, 1]'.  If
the input is not a polynomial in `x', the call to `poly' is left in
symbolic form.  If the input does not involve the variable `x', the
input is returned in a list of length one, representing a polynomial
with only a constant coefficient.  The call `poly(x, x)' returns the
vector `[0, 1]'.  The last element of the returned vector is guaranteed
to be nonzero; note that `poly(0, x)' returns the empty vector `[]'.
Note also that `x' may actually be any formula; for example,
`poly(sin(x)^2 - sin(x) + 3, sin(x))' returns `[3, -1, 1]'.

   To get the `x^k' coefficient of polynomial `p', use `poly(p,
x)_(k+1)'.  To get the degree of polynomial `p', use `vlen(poly(p, x))
- 1'.  For example, `poly((x+1)^4, x)' returns `[1, 4, 6, 4, 1]', so
`poly((x+1)^4, x)_(2+1)' gives the `x^2' coefficient of this
polynomial, 6.

   One important feature of the solver is its ability to recognize
formulas which are "essentially" polynomials.  This ability is made
available to the user through the `gpoly' function, which is used just
like `poly':  `gpoly(EXPR, VAR)'.  If EXPR is a polynomial in some term
which includes VAR, then this function will return a vector `[X, C, A]'
where X is the term that depends on VAR, C is a vector of polynomial
coefficients (like the one returned by `poly'), and A is a multiplier
which is usually 1.  Basically, `EXPR = A*(C_1 + C_2 X + C_3 X^2 +
...)'.  The last element of C is guaranteed to be non-zero, and C will
not equal `[1]' (i.e., the trivial decomposition EXPR = X is not
considered a polynomial).  One side effect is that `gpoly(x, x)' and
`gpoly(6, x)', both of which might be expected to recognize their
arguments as polynomials, will not because the decomposition is
considered trivial.

   For example, `gpoly((x-2)^2, x)' returns `[x, [4, -4, 1], 1]', since
the expanded form of this polynomial is `4 - 4 x + x^2'.

   The term X may itself be a polynomial in VAR.  This is done to
reduce the size of the C vector.  For example, `gpoly(x^4 + x^2 - 1,
x)' returns `[x^2, [-1, 1, 1], 1]', since a quadratic polynomial in
`x^2' is easier to solve than a quartic polynomial in `x'.

   A few more examples of the kinds of polynomials `gpoly' can discover:

     sin(x) - 1               [sin(x), [-1, 1], 1]
     x + 1/x - 1              [x, [1, -1, 1], 1/x]
     x + 1/x                  [x^2, [1, 1], 1/x]
     x^3 + 2 x                [x^2, [2, 1], x]
     x + x^2:3 + sqrt(x)      [x^1:6, [1, 1, 0, 1], x^1:2]
     x^(2a) + 2 x^a + 5       [x^a, [5, 2, 1], 1]
     (exp(-x) + exp(x)) / 2   [e^(2 x), [0.5, 0.5], e^-x]

   The `poly' and `gpoly' functions accept a third integer argument
which specifies the largest degree of polynomial that is acceptable.
If this is `n', then only C vectors of length `n+1' or less will be
returned.  Otherwise, the `poly' or `gpoly' call will remain in
symbolic form.  For example, the equation solver can handle quartics
and smaller polynomials, so it calls `gpoly(EXPR, VAR, 4)' to discover
whether EXPR can be treated by its linear, quadratic, cubic, or quartic
formulas.

   The `pdeg' function computes the degree of a polynomial; `pdeg(p,x)'
is the highest power of `x' that appears in `p'.  This is the same as
`vlen(poly(p,x))-1', but is much more efficient.  If `p' is constant
with respect to `x', then `pdeg(p,x) = 0'.  If `p' is not a polynomial
in `x' (e.g., `pdeg(2 cos(x), x)', the function remains unevaluated.
It is possible to omit the second argument `x', in which case `pdeg(p)'
returns the highest total degree of any term of the polynomial,
counting all variables that appear in `p'.  Note that `pdeg(c) =
pdeg(c,x) = 0' for any nonzero constant `c'; the degree of the constant
zero is considered to be `-inf' (minus infinity).

   The `plead' function finds the leading term of a polynomial.  Thus
`plead(p,x)' is equivalent to `poly(p,x)_vlen(poly(p,x))', though again
more efficient.  In particular, `plead((2x+1)^10, x)' returns 1024
without expanding out the list of coefficients.  The value of
`plead(p,x)' will be zero only if `p = 0'.

   The `pcont' function finds the "content" of a polynomial.  This is
the greatest common divisor of all the coefficients of the polynomial.
With two arguments, `pcont(p,x)' effectively uses `poly(p,x)' to get a
list of coefficients, then uses `pgcd' (the polynomial GCD function) to
combine these into an answer.  For example, `pcont(4 x y^2 + 6 x^2 y,
x)' is `2 y'.  The content is basically the "biggest" polynomial that
can be divided into `p' exactly.  The sign of the content is the same
as the sign of the leading coefficient.

   With only one argument, `pcont(p)' computes the numerical content of
the polynomial, i.e., the `gcd' of the numerical coefficients of all
the terms in the formula.  Note that `gcd' is defined on rational
numbers as well as integers; it computes the `gcd' of the numerators
and the `lcm' of the denominators.  Thus `pcont(4:3 x y^2 + 6 x^2 y)'
returns 2:3.  Dividing the polynomial by this number will clear all the
denominators, as well as dividing by any common content in the
numerators.  The numerical content of a polynomial is negative only if
all the coefficients in the polynomial are negative.

   The `pprim' function finds the "primitive part" of a polynomial,
which is simply the polynomial divided (using `pdiv' if necessary) by
its content.  If the input polynomial has rational coefficients, the
result will have integer coefficients in simplest terms.

File: calc,  Node: Numerical Solutions,  Next: Curve Fitting,  Prev: Solving Equations,  Up: Algebra

12.7 Numerical Solutions
========================

Not all equations can be solved symbolically.  The commands in this
section use numerical algorithms that can find a solution to a specific
instance of an equation to any desired accuracy.  Note that the
numerical commands are slower than their algebraic cousins; it is a
good idea to try `a S' before resorting to these commands.

   (*Note Curve Fitting::, for some other, more specialized, operations
on numerical data.)

* Menu:

* Root Finding::
* Minimization::
* Numerical Systems of Equations::

File: calc,  Node: Root Finding,  Next: Minimization,  Prev: Numerical Solutions,  Up: Numerical Solutions

12.7.1 Root Finding
-------------------

The `a R' (`calc-find-root') [`root'] command finds a numerical
solution (or "root") of an equation.  (This command treats inequalities
the same as equations.  If the input is any other kind of formula, it
is interpreted as an equation of the form `X = 0'.)

   The `a R' command requires an initial guess on the top of the stack,
and a formula in the second-to-top position.  It prompts for a solution
variable, which must appear in the formula.  All other variables that
appear in the formula must have assigned values, i.e., when a value is
assigned to the solution variable and the formula is evaluated with
`=', it should evaluate to a number.  Any assigned value for the
solution variable itself is ignored and unaffected by this command.

   When the command completes, the initial guess is replaced on the
stack by a vector of two numbers:  The value of the solution variable
that solves the equation, and the difference between the lefthand and
righthand sides of the equation at that value.  Ordinarily, the second
number will be zero or very nearly zero.  (Note that Calc uses a
slightly higher precision while finding the root, and thus the second
number may be slightly different from the value you would compute from
the equation yourself.)

   The `v h' (`calc-head') command is a handy way to extract the first
element of the result vector, discarding the error term.

   The initial guess can be a real number, in which case Calc searches
for a real solution near that number, or a complex number, in which
case Calc searches the whole complex plane near that number for a
solution, or it can be an interval form which restricts the search to
real numbers inside that interval.

   Calc tries to use `a d' to take the derivative of the equation.  If
this succeeds, it uses Newton's method.  If the equation is not
differentiable Calc uses a bisection method.  (If Newton's method
appears to be going astray, Calc switches over to bisection if it can,
or otherwise gives up.  In this case it may help to try again with a
slightly different initial guess.)  If the initial guess is a complex
number, the function must be differentiable.

   If the formula (or the difference between the sides of an equation)
is negative at one end of the interval you specify and positive at the
other end, the root finder is guaranteed to find a root.  Otherwise,
Calc subdivides the interval into small parts looking for positive and
negative values to bracket the root.  When your guess is an interval,
Calc will not look outside that interval for a root.

   The `H a R' [`wroot'] command is similar to `a R', except that if
the initial guess is an interval for which the function has the same
sign at both ends, then rather than subdividing the interval Calc
attempts to widen it to enclose a root.  Use this mode if you are not
sure if the function has a root in your interval.

   If the function is not differentiable, and you give a simple number
instead of an interval as your initial guess, Calc uses this widening
process even if you did not type the Hyperbolic flag.  (If the function
_is_ differentiable, Calc uses Newton's method which does not require a
bounding interval in order to work.)

   If Calc leaves the `root' or `wroot' function in symbolic form on
the stack, it will normally display an explanation for why no root was
found.  If you miss this explanation, press `w' (`calc-why') to get it
back.

File: calc,  Node: Minimization,  Next: Numerical Systems of Equations,  Prev: Root Finding,  Up: Numerical Solutions

12.7.2 Minimization
-------------------

The `a N' (`calc-find-minimum') [`minimize'] command finds a minimum
value for a formula.  It is very similar in operation to `a R'
(`calc-find-root'):  You give the formula and an initial guess on the
stack, and are prompted for the name of a variable.  The guess may be
either a number near the desired minimum, or an interval enclosing the
desired minimum.  The function returns a vector containing the value of
the variable which minimizes the formula's value, along with the
minimum value itself.

   Note that this command looks for a _local_ minimum.  Many functions
have more than one minimum; some, like `x sin(x)', have infinitely
many.  In fact, there is no easy way to define the "global" minimum of
`x sin(x)' but Calc can still locate any particular local minimum for
you.  Calc basically goes downhill from the initial guess until it
finds a point at which the function's value is greater both to the left
and to the right.  Calc does not use derivatives when minimizing a
function.

   If your initial guess is an interval and it looks like the minimum
occurs at one or the other endpoint of the interval, Calc will return
that endpoint only if that endpoint is closed; thus, minimizing `17 x'
over `[2..3]' will return `[2, 38]', but minimizing over `(2..3]' would
report no minimum found.  In general, you should use closed intervals
to find literally the minimum value in that range of `x', or open
intervals to find the local minimum, if any, that happens to lie in
that range.

   Most functions are smooth and flat near their minimum values.
Because of this flatness, if the current precision is, say, 12 digits,
the variable can only be determined meaningfully to about six digits.
Thus you should set the precision to twice as many digits as you need
in your answer.

   The `H a N' [`wminimize'] command, analogously to `H a R', expands
the guess interval to enclose a minimum rather than requiring that the
minimum lie inside the interval you supply.

   The `a X' (`calc-find-maximum') [`maximize'] and `H a X'
[`wmaximize'] commands effectively minimize the negative of the formula
you supply.

   The formula must evaluate to a real number at all points inside the
interval (or near the initial guess if the guess is a number).  If the
initial guess is a complex number the variable will be minimized over
the complex numbers; if it is real or an interval it will be minimized
over the reals.

File: calc,  Node: Numerical Systems of Equations,  Prev: Minimization,  Up: Numerical Solutions

12.7.3 Systems of Equations
---------------------------

The `a R' command can also solve systems of equations.  In this case,
the equation should instead be a vector of equations, the guess should
instead be a vector of numbers (intervals are not supported), and the
variable should be a vector of variables.  You can omit the brackets
while entering the list of variables.  Each equation must be
differentiable by each variable for this mode to work.  The result will
be a vector of two vectors:  The variable values that solved the system
of equations, and the differences between the sides of the equations
with those variable values.  There must be the same number of equations
as variables.  Since only plain numbers are allowed as guesses, the
Hyperbolic flag has no effect when solving a system of equations.

   It is also possible to minimize over many variables with `a N' (or
maximize with `a X').  Once again the variable name should be replaced
by a vector of variables, and the initial guess should be an
equal-sized vector of initial guesses.  But, unlike the case of
multidimensional `a R', the formula being minimized should still be a
single formula, _not_ a vector.  Beware that multidimensional
minimization is currently _very_ slow.

File: calc,  Node: Curve Fitting,  Next: Summations,  Prev: Numerical Solutions,  Up: Algebra

12.8 Curve Fitting
==================

The `a F' command fits a set of data to a "model formula", such as `y =
m x + b' where `m' and `b' are parameters to be determined.  For a
typical set of measured data there will be no single `m' and `b' that
exactly fit the data; in this case, Calc chooses values of the
parameters that provide the closest possible fit.  The model formula
can be entered in various ways after the key sequence `a F' is pressed.

   If the letter `P' is pressed after `a F' but before the model
description is entered, the data as well as the model formula will be
plotted after the formula is determined.  This will be indicated by a
"P" in the minibuffer after the help message.

* Menu:

* Linear Fits::
* Polynomial and Multilinear Fits::
* Error Estimates for Fits::
* Standard Nonlinear Models::
* Curve Fitting Details::
* Interpolation::

File: calc,  Node: Linear Fits,  Next: Polynomial and Multilinear Fits,  Prev: Curve Fitting,  Up: Curve Fitting

12.8.1 Linear Fits
------------------

The `a F' (`calc-curve-fit') [`fit'] command attempts to fit a set of
data (`x' and `y' vectors of numbers) to a straight line, polynomial,
or other function of `x'.  For the moment we will consider only the
case of fitting to a line, and we will ignore the issue of whether or
not the model was in fact a good fit for the data.

   In a standard linear least-squares fit, we have a set of `(x,y)'
data points that we wish to fit to the model `y = m x + b' by adjusting
the parameters `m' and `b' to make the `y' values calculated from the
formula be as close as possible to the actual `y' values in the data
set.  (In a polynomial fit, the model is instead, say, `y = a x^3 + b
x^2 + c x + d'.  In a multilinear fit, we have data points of the form
`(x_1,x_2,x_3,y)' and our model is `y = a x_1 + b x_2 + c x_3 + d'.
These will be discussed later.)

   In the model formula, variables like `x' and `x_2' are called the
"independent variables", and `y' is the "dependent variable".
Variables like `m', `a', and `b' are called the "parameters" of the
model.

   The `a F' command takes the data set to be fitted from the stack.
By default, it expects the data in the form of a matrix.  For example,
for a linear or polynomial fit, this would be a 2xN matrix where the
first row is a list of `x' values and the second row has the
corresponding `y' values.  For the multilinear fit shown above, the
matrix would have four rows (`x_1', `x_2', `x_3', and `y',
respectively).

   If you happen to have an Nx2 matrix instead of a 2xN matrix, just
press `v t' first to transpose the matrix.

   After you type `a F', Calc prompts you to select a model.  For a
linear fit, press the digit `1'.

   Calc then prompts for you to name the variables.  By default it
chooses high letters like `x' and `y' for independent variables and low
letters like `a' and `b' for parameters.  (The dependent variable
doesn't need a name.)  The two kinds of variables are separated by a
semicolon.  Since you generally care more about the names of the
independent variables than of the parameters, Calc also allows you to
name only those and let the parameters use default names.

   For example, suppose the data matrix

     [ [ 1, 2, 3, 4,  5  ]
       [ 5, 7, 9, 11, 13 ] ]

is on the stack and we wish to do a simple linear fit.  Type `a F',
then `1' for the model, then <RET> to use the default names.  The
result will be the formula `3. + 2. x' on the stack.  Calc has created
the model expression `a + b x', then found the optimal values of `a'
and `b' to fit the data.  (In this case, it was able to find an exact
fit.)  Calc then substituted those values for `a' and `b' in the model
formula.

   The `a F' command puts two entries in the trail.  One is, as always,
a copy of the result that went to the stack; the other is a vector of
the actual parameter values, written as equations: `[a = 3, b = 2]', in
case you'd rather read them in a list than pick them out of the
formula.  (You can type `t y' to move this vector to the stack; see
*note Trail Commands::.

   Specifying a different independent variable name will affect the
resulting formula: `a F 1 k <RET>' produces `3 + 2 k'.  Changing the
parameter names (say, `a F 1 k;b,m <RET>') will affect the equations
that go into the trail.

   To see what happens when the fit is not exact, we could change the
number 13 in the data matrix to 14 and try the fit again.  The result
is:

     2.6 + 2.2 x

   Evaluating this formula, say with `v x 5 <RET> <TAB> V M $ <RET>',
shows a reasonably close match to the y-values in the data.

     [4.8, 7., 9.2, 11.4, 13.6]

   Since there is no line which passes through all the N data points,
Calc has chosen a line that best approximates the data points using the
method of least squares.  The idea is to define the "chi-square" error
measure

     chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)

which is clearly zero if `a + b x' exactly fits all data points, and
increases as various `a + b x_i' values fail to match the corresponding
`y_i' values.  There are several reasons why the summand is squared,
one of them being to ensure that `chi^2 >= 0'.  Least-squares fitting
simply chooses the values of `a' and `b' for which the error `chi^2' is
as small as possible.

   Other kinds of models do the same thing but with a different model
formula in place of `a + b x_i'.

   A numeric prefix argument causes the `a F' command to take the data
in some other form than one big matrix.  A positive argument N will
take N items from the stack, corresponding to the N rows of a data
matrix.  In the linear case, N must be 2 since there is always one
independent variable and one dependent variable.

   A prefix of zero or plain `C-u' is a compromise; Calc takes two
items from the stack, an N-row matrix of `x' values, and a vector of
`y' values.  If there is only one independent variable, the `x' values
can be either a one-row matrix or a plain vector, in which case the
`C-u' prefix is the same as a `C-u 2' prefix.

File: calc,  Node: Polynomial and Multilinear Fits,  Next: Error Estimates for Fits,  Prev: Linear Fits,  Up: Curve Fitting

12.8.2 Polynomial and Multilinear Fits
--------------------------------------

To fit the data to higher-order polynomials, just type one of the
digits `2' through `9' when prompted for a model.  For example, we
could fit the original data matrix from the previous section (with 13,
not 14) to a parabola instead of a line by typing `a F 2 <RET>'.

     2.00000000001 x - 1.5e-12 x^2 + 2.99999999999

   Note that since the constant and linear terms are enough to fit the
data exactly, it's no surprise that Calc chose a tiny contribution for
`x^2'.  (The fact that it's not exactly zero is due only to roundoff
error.  Since our data are exact integers, we could get an exact answer
by typing `m f' first to get Fraction mode.  Then the `x^2' term would
vanish altogether.  Usually, though, the data being fitted will be
approximate floats so Fraction mode won't help.)

   Doing the `a F 2' fit on the data set with 14 instead of 13 gives a
much larger `x^2' contribution, as Calc bends the line slightly to
improve the fit.

     0.142857142855 x^2 + 1.34285714287 x + 3.59999999998

   An important result from the theory of polynomial fitting is that it
is always possible to fit N data points exactly using a polynomial of
degree N-1, sometimes called an "interpolating polynomial".  Using the
modified (14) data matrix, a model number of 4 gives a polynomial that
exactly matches all five data points:

     0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.

   The actual coefficients we get with a precision of 12, like
`0.0416666663588', clearly suffer from loss of precision.  It is a good
idea to increase the working precision to several digits beyond what
you need when you do a fitting operation.  Or, if your data are exact,
use Fraction mode to get exact results.

   You can type `i' instead of a digit at the model prompt to fit the
data exactly to a polynomial.  This just counts the number of columns
of the data matrix to choose the degree of the polynomial automatically.

   Fitting data "exactly" to high-degree polynomials is not always a
good idea, though.  High-degree polynomials have a tendency to wiggle
uncontrollably in between the fitting data points.  Also, if the
exact-fit polynomial is going to be used to interpolate or extrapolate
the data, it is numerically better to use the `a p' command described
below.  *Note Interpolation::.

   Another generalization of the linear model is to assume the `y'
values are a sum of linear contributions from several `x' values.  This
is a "multilinear" fit, and it is also selected by the `1' digit key.
(Calc decides whether the fit is linear or multilinear by counting the
rows in the data matrix.)

   Given the data matrix,

     [ [  1,   2,   3,    4,   5  ]
       [  7,   2,   3,    5,   2  ]
       [ 14.5, 15, 18.5, 22.5, 24 ] ]

the command `a F 1 <RET>' will call the first row `x' and the second
row `y', and will fit the values in the third row to the model `a + b x
+ c y'.

     8. + 3. x + 0.5 y

   Calc can do multilinear fits with any number of independent variables
(i.e., with any number of data rows).

   Yet another variation is "homogeneous" linear models, in which the
constant term is known to be zero.  In the linear case, this means the
model formula is simply `a x'; in the multilinear case, the model might
be `a x + b y + c z'; and in the polynomial case, the model could be `a
x + b x^2 + c x^3'.  You can get a homogeneous linear or multilinear
model by pressing the letter `h' followed by a regular model key, like
`1' or `2'.  This will be indicated by an "h" in the minibuffer after
the help message.

   It is certainly possible to have other constrained linear models,
like `2.3 + a x' or `a - 4 x'.  While there is no single key to select
models like these, a later section shows how to enter any desired model
by hand.  In the first case, for example, you would enter `a F ' 2.3 +
a x'.

   Another class of models that will work but must be entered by hand
are multinomial fits, e.g., `a + b x + c y + d x^2 + e y^2 + f x y'.

File: calc,  Node: Error Estimates for Fits,  Next: Standard Nonlinear Models,  Prev: Polynomial and Multilinear Fits,  Up: Curve Fitting

12.8.3 Error Estimates for Fits
-------------------------------

With the Hyperbolic flag, `H a F' [`efit'] performs the same fitting
operation as `a F', but reports the coefficients as error forms instead
of plain numbers.  Fitting our two data matrices (first with 13, then
with 14) to a line with `H a F' gives the results,

     3. + 2. x
     2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x

   In the first case the estimated errors are zero because the linear
fit is perfect.  In the second case, the errors are nonzero but
moderately small, because the data are still very close to linear.

   It is also possible for the _input_ to a fitting operation to
contain error forms.  The data values must either all include errors or
all be plain numbers.  Error forms can go anywhere but generally go on
the numbers in the last row of the data matrix.  If the last row
contains error forms `Y_I +/- SIGMA_I', then the `chi^2' statistic is
now,

     chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)

so that data points with larger error estimates contribute less to the
fitting operation.

   If there are error forms on other rows of the data matrix, all the
errors for a given data point are combined; the square root of the sum
of the squares of the errors forms the `sigma_i' used for the data
point.

   Both `a F' and `H a F' can accept error forms in the input matrix,
although if you are concerned about error analysis you will probably
use `H a F' so that the output also contains error estimates.

   If the input contains error forms but all the `sigma_i' values are
the same, it is easy to see that the resulting fitted model will be the
same as if the input did not have error forms at all (`chi^2' is simply
scaled uniformly by `1 / sigma^2', which doesn't affect where it has a
minimum).  But there _will_ be a difference in the estimated errors of
the coefficients reported by `H a F'.

   Consult any text on statistical modeling of data for a discussion of
where these error estimates come from and how they should be
interpreted.

   With the Inverse flag, `I a F' [`xfit'] produces even more
information.  The result is a vector of six items:

  1. The model formula with error forms for its coefficients or
     parameters.  This is the result that `H a F' would have produced.

  2. A vector of "raw" parameter values for the model.  These are the
     polynomial coefficients or other parameters as plain numbers, in
     the same order as the parameters appeared in the final prompt of
     the `I a F' command.  For polynomials of degree `d', this vector
     will have length `M = d+1' with the constant term first.

  3. The covariance matrix `C' computed from the fit.  This is an MxM
     symmetric matrix; the diagonal elements `C_j_j' are the variances
     `sigma_j^2' of the parameters.  The other elements are covariances
     `sigma_i_j^2' that describe the correlation between pairs of
     parameters.  (A related set of numbers, the "linear correlation
     coefficients" `r_i_j', are defined as `sigma_i_j^2 / sigma_i
     sigma_j'.)

  4. A vector of `M' "parameter filter" functions whose meanings are
     described below.  If no filters are necessary this will instead be
     an empty vector; this is always the case for the polynomial and
     multilinear fits described so far.

  5. The value of `chi^2' for the fit, calculated by the formulas shown
     above.  This gives a measure of the quality of the fit;
     statisticians consider `chi^2 = N - M' to indicate a moderately
     good fit (where again `N' is the number of data points and `M' is
     the number of parameters).

  6. A measure of goodness of fit expressed as a probability `Q'.  This
     is computed from the `utpc' probability distribution function using
     `chi^2' with `N - M' degrees of freedom.  A value of 0.5 implies a
     good fit; some texts recommend that often `Q = 0.1' or even 0.001
     can signify an acceptable fit.  In particular, `chi^2' statistics
     assume the errors in your inputs follow a normal (Gaussian)
     distribution; if they don't, you may have to accept smaller values
     of `Q'.

     The `Q' value is computed only if the input included error
     estimates.  Otherwise, Calc will report the symbol `nan' for `Q'.
     The reason is that in this case the `chi^2' value has effectively
     been used to estimate the original errors in the input, and thus
     there is no redundant information left over to use for a
     confidence test.

File: calc,  Node: Standard Nonlinear Models,  Next: Curve Fitting Details,  Prev: Error Estimates for Fits,  Up: Curve Fitting

12.8.4 Standard Nonlinear Models
--------------------------------

The `a F' command also accepts other kinds of models besides lines and
polynomials.  Some common models have quick single-key abbreviations;
others must be entered by hand as algebraic formulas.

   Here is a complete list of the standard models recognized by `a F':

`1'
     Linear or multilinear.  a + b x + c y + d z.

`2-9'
     Polynomials.  a + b x + c x^2 + d x^3.

`e'
     Exponential.  a exp(b x) exp(c y).

`E'
     Base-10 exponential.  a 10^(b x) 10^(c y).

`x'
     Exponential (alternate notation).  exp(a + b x + c y).

`X'
     Base-10 exponential (alternate).  10^(a + b x + c y).

`l'
     Logarithmic.  a + b ln(x) + c ln(y).

`L'
     Base-10 logarithmic.  a + b log10(x) + c log10(y).

`^'
     General exponential.  a b^x c^y.

`p'
     Power law.  a x^b y^c.

`q'
     Quadratic.  a + b (x-c)^2 + d (x-e)^2.

`g'
     Gaussian.  (a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2).

`s'
     Logistic _s_ curve.  a/(1 + exp(b (x - c))).

`b'
     Logistic bell curve.  a exp(b (x - c))/(1 + exp(b (x - c)))^2.

`o'
     Hubbert linearization.  (y/x) = a (1 - x/b).

   All of these models are used in the usual way; just press the
appropriate letter at the model prompt, and choose variable names if
you wish.  The result will be a formula as shown in the above table,
with the best-fit values of the parameters substituted.  (You may find
it easier to read the parameter values from the vector that is placed
in the trail.)

   All models except Gaussian, logistics, Hubbert and polynomials can
generalize as shown to any number of independent variables.  Also, all
the built-in models except for the logistic and Hubbert curves have an
additive or multiplicative parameter shown as `a' in the above table
which can be replaced by zero or one, as appropriate, by typing `h'
before the model key.

   Note that many of these models are essentially equivalent, but
express the parameters slightly differently.  For example, `a b^x' and
the other two exponential models are all algebraic rearrangements of
each other.  Also, the "quadratic" model is just a degree-2 polynomial
with the parameters expressed differently.  Use whichever form best
matches the problem.

   The HP-28/48 calculators support four different models for curve
fitting, called `LIN', `LOG', `EXP', and `PWR'.  These correspond to
Calc models `a + b x', `a + b ln(x)', `a exp(b x)', and `a x^b',
respectively.  In each case, `a' is what the HP-48 identifies as the
"intercept," and `b' is what it calls the "slope."

   If the model you want doesn't appear on this list, press `'' (the
apostrophe key) at the model prompt to enter any algebraic formula,
such as `m x - b', as the model.  (Not all models will work,
though--see the next section for details.)

   The model can also be an equation like `y = m x + b'.  In this case,
Calc thinks of all the rows of the data matrix on equal terms; this
model effectively has two parameters (`m' and `b') and two independent
variables (`x' and `y'), with no "dependent" variables.  Model equations
do not need to take this `y =' form.  For example, the implicit line
equation `a x + b y = 1' works fine as a model.

   When you enter a model, Calc makes an alphabetical list of all the
variables that appear in the model.  These are used for the default
parameters, independent variables, and dependent variable (in that
order).  If you enter a plain formula (not an equation), Calc assumes
the dependent variable does not appear in the formula and thus does not
need a name.

   For example, if the model formula has the variables `a,mu,sigma,t,x',
and the data matrix has three rows (meaning two independent variables),
Calc will use `a,mu,sigma' as the default parameters, and the data rows
will be named `t' and `x', respectively.  If you enter an equation
instead of a plain formula, Calc will use `a,mu' as the parameters, and
`sigma,t,x' as the three independent variables.

   You can, of course, override these choices by entering something
different at the prompt.  If you leave some variables out of the list,
those variables must have stored values and those stored values will be
used as constants in the model.  (Stored values for the parameters and
independent variables are ignored by the `a F' command.)  If you list
only independent variables, all the remaining variables in the model
formula will become parameters.

   If there are `$' signs in the model you type, they will stand for
parameters and all other variables (in alphabetical order) will be
independent.  Use `$' for one parameter, `$$' for another, and so on.
Thus `$ x + $$' is another way to describe a linear model.

   If you type a `$' instead of `'' at the model prompt itself, Calc
will take the model formula from the stack.  (The data must then appear
at the second stack level.)  The same conventions are used to choose
which variables in the formula are independent by default and which are
parameters.

   Models taken from the stack can also be expressed as vectors of two
or three elements, `[MODEL, VARS]' or `[MODEL, VARS, PARAMS]'.  Each of
VARS and PARAMS may be either a variable or a vector of variables.  (If
PARAMS is omitted, all variables in MODEL except those listed as VARS
are parameters.)

   When you enter a model manually with `'', Calc puts a 3-vector
describing the model in the trail so you can get it back if you wish.

   Finally, you can store a model in one of the Calc variables `Model1'
or `Model2', then use this model by typing `a F u' or `a F U'
(respectively).  The value stored in the variable can be any of the
formats that `a F $' would accept for a model on the stack.

   Calc uses the principal values of inverse functions like `ln' and
`arcsin' when doing fits.  For example, when you enter the model `y =
sin(a t + b)' Calc actually uses the easier form `arcsin(y) = a t + b'.
The `arcsin' function always returns results in the range from -90 to
90 degrees (or the equivalent range in radians).  Suppose you had data
that you believed to represent roughly three oscillations of a sine
wave, so that the argument of the sine might go from zero to 3*360
degrees.  The above model would appear to be a good way to determine the
true frequency and phase of the sine wave, but in practice it would
fail utterly.  The righthand side of the actual model `arcsin(y) = a t
+ b' will grow smoothly with `t', but the lefthand side will bounce
back and forth between -90 and 90.  No values of `a' and `b' can make
the two sides match, even approximately.

   There is no good solution to this problem at present.  You could
restrict your data to small enough ranges so that the above problem
doesn't occur (i.e., not straddling any peaks in the sine wave).  Or,
in this case, you could use a totally different method such as Fourier
analysis, which is beyond the scope of the `a F' command.
(Unfortunately, Calc does not currently have any facilities for taking
Fourier and related transforms.)

File: calc,  Node: Curve Fitting Details,  Next: Interpolation,  Prev: Standard Nonlinear Models,  Up: Curve Fitting

12.8.5 Curve Fitting Details
----------------------------

Calc's internal least-squares fitter can only handle multilinear
models.  More precisely, it can handle any model of the form `a
f(x,y,z) + b g(x,y,z) + c h(x,y,z)', where `a,b,c' are the parameters
and `x,y,z' are the independent variables (of course there can be any
number of each, not just three).

   In a simple multilinear or polynomial fit, it is easy to see how to
convert the model into this form.  For example, if the model is `a + b
x + c x^2', then `f(x) = 1', `g(x) = x', and `h(x) = x^2' are suitable
functions.

   For most other models, Calc uses a variety of algebraic manipulations
to try to put the problem into the form

     Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)

where `Y,A,B,C,F,G,H' are arbitrary functions.  It computes `Y', `F',
`G', and `H' for all the data points, does a standard linear fit to
find the values of `A', `B', and `C', then uses the equation solver to
solve for `a,b,c' in terms of `A,B,C'.

   A remarkable number of models can be cast into this general form.
We'll look at two examples here to see how it works.  The power-law
model `y = a x^b' with two independent variables and two parameters can
be rewritten as follows:

     y = a x^b
     y = a exp(b ln(x))
     y = exp(ln(a) + b ln(x))
     ln(y) = ln(a) + b ln(x)

which matches the desired form with `Y = ln(y)', `A = ln(a)', `F = 1',
`B = b', and `G = ln(x)'.  Calc thus computes the logarithms of your
`y' and `x' values, does a linear fit for `A' and `B', then solves to
get `a = exp(A)' and `b = B'.

   Another interesting example is the "quadratic" model, which can be
handled by expanding according to the distributive law.

     y = a + b*(x - c)^2
     y = a + b c^2 - 2 b c x + b x^2

which matches with `Y = y', `A = a + b c^2', `F = 1', `B = -2 b c', `G
= x' (the -2 factor could just as easily have been put into `G' instead
of `B'), `C = b', and `H = x^2'.

   The Gaussian model looks quite complicated, but a closer examination
shows that it's actually similar to the quadratic model but with an
exponential that can be brought to the top and moved into `Y'.

   The logistic models cannot be put into general linear form.  For
these models, and the Hubbert linearization, Calc computes a rough
approximation for the parameters, then uses the Levenberg-Marquardt
iterative method to refine the approximations.

   Another model that cannot be put into general linear form is a
Gaussian with a constant background added on, i.e., `d' + the regular
Gaussian formula.  If you have a model like this, your best bet is to
replace enough of your parameters with constants to make the model
linearizable, then adjust the constants manually by doing a series of
fits.  You can compare the fits by graphing them, by examining the
goodness-of-fit measures returned by `I a F', or by some other method
suitable to your application.  Note that some models can be linearized
in several ways.  The Gaussian-plus-D model can be linearized by
setting `d' (the background) to a constant, or by setting `b' (the
standard deviation) and `c' (the mean) to constants.

   To fit a model with constants substituted for some parameters, just
store suitable values in those parameter variables, then omit them from
the list of parameters when you answer the variables prompt.

   A last desperate step would be to use the general-purpose `minimize'
function rather than `fit'.  After all, both functions solve the
problem of minimizing an expression (the `chi^2' sum) by adjusting
certain parameters in the expression.  The `a F' command is able to use
a vastly more efficient algorithm due to its special knowledge about
linear chi-square sums, but the `a N' command can do the same thing by
brute force.

   A compromise would be to pick out a few parameters without which the
fit is linearizable, and use `minimize' on a call to `fit' which
efficiently takes care of the rest of the parameters.  The thing to be
minimized would be the value of `chi^2' returned as the fifth result of
the `xfit' function:

     minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)

where `gaus' represents the Gaussian model with background, `data'
represents the data matrix, and `guess' represents the initial guess
for `d' that `minimize' requires.  This operation will only be, shall
we say, extraordinarily slow rather than astronomically slow (as would
be the case if `minimize' were used by itself to solve the problem).

   The `I a F' [`xfit'] command is somewhat trickier when nonlinear
models are used.  The second item in the result is the vector of "raw"
parameters `A', `B', `C'.  The covariance matrix is written in terms of
those raw parameters.  The fifth item is a vector of "filter"
expressions.  This is the empty vector `[]' if the raw parameters were
the same as the requested parameters, i.e., if `A = a', `B = b', and so
on (which is always true if the model is already linear in the
parameters as written, e.g., for polynomial fits).  If the parameters
had to be rearranged, the fifth item is instead a vector of one formula
per parameter in the original model.  The raw parameters are expressed
in these "filter" formulas as `fitdummy(1)' for `A', `fitdummy(2)' for
`B', and so on.

   When Calc needs to modify the model to return the result, it replaces
`fitdummy(1)' in all the filters with the first item in the raw
parameters list, and so on for the other raw parameters, then evaluates
the resulting filter formulas to get the actual parameter values to be
substituted into the original model.  In the case of `H a F' and `I a
F' where the parameters must be error forms, Calc uses the square roots
of the diagonal entries of the covariance matrix as error values for
the raw parameters, then lets Calc's standard error-form arithmetic
take it from there.

   If you use `I a F' with a nonlinear model, be sure to remember that
the covariance matrix is in terms of the raw parameters, _not_ the
actual requested parameters.  It's up to you to figure out how to
interpret the covariances in the presence of nontrivial filter
functions.

   Things are also complicated when the input contains error forms.
Suppose there are three independent and dependent variables, `x', `y',
and `z', one or more of which are error forms in the data.  Calc
combines all the error values by taking the square root of the sum of
the squares of the errors.  It then changes `x' and `y' to be plain
numbers, and makes `z' into an error form with this combined error.
The `Y(x,y,z)' part of the linearized model is evaluated, and the
result should be an error form.  The error part of that result is used
for `sigma_i' for the data point.  If for some reason `Y(x,y,z)' does
not return an error form, the combined error from `z' is used directly
for `sigma_i'.  Finally, `z' is also stripped of its error for use in
computing `F(x,y,z)', `G(x,y,z)' and so on; the righthand side of the
linearized model is computed in regular arithmetic with no error forms.

   (While these rules may seem complicated, they are designed to do the
most reasonable thing in the typical case that `Y(x,y,z)' depends only
on the dependent variable `z', and in fact is often simply equal to
`z'.  For common cases like polynomials and multilinear models, the
combined error is simply used as the `sigma' for the data point with no
further ado.)

   It may be the case that the model you wish to use is linearizable,
but Calc's built-in rules are unable to figure it out.  Calc uses its
algebraic rewrite mechanism to linearize a model.  The rewrite rules
are kept in the variable `FitRules'.  You can edit this variable using
the `s e FitRules' command; in fact, there is a special `s F' command
just for editing `FitRules'.  *Note Operations on Variables::.

   *Note Rewrite Rules::, for a discussion of rewrite rules.

   Calc uses `FitRules' as follows.  First, it converts the model to an
equation if necessary and encloses the model equation in a call to the
function `fitmodel' (which is not actually a defined function in Calc;
it is only used as a placeholder by the rewrite rules).  Parameter
variables are renamed to function calls `fitparam(1)', `fitparam(2)',
and so on, and independent variables are renamed to `fitvar(1)',
`fitvar(2)', etc.  The dependent variable is the highest-numbered
`fitvar'.  For example, the power law model `a x^b' is converted to `y
= a x^b', then to

     fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))

   Calc then applies the rewrites as if by `C-u 0 a r FitRules'.  (The
zero prefix means that rewriting should continue until no further
changes are possible.)

   When rewriting is complete, the `fitmodel' call should have been
replaced by a `fitsystem' call that looks like this:

     fitsystem(Y, FGH, ABC)

where Y is a formula that describes the function `Y(x,y,z)', FGH is the
vector of formulas `[F(x,y,z), G(x,y,z), H(x,y,z)]', and ABC is the
vector of parameter filters which refer to the raw parameters as
`fitdummy(1)' for `A', `fitdummy(2)' for `B', etc.  While the number of
raw parameters (the length of the FGH vector) is usually the same as
the number of original parameters (the length of the ABC vector), this
is not required.

   The power law model eventually boils down to

     fitsystem(ln(fitvar(2)),
               [1, ln(fitvar(1))],
               [exp(fitdummy(1)), fitdummy(2)])

   The actual implementation of `FitRules' is complicated; it proceeds
in four phases.  First, common rearrangements are done to try to bring
linear terms together and to isolate functions like `exp' and `ln'
either all the way "out" (so that they can be put into Y) or all the
way "in" (so that they can be put into ABC or FGH).  In particular, all
non-constant powers are converted to logs-and-exponentials form, and
the distributive law is used to expand products of sums.  Quotients are
rewritten to use the `fitinv' function, where `fitinv(x)' represents
`1/x' while the `FitRules' are operating.  (The use of `fitinv' makes
recognition of linear-looking forms easier.)  If you modify `FitRules',
you will probably only need to modify the rules for this phase.

   Phase two, whose rules can actually also apply during phases one and
three, first rewrites `fitmodel' to a two-argument form `fitmodel(Y,
MODEL)', where Y is initially zero and MODEL has been changed from `a=b'
to `a-b' form.  It then tries to peel off invertible functions from the
outside of MODEL and put them into Y instead, calling the equation
solver to invert the functions.  Finally, when this is no longer
possible, the `fitmodel' is changed to a four-argument `fitsystem',
where the fourth argument is MODEL and the FGH and ABC vectors are
initially empty.  (The last vector is really ABC, corresponding to raw
parameters, for now.)

   Phase three converts a sum of items in the MODEL to a sum of
`fitpart(A, B, C)' terms which represent terms `A*B*C' of the sum,
where A is all factors that do not involve any variables, B is all
factors that involve only parameters, and C is the factors that involve
only independent variables.  (If this decomposition is not possible,
the rule set will not complete and Calc will complain that the model is
too complex.)  Then `fitpart's with equal B or C components are merged
back together using the distributive law in order to minimize the
number of raw parameters needed.

   Phase four moves the `fitpart' terms into the FGH and ABC vectors.
Also, some of the algebraic expansions that were done in phase 1 are
undone now to make the formulas more computationally efficient.
Finally, it calls the solver one more time to convert the ABC vector to
an ABC vector, and removes the fourth MODEL argument (which by now will
be zero) to obtain the three-argument `fitsystem' that the linear
least-squares solver wants to see.

   Two functions which are useful in connection with `FitRules' are
`hasfitparams(x)' and `hasfitvars(x)', which check whether `x' refers
to any parameters or independent variables, respectively.
Specifically, these functions return "true" if the argument contains
any `fitparam' (or `fitvar') function calls, and "false" otherwise.
(Recall that "true" means a nonzero number, and "false" means zero.
The actual nonzero number returned is the largest N from all the
`fitparam(N)'s or `fitvar(N)'s, respectively, that appear in the
formula.)

   The `fit' function in algebraic notation normally takes four
arguments, `fit(MODEL, VARS, PARAMS, DATA)', where MODEL is the model
formula as it would be typed after `a F '', VARS is the independent
variable or a vector of independent variables, PARAMS likewise gives
the parameter(s), and DATA is the data matrix.  Note that the length of
VARS must be equal to the number of rows in DATA if MODEL is an
equation, or one less than the number of rows if MODEL is a plain
formula.  (Actually, a name for the dependent variable is allowed but
will be ignored in the plain-formula case.)

   If PARAMS is omitted, the parameters are all variables in MODEL
except those that appear in VARS.  If VARS is also omitted, Calc sorts
all the variables that appear in MODEL alphabetically and uses the
higher ones for VARS and the lower ones for PARAMS.

   Alternatively, `fit(MODELVEC, DATA)' is allowed where MODELVEC is a
2- or 3-vector describing the model and variables, as discussed
previously.

   If Calc is unable to do the fit, the `fit' function is left in
symbolic form, ordinarily with an explanatory message.  The message
will be "Model expression is too complex" if the linearizer was unable
to put the model into the required form.

   The `efit' (corresponding to `H a F') and `xfit' (for `I a F')
functions are completely analogous.

File: calc,  Node: Interpolation,  Prev: Curve Fitting Details,  Up: Curve Fitting

12.8.6 Polynomial Interpolation
-------------------------------

The `a p' (`calc-poly-interp') [`polint'] command does a polynomial
interpolation at a particular `x' value.  It takes two arguments from
the stack:  A data matrix of the sort used by `a F', and a single
number which represents the desired `x' value.  Calc effectively does
an exact polynomial fit as if by `a F i', then substitutes the `x'
value into the result in order to get an approximate `y' value based on
the fit.  (Calc does not actually use `a F i', however; it uses a
direct method which is both more efficient and more numerically stable.)

   The result of `a p' is actually a vector of two values:  The `y'
value approximation, and an error measure `dy' that reflects Calc's
estimation of the probable error of the approximation at that value of
`x'.  If the input `x' is equal to any of the `x' values in the data
matrix, the output `y' will be the corresponding `y' value from the
matrix, and the output `dy' will be exactly zero.

   A prefix argument of 2 causes `a p' to take separate x- and
y-vectors from the stack instead of one data matrix.

   If `x' is a vector of numbers, `a p' will return a matrix of
interpolated results for each of those `x' values.  (The matrix will
have two columns, the `y' values and the `dy' values.)  If `x' is a
formula instead of a number, the `polint' function remains in symbolic
form; use the `a "' command to expand it out to a formula that
describes the fit in symbolic terms.

   In all cases, the `a p' command leaves the data vectors or matrix on
the stack.  Only the `x' value is replaced by the result.

   The `H a p' [`ratint'] command does a rational function
interpolation.  It is used exactly like `a p', except that it uses as
its model the quotient of two polynomials.  If there are `N' data
points, the numerator and denominator polynomials will each have degree
`N/2' (if `N' is odd, the denominator will have degree one higher than
the numerator).

   Rational approximations have the advantage that they can accurately
describe functions that have poles (points at which the function's value
goes to infinity, so that the denominator polynomial of the
approximation goes to zero).  If `x' corresponds to a pole of the
fitted rational function, then the result will be a division by zero.
If Infinite mode is enabled, the result will be `[uinf, uinf]'.

   There is no way to get the actual coefficients of the rational
function used by `H a p'.  (The algorithm never generates these
coefficients explicitly, and quotients of polynomials are beyond `a F''s
capabilities to fit.)

File: calc,  Node: Summations,  Next: Logical Operations,  Prev: Curve Fitting,  Up: Algebra

12.9 Summations
===============

The `a +' (`calc-summation') [`sum'] command computes the sum of a
formula over a certain range of index values.  The formula is taken
from the top of the stack; the command prompts for the name of the
summation index variable, the lower limit of the sum (any formula), and
the upper limit of the sum.  If you enter a blank line at any of these
prompts, that prompt and any later ones are answered by reading
additional elements from the stack.  Thus, `' k^2 <RET> ' k <RET> 1
<RET> 5 <RET> a + <RET>' produces the result 55.

   The choice of index variable is arbitrary, but it's best not to use
a variable with a stored value.  In particular, while `i' is often a
favorite index variable, it should be avoided in Calc because `i' has
the imaginary constant `(0, 1)' as a value.  If you pressed `=' on a
sum over `i', it would be changed to a nonsensical sum over the
"variable" `(0, 1)'!  If you really want to use `i' as an index
variable, use `s u i <RET>' first to "unstore" this variable.  (*Note
Storing Variables::.)

   A numeric prefix argument steps the index by that amount rather than
by one.  Thus `' a_k <RET> C-u -2 a + k <RET> 10 <RET> 0 <RET>' yields
`a_10 + a_8 + a_6 + a_4 + a_2 + a_0'.  A prefix argument of plain `C-u'
causes `a +' to prompt for the step value, in which case you can enter
any formula or enter a blank line to take the step value from the
stack.  With the `C-u' prefix, `a +' can take up to five arguments from
the stack:  The formula, the variable, the lower limit, the upper
limit, and (at the top of the stack), the step value.

   Calc knows how to do certain sums in closed form.  For example,
`sum(6 k^2, k, 1, n) = 2 n^3 + 3 n^2 + n'.  In particular, this is
possible if the formula being summed is polynomial or exponential in
the index variable.  Sums of logarithms are transformed into logarithms
of products.  Sums of trigonometric and hyperbolic functions are
transformed to sums of exponentials and then done in closed form.
Also, of course, sums in which the lower and upper limits are both
numbers can always be evaluated just by grinding them out, although
Calc will use closed forms whenever it can for the sake of efficiency.

   The notation for sums in algebraic formulas is `sum(EXPR, VAR, LOW,
HIGH, STEP)'.  If STEP is omitted, it defaults to one.  If HIGH is
omitted, LOW is actually the upper limit and the lower limit is one.
If LOW is also omitted, the limits are `-inf' and `inf', respectively.

   Infinite sums can sometimes be evaluated:  `sum(.5^k, k, 1, inf)'
returns `1'.  This is done by evaluating the sum in closed form (to `1.
- 0.5^n' in this case), then evaluating this formula with `n' set to
`inf'.  Calc's usual rules for "infinite" arithmetic can find the
answer from there.  If infinite arithmetic yields a `nan', or if the
sum cannot be solved in closed form, Calc leaves the `sum' function in
symbolic form.  *Note Infinities::.

   As a special feature, if the limits are infinite (or omitted, as
described above) but the formula includes vectors subscripted by
expressions that involve the iteration variable, Calc narrows the
limits to include only the range of integers which result in valid
subscripts for the vector.  For example, the sum `sum(k
[a,b,c,d,e,f,g]_(2k),k)' evaluates to `b + 2 d + 3 f'.

   The limits of a sum do not need to be integers.  For example,
`sum(a_k, k, 0, 2 n, n)' produces `a_0 + a_n + a_(2 n)'.  Calc computes
the number of iterations using the formula `1 + (HIGH - LOW) / STEP',
which must, after simplification as if by `a s', evaluate to an integer.

   If the number of iterations according to the above formula does not
come out to an integer, the sum is invalid and will be left in symbolic
form.  However, closed forms are still supplied, and you are on your
honor not to misuse the resulting formulas by substituting mismatched
bounds into them.  For example, `sum(k, k, 1, 10, 2)' is invalid, but
Calc will go ahead and evaluate the closed form solution for the limits
1 and 10 to get the rather dubious answer, 29.25.

   If the lower limit is greater than the upper limit (assuming a
positive step size), the result is generally zero.  However, Calc only
guarantees a zero result when the upper limit is exactly one step less
than the lower limit, i.e., if the number of iterations is -1.  Thus
`sum(f(k), k, n, n-1)' is zero but the sum from `n' to `n-2' may report
a nonzero value if Calc used a closed form solution.

   Calc's logical predicates like `a < b' return 1 for "true" and 0 for
"false."  *Note Logical Operations::.  This can be used to advantage
for building conditional sums.  For example, `sum(prime(k)*k^2, k, 1,
20)' is the sum of the squares of all prime numbers from 1 to 20; the
`prime' predicate returns 1 if its argument is prime and 0 otherwise.
You can read this expression as "the sum of `k^2', where `k' is prime."
Indeed, `sum(prime(k)*k^2, k)' would represent the sum of _all_ primes
squared, since the limits default to plus and minus infinity, but there
are no such sums that Calc's built-in rules can do in closed form.

   As another example, `sum((k != k_0) * f(k), k, 1, n)' is the sum of
`f(k)' for all `k' from 1 to `n', excluding one value `k_0'.  Slightly
more tricky is the summand `(k != k_0) / (k - k_0)', which is an
attempt to describe the sum of all `1/(k-k_0)' except at `k = k_0',
where this would be a division by zero.  But at `k = k_0', this formula
works out to the indeterminate form `0 / 0', which Calc will not assume
is zero.  Better would be to use `(k != k_0) ? 1/(k-k_0) : 0'; the `?
:' operator does an "if-then-else" test:  This expression says, "if `k
!= k_0', then `1/(k-k_0)', else zero."  Now the formula `1/(k-k_0)'
will not even be evaluated by Calc when `k = k_0'.

   The `a -' (`calc-alt-summation') [`asum'] command computes an
alternating sum.  Successive terms of the sequence are given
alternating signs, with the first term (corresponding to the lower
index value) being positive.  Alternating sums are converted to normal
sums with an extra term of the form `(-1)^(k-LOW)'.  This formula is
adjusted appropriately if the step value is other than one.  For
example, the Taylor series for the sine function is `asum(x^k / k!, k,
1, inf, 2)'.  (Calc cannot evaluate this infinite series, but it can
approximate it if you replace `inf' with any particular odd number.)
Calc converts this series to a regular sum with a step of one, namely
`sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)'.

   The `a *' (`calc-product') [`prod'] command is the analogous way to
take a product of many terms.  Calc also knows some closed forms for
products, such as `prod(k, k, 1, n) = n!'.  Conditional products can be
written `prod(k^prime(k), k, 1, n)' or `prod(prime(k) ? k : 1, k, 1,
n)'.

   The `a T' (`calc-tabulate') [`table'] command evaluates a formula at
a series of iterated index values, just like `sum' and `prod', but its
result is simply a vector of the results.  For example, `table(a_i, i,
1, 7, 2)' produces `[a_1, a_3, a_5, a_7]'.

File: calc,  Node: Logical Operations,  Next: Rewrite Rules,  Prev: Summations,  Up: Algebra

12.10 Logical Operations
========================

The following commands and algebraic functions return true/false values,
where 1 represents "true" and 0 represents "false."  In cases where a
truth value is required (such as for the condition part of a rewrite
rule, or as the condition for a `Z [ Z ]' control structure), any
nonzero value is accepted to mean "true."  (Specifically, anything for
which `dnonzero' returns 1 is "true," and anything for which `dnonzero'
returns 0 or cannot decide is assumed "false."  Note that this means
that `Z [ Z ]' will execute the "then" portion if its condition is
provably true, but it will execute the "else" portion for any condition
like `a = b' that is not provably true, even if it might be true.
Algebraic functions that have conditions as arguments, like `? :' and
`&&', remain unevaluated if the condition is neither provably true nor
provably false.  *Note Declarations::.)

   The `a =' (`calc-equal-to') command, or `eq(a,b)' function (which
can also be written `a = b' or `a == b' in an algebraic formula) is
true if `a' and `b' are equal, either because they are identical
expressions, or because they are numbers which are numerically equal.
(Thus the integer 1 is considered equal to the float 1.0.)  If the
equality of `a' and `b' cannot be determined, the comparison is left in
symbolic form.  Note that as a command, this operation pops two values
from the stack and pushes back either a 1 or a 0, or a formula `a = b'
if the values' equality cannot be determined.

   Many Calc commands use `=' formulas to represent "equations".  For
example, the `a S' (`calc-solve-for') command rearranges an equation to
solve for a given variable.  The `a M' (`calc-map-equation') command
can be used to apply any function to both sides of an equation; for
example, `2 a M *' multiplies both sides of the equation by two.  Note
that just `2 *' would not do the same thing; it would produce the
formula `2 (a = b)' which represents 2 if the equality is true or zero
if not.

   The `eq' function with more than two arguments (e.g., `C-u 3 a =' or
`a = b = c') tests if all of its arguments are equal.  In algebraic
notation, the `=' operator is unusual in that it is neither left- nor
right-associative:  `a = b = c' is not the same as `(a = b) = c' or `a
= (b = c)' (which each compare one variable with the 1 or 0 that
results from comparing two other variables).

   The `a #' (`calc-not-equal-to') command, or `neq(a,b)' or `a != b'
function, is true if `a' and `b' are not equal.  This also works with
more than two arguments; `a != b != c != d' tests that all four of `a',
`b', `c', and `d' are distinct numbers.

   The `a <' (`calc-less-than') [`lt(a,b)' or `a < b'] operation is
true if `a' is less than `b'.  Similar functions are `a >'
(`calc-greater-than') [`gt(a,b)' or `a > b'], `a [' (`calc-less-equal')
[`leq(a,b)' or `a <= b'], and `a ]' (`calc-greater-equal') [`geq(a,b)'
or `a >= b'].

   While the inequality functions like `lt' do not accept more than two
arguments, the syntax `a <= b < c' is translated to an equivalent
expression involving intervals: `b in [a .. c)'.  (See the description
of `in' below.)  All four combinations of `<' and `<=' are allowed, or
any of the four combinations of `>' and `>='.  Four-argument
constructions like `a < b < c < d', and mixtures like `a < b = c' that
involve both equalities and inequalities, are not allowed.

   The `a .' (`calc-remove-equal') [`rmeq'] command extracts the
righthand side of the equation or inequality on the top of the stack.
It also works elementwise on vectors.  For example, if `[x = 2.34, y =
z / 2]' is on the stack, then `a .' produces `[2.34, z / 2]'.  As a
special case, if the righthand side is a variable and the lefthand side
is a number (as in `2.34 = x'), then Calc keeps the lefthand side
instead.  Finally, this command works with assignments `x := 2.34' as
well as equations, always taking the righthand side, and for `=>'
(evaluates-to) operators, always taking the lefthand side.

   The `a &' (`calc-logical-and') [`land(a,b)' or `a && b'] function is
true if both of its arguments are true, i.e., are non-zero numbers.  In
this case, the result will be either `a' or `b', chosen arbitrarily.
If either argument is zero, the result is zero.  Otherwise, the formula
is left in symbolic form.

   The `a |' (`calc-logical-or') [`lor(a,b)' or `a || b'] function is
true if either or both of its arguments are true (nonzero).  The result
is whichever argument was nonzero, choosing arbitrarily if both are
nonzero.  If both `a' and `b' are zero, the result is zero.

   The `a !' (`calc-logical-not') [`lnot(a)' or `! a'] function is true
if `a' is false (zero), or false if `a' is true (nonzero).  It is left
in symbolic form if `a' is not a number.

   The `a :' (`calc-logical-if') [`if(a,b,c)' or `a ? b : c'] function
is equal to either `b' or `c' if `a' is a nonzero number or zero,
respectively.  If `a' is not a number, the test is left in symbolic
form and neither `b' nor `c' is evaluated in any way.  In algebraic
formulas, this is one of the few Calc functions whose arguments are not
automatically evaluated when the function itself is evaluated.  The
others are `lambda', `quote', and `condition'.

   One minor surprise to watch out for is that the formula `a?3:4' will
not work because the `3:4' is parsed as a fraction instead of as three
separate symbols.  Type something like `a ? 3 : 4' or `a?(3):4' instead.

   As a special case, if `a' evaluates to a vector, then both `b' and
`c' are evaluated; the result is a vector of the same length as `a'
whose elements are chosen from corresponding elements of `b' and `c'
according to whether each element of `a' is zero or nonzero.  Each of
`b' and `c' must be either a vector of the same length as `a', or a
non-vector which is matched with all elements of `a'.

   The `a {' (`calc-in-set') [`in(a,b)'] function is true if the number
`a' is in the set of numbers represented by `b'.  If `b' is an interval
form, `a' must be one of the values encompassed by the interval.  If
`b' is a vector, `a' must be equal to one of the elements of the
vector.  (If any vector elements are intervals, `a' must be in any of
the intervals.)  If `b' is a plain number, `a' must be numerically
equal to `b'.  *Note Set Operations::, for a group of commands that
manipulate sets of this sort.

   The `typeof(a)' function produces an integer or variable which
characterizes `a'.  If `a' is a number, vector, or variable, the result
will be one of the following numbers:

      1   Integer
      2   Fraction
      3   Floating-point number
      4   HMS form
      5   Rectangular complex number
      6   Polar complex number
      7   Error form
      8   Interval form
      9   Modulo form
     10   Date-only form
     11   Date/time form
     12   Infinity (inf, uinf, or nan)
     100  Variable
     101  Vector (but not a matrix)
     102  Matrix

   Otherwise, `a' is a formula, and the result is a variable which
represents the name of the top-level function call.

   The `integer(a)' function returns true if `a' is an integer.  The
`real(a)' function is true if `a' is a real number, either integer,
fraction, or float.  The `constant(a)' function returns true if `a' is
any of the objects for which `typeof' would produce an integer code
result except for variables, and provided that the components of an
object like a vector or error form are themselves constant.  Note that
infinities do not satisfy any of these tests, nor do special constants
like `pi' and `e'.

   *Note Declarations::, for a set of similar functions that recognize
formulas as well as actual numbers.  For example, `dint(floor(x))' is
true because `floor(x)' is provably integer-valued, but
`integer(floor(x))' does not because `floor(x)' is not literally an
integer constant.

   The `refers(a,b)' function is true if the variable (or
sub-expression) `b' appears in `a', or false otherwise.  Unlike the
other tests described here, this function returns a definite "no" answer
even if its arguments are still in symbolic form.  The only case where
`refers' will be left unevaluated is if `a' is a plain variable
(different from `b').

   The `negative(a)' function returns true if `a' "looks" negative,
because it is a negative number, because it is of the form `-x', or
because it is a product or quotient with a term that looks negative.
This is most useful in rewrite rules.  Beware that `negative(a)'
evaluates to 1 or 0 for _any_ argument `a', so it can only be stored in
a formula if the default simplifications are turned off first with `m
O' (or if it appears in an unevaluated context such as a rewrite rule
condition).

   The `variable(a)' function is true if `a' is a variable, or false if
not.  If `a' is a function call, this test is left in symbolic form.
Built-in variables like `pi' and `inf' are considered variables like
any others by this test.

   The `nonvar(a)' function is true if `a' is a non-variable.  If its
argument is a variable it is left unsimplified; it never actually
returns zero.  However, since Calc's condition-testing commands
consider "false" anything not provably true, this is often good enough.

   The functions `lin', `linnt', `islin', and `islinnt' check if an
expression is "linear," i.e., can be written in the form `a + b x' for
some constants `a' and `b', and some variable or subformula `x'.  The
function `islin(f,x)' checks if formula `f' is linear in `x', returning
1 if so.  For example, `islin(x,x)', `islin(-x,x)', `islin(3,x)', and
`islin(x y / 3 - 2, x)' all return 1.  The `lin(f,x)' function is
similar, except that instead of returning 1 it returns the vector `[a,
b, x]'.  For the above examples, this vector would be `[0, 1, x]', `[0,
-1, x]', `[3, 0, x]', and `[-2, y/3, x]', respectively.  Both `lin' and
`islin' generally remain unevaluated for expressions which are not
linear, e.g., `lin(2 x^2, x)' and `lin(sin(x), x)'.  The second
argument can also be a formula; `islin(2 + 3 sin(x), sin(x))' returns
true.

   The `linnt' and `islinnt' functions perform a similar check, but
require a "non-trivial" linear form, which means that the `b'
coefficient must be non-zero.  For example, `lin(2,x)' returns `[2, 0,
x]' and `lin(y,x)' returns `[y, 0, x]', but `linnt(2,x)' and
`linnt(y,x)' are left unevaluated (in other words, these formulas are
considered to be only "trivially" linear in `x').

   All four linearity-testing functions allow you to omit the second
argument, in which case the input may be linear in any non-constant
formula.  Here, the `a=0', `b=1' case is also considered trivial, and
only constant values for `a' and `b' are recognized.  Thus, `lin(2 x
y)' returns `[0, 2, x y]', `lin(2 - x y)' returns `[2, -1, x y]', and
`lin(x y)' returns `[0, 1, x y]'.  The `linnt' function would allow the
first two cases but not the third.  Also, neither `lin' nor `linnt'
accept plain constants as linear in the one-argument case: `islin(2,x)'
is true, but `islin(2)' is false.

   The `istrue(a)' function returns 1 if `a' is a nonzero number or
provably nonzero formula, or 0 if `a' is anything else.  Calls to
`istrue' can only be manipulated if `m O' mode is used to make sure
they are not evaluated prematurely.  (Note that declarations are used
when deciding whether a formula is true; `istrue' returns 1 when
`dnonzero' would return 1, and it returns 0 when `dnonzero' would
return 0 or leave itself in symbolic form.)

File: calc,  Node: Rewrite Rules,  Prev: Logical Operations,  Up: Algebra

12.11 Rewrite Rules
===================

The `a r' (`calc-rewrite') [`rewrite'] command makes substitutions in a
formula according to a specified pattern or patterns known as "rewrite
rules".  Whereas `a b' (`calc-substitute') matches literally, so that
substituting `sin(x)' with `cos(x)' matches only the `sin' function
applied to the variable `x', rewrite rules match general kinds of
formulas; rewriting using the rule `sin(x) := cos(x)' matches `sin' of
any argument and replaces it with `cos' of that same argument.  The
only significance of the name `x' is that the same name is used on both
sides of the rule.

   Rewrite rules rearrange formulas already in Calc's memory.  *Note
Syntax Tables::, to read about "syntax rules", which are similar to
algebraic rewrite rules but operate when new algebraic entries are
being parsed, converting strings of characters into Calc formulas.

* Menu:

* Entering Rewrite Rules::
* Basic Rewrite Rules::
* Conditional Rewrite Rules::
* Algebraic Properties of Rewrite Rules::
* Other Features of Rewrite Rules::
* Composing Patterns in Rewrite Rules::
* Nested Formulas with Rewrite Rules::
* Multi-Phase Rewrite Rules::
* Selections with Rewrite Rules::
* Matching Commands::
* Automatic Rewrites::
* Debugging Rewrites::
* Examples of Rewrite Rules::

File: calc,  Node: Entering Rewrite Rules,  Next: Basic Rewrite Rules,  Prev: Rewrite Rules,  Up: Rewrite Rules

12.11.1 Entering Rewrite Rules
------------------------------

Rewrite rules normally use the "assignment" operator `OLD := NEW'.
This operator is equivalent to the function call `assign(old, new)'.
The `assign' function is undefined by itself in Calc, so an assignment
formula such as a rewrite rule will be left alone by ordinary Calc
commands.  But certain commands, like the rewrite system, interpret
assignments in special ways.

   For example, the rule `sin(x)^2 := 1-cos(x)^2' says to replace every
occurrence of the sine of something, squared, with one minus the square
of the cosine of that same thing.  All by itself as a formula on the
stack it does nothing, but when given to the `a r' command it turns
that command into a sine-squared-to-cosine-squared converter.

   To specify a set of rules to be applied all at once, make a vector of
rules.

   When `a r' prompts you to enter the rewrite rules, you can answer in
several ways:

  1. With a rule:  `f(x) := g(x) <RET>'.

  2. With a vector of rules:  `[f1(x) := g1(x), f2(x) := g2(x)] <RET>'.
     (You can omit the enclosing square brackets if you wish.)

  3. With the name of a variable that contains the rule or rules vector:
     `myrules <RET>'.

  4. With any formula except a rule, a vector, or a variable name; this
     will be interpreted as the OLD half of a rewrite rule, and you
     will be prompted a second time for the NEW half: `f(x) <RET> g(x)
     <RET>'.

  5. With a blank line, in which case the rule, rules vector, or
     variable will be taken from the top of the stack (and the formula
     to be rewritten will come from the second-to-top position).

   If you enter the rules directly (as opposed to using rules stored in
a variable), those rules will be put into the Trail so that you can
retrieve them later.  *Note Trail Commands::.

   It is most convenient to store rules you use often in a variable and
invoke them by giving the variable name.  The `s e'
(`calc-edit-variable') command is an easy way to create or edit a rule
set stored in a variable.  You may also wish to use `s p'
(`calc-permanent-variable') to save your rules permanently; *note
Operations on Variables::.

   Rewrite rules are compiled into a special internal form for faster
matching.  If you enter a rule set directly it must be recompiled every
time.  If you store the rules in a variable and refer to them through
that variable, they will be compiled once and saved away along with the
variable for later reference.  This is another good reason to store
your rules in a variable.

   Calc also accepts an obsolete notation for rules, as vectors `[OLD,
NEW]'.  But because it is easily confused with a vector of two rules,
the use of this notation is no longer recommended.

File: calc,  Node: Basic Rewrite Rules,  Next: Conditional Rewrite Rules,  Prev: Entering Rewrite Rules,  Up: Rewrite Rules

12.11.2 Basic Rewrite Rules
---------------------------

To match a particular formula `x' with a particular rewrite rule `OLD
:= NEW', Calc compares the structure of `x' with the structure of OLD.
Variables that appear in OLD are treated as "meta-variables"; the
corresponding positions in `x' may contain any sub-formulas.  For
example, the pattern `f(x,y)' would match the expression `f(12, a+1)'
with the meta-variable `x' corresponding to 12 and with `y'
corresponding to `a+1'.  However, this pattern would not match `f(12)'
or `g(12, a+1)', since there is no assignment of the meta-variables
that will make the pattern match these expressions.  Notice that if the
pattern is a single meta-variable, it will match any expression.

   If a given meta-variable appears more than once in OLD, the
corresponding sub-formulas of `x' must be identical.  Thus the pattern
`f(x,x)' would match `f(12, 12)' and `f(a+1, a+1)' but not `f(12, a+1)'
or `f(a+b, b+a)'.  (*Note Conditional Rewrite Rules::, for a way to
match the latter.)

   Things other than variables must match exactly between the pattern
and the target formula.  To match a particular variable exactly, use
the pseudo-function `quote(v)' in the pattern.  For example, the
pattern `x+quote(y)' matches `x+y', `2+y', or `sin(a)+y'.

   The special variable names `e', `pi', `i', `phi', `gamma', `inf',
`uinf', and `nan' always match literally.  Thus the pattern `sin(d + e
+ f)' acts exactly like `sin(d + quote(e) + f)'.

   If the OLD pattern is found to match a given formula, that formula
is replaced by NEW, where any occurrences in NEW of meta-variables from
the pattern are replaced with the sub-formulas that they matched.
Thus, applying the rule `f(x,y) := g(y+x,x)' to `f(12, a+1)' would
produce `g(a+13, 12)'.

   The normal `a r' command applies rewrite rules over and over
throughout the target formula until no further changes are possible (up
to a limit of 100 times).  Use `C-u 1 a r' to make only one change at a
time.

File: calc,  Node: Conditional Rewrite Rules,  Next: Algebraic Properties of Rewrite Rules,  Prev: Basic Rewrite Rules,  Up: Rewrite Rules

12.11.3 Conditional Rewrite Rules
---------------------------------

A rewrite rule can also be "conditional", written in the form `OLD :=
NEW :: COND'.  (There is also the obsolete form `[OLD, NEW, COND]'.)
If a COND part is present in the rule, this is an additional condition
that must be satisfied before the rule is accepted.  Once OLD has been
successfully matched to the target expression, COND is evaluated (with
all the meta-variables substituted for the values they matched) and
simplified with `a s' (`calc-simplify').  If the result is a nonzero
number or any other object known to be nonzero (*note Declarations::),
the rule is accepted.  If the result is zero or if it is a symbolic
formula that is not known to be nonzero, the rule is rejected.  *Note
Logical Operations::, for a number of functions that return 1 or 0
according to the results of various tests.

   For example, the formula `n > 0' simplifies to 1 or 0 if `n' is
replaced by a positive or nonpositive number, respectively (or if `n'
has been declared to be positive or nonpositive).  Thus, the rule
`f(x,y) := g(y+x,x) :: x+y > 0' would apply to `f(0, 4)' but not to
`f(-3, 2)' or `f(12, a+1)' (assuming no outstanding declarations for
`a').  In the case of `f(-3, 2)', the condition can be shown not to be
satisfied; in the case of `f(12, a+1)', the condition merely cannot be
shown to be satisfied, but that is enough to reject the rule.

   While Calc will use declarations to reason about variables in the
formula being rewritten, declarations do not apply to meta-variables.
For example, the rule `f(a) := g(a+1)' will match for any values of
`a', such as complex numbers, vectors, or formulas, even if `a' has
been declared to be real or scalar.  If you want the meta-variable `a'
to match only literal real numbers, use `f(a) := g(a+1) :: real(a)'.
If you want `a' to match only reals and formulas which are provably
real, use `dreal(a)' as the condition.

   The `::' operator is a shorthand for the `condition' function; `OLD
:= NEW :: COND' is equivalent to the formula `condition(assign(OLD,
NEW), COND)'.

   If you have several conditions, you can use `... :: c1 :: c2 :: c3'
or `... :: c1 && c2 && c3'.  The two are entirely equivalent.

   It is also possible to embed conditions inside the pattern: `f(x ::
x>0, y) := g(y+x, x)'.  This is purely a notational convenience,
though; where a condition appears in a rule has no effect on when it is
tested.  The rewrite-rule compiler automatically decides when it is
best to test each condition while a rule is being matched.

   Certain conditions are handled as special cases by the rewrite rule
system and are tested very efficiently:  Where `x' is any
meta-variable, these conditions are `integer(x)', `real(x)',
`constant(x)', `negative(x)', `x >= y' where `y' is either a constant
or another meta-variable and `>=' may be replaced by any of the six
relational operators, and `x % a = b' where `a' and `b' are constants.
Other conditions, like `x >= y+1' or `dreal(x)', will be less efficient
to check since Calc must bring the whole evaluator and simplifier into
play.

   An interesting property of `::' is that neither of its arguments
will be touched by Calc's default simplifications.  This is important
because conditions often are expressions that cannot safely be
evaluated early.  For example, the `typeof' function never remains in
symbolic form; entering `typeof(a)' will put the number 100 (the type
code for variables like `a') on the stack.  But putting the condition
`... :: typeof(a) = 6' on the stack is safe since `::' prevents the
`typeof' from being evaluated until the condition is actually used by
the rewrite system.

   Since `::' protects its lefthand side, too, you can use a dummy
condition to protect a rule that must itself not evaluate early.  For
example, it's not safe to put `a(f,x) := apply(f, [x])' on the stack
because it will immediately evaluate to `a(f,x) := f(x)', where the
meta-variable-ness of `f' on the righthand side has been lost.  But
`a(f,x) := apply(f, [x]) :: 1' is safe, and of course the condition `1'
is always true (nonzero) so it has no effect on the functioning of the
rule.  (The rewrite compiler will ensure that it doesn't even impact
the speed of matching the rule.)

File: calc,  Node: Algebraic Properties of Rewrite Rules,  Next: Other Features of Rewrite Rules,  Prev: Conditional Rewrite Rules,  Up: Rewrite Rules

12.11.4 Algebraic Properties of Rewrite Rules
---------------------------------------------

The rewrite mechanism understands the algebraic properties of functions
like `+' and `*'.  In particular, pattern matching takes the
associativity and commutativity of the following functions into account:

     + - *  = !=  && ||  and or xor  vint vunion vxor  gcd lcm  max min  beta

   For example, the rewrite rule:

     a x + b x  :=  (a + b) x

will match formulas of the form,

     a x + b x,  x a + x b,  a x + x b,  x a + b x

   Rewrites also understand the relationship between the `+' and `-'
operators.  The above rewrite rule will also match the formulas,

     a x - b x,  x a - x b,  a x - x b,  x a - b x

by matching `b' in the pattern to `-b' from the formula.

   Applied to a sum of many terms like `r + a x + s + b x + t', this
pattern will check all pairs of terms for possible matches.  The rewrite
will take whichever suitable pair it discovers first.

   In general, a pattern using an associative operator like `a + b'
will try 2 N different ways to match a sum of N terms like `x + y + z -
w'.  First, `a' is matched against each of `x', `y', `z', and `-w' in
turn, with `b' being matched to the remainders `y + z - w', `x + z -
w', etc.  If none of these succeed, then `b' is matched against each of
the four terms with `a' matching the remainder.  Half-and-half matches,
like `(x + y) + (z - w)', are not tried.

   Note that `*' is not commutative when applied to matrices, but
rewrite rules pretend that it is.  If you type `m v' to enable Matrix
mode (*note Matrix Mode::), rewrite rules will match `*' literally,
ignoring its usual commutativity property.  (In the current
implementation, the associativity also vanishes--it is as if the
pattern had been enclosed in a `plain' marker; see below.)  If you are
applying rewrites to formulas with matrices, it's best to enable Matrix
mode first to prevent algebraically incorrect rewrites from occurring.

   The pattern `-x' will actually match any expression.  For example,
the rule

     f(-x)  :=  -f(x)

will rewrite `f(a)' to `-f(-a)'.  To avoid this, either use a `plain'
marker as described below, or add a `negative(x)' condition.  The
`negative' function is true if its argument "looks" negative, for
example, because it is a negative number or because it is a formula
like `-x'.  The new rule using this condition is:

     f(x)  :=  -f(-x)  :: negative(x)    or, equivalently,
     f(-x)  :=  -f(x)  :: negative(-x)

   In the same way, the pattern `x - y' will match the sum `a + b' by
matching `y' to `-b'.

   The pattern `a b' will also match the formula `x/y' if `y' is a
number.  Thus the rule `a x + b x := (a+b) x' will also convert `a x +
x / 2' to `(a + 0.5) x' (or `(a + 1:2) x', depending on the current
fraction mode).

   Calc will _not_ take other liberties with `*', `/', and `^'.  For
example, the pattern `f(a b)' will not match `f(x^2)', and `f(a + b)'
will not match `f(2 x)', even though conceivably these patterns could
match with `a = b = x'.  Nor will `f(a b)' match `f(x / y)' if `y' is
not a constant, even though it could be considered to match with `a = x'
and `b = 1/y'.  The reasons are partly for efficiency, and partly
because while few mathematical operations are substantively different
for addition and subtraction, often it is preferable to treat the cases
of multiplication, division, and integer powers separately.

   Even more subtle is the rule set

     [ f(a) + f(b) := f(a + b),  -f(a) := f(-a) ]

attempting to match `f(x) - f(y)'.  You might think that Calc will view
this subtraction as `f(x) + (-f(y))' and then apply the above two rules
in turn, but actually this will not work because Calc only does this
when considering rules for `+' (like the first rule in this set).  So
it will see first that `f(x) + (-f(y))' does not match `f(a) + f(b)'
for any assignments of the meta-variables, and then it will see that
`f(x) - f(y)' does not match `-f(a)' for any assignment of `a'.
Because Calc tries only one rule at a time, it will not be able to
rewrite `f(x) - f(y)' with this rule set.  An explicit `f(a) - f(b)'
rule will have to be added.

   Another thing patterns will _not_ do is break up complex numbers.
The pattern `myconj(a + b i) := a - b i' will work for formulas
involving the special constant `i' (such as `3 - 4 i'), but it will not
match actual complex numbers like `(3, -4)'.  A version of the above
rule for complex numbers would be

     myconj(a)  :=  re(a) - im(a) (0,1)  :: im(a) != 0

(Because the `re' and `im' functions understand the properties of the
special constant `i', this rule will also work for `3 - 4 i'.  In fact,
this particular rule would probably be better without the `im(a) != 0'
condition, since if `im(a) = 0' the righthand side of the rule will
still give the correct answer for the conjugate of a real number.)

   It is also possible to specify optional arguments in patterns.  The
rule

     opt(a) x + opt(b) (x^opt(c) + opt(d))  :=  f(a, b, c, d)

will match the formula

     5 (x^2 - 4) + 3 x

in a fairly straightforward manner, but it will also match reduced
formulas like

     x + x^2,    2(x + 1) - x,    x + x

producing, respectively,

     f(1, 1, 2, 0),   f(-1, 2, 1, 1),   f(1, 1, 1, 0)

   (The latter two formulas can be entered only if default
simplifications have been turned off with `m O'.)

   The default value for a term of a sum is zero.  The default value
for a part of a product, for a power, or for the denominator of a
quotient, is one.  Also, `-x' matches the pattern `opt(a) b' with `a =
-1'.

   In particular, the distributive-law rule can be refined to

     opt(a) x + opt(b) x  :=  (a + b) x

so that it will convert, e.g., `a x - x', to `(a - 1) x'.

   The pattern `opt(a) + opt(b) x' matches almost any formulas which
are linear in `x'.  You can also use the `lin' and `islin' functions
with rewrite conditions to test for this; *note Logical Operations::.
These functions are not as convenient to use in rewrite rules, but they
recognize more kinds of formulas as linear: `x/z' is considered linear
with `b = 1/z' by `lin', but it will not match the above pattern
because that pattern calls for a multiplication, not a division.

   As another example, the obvious rule to replace `sin(x)^2 + cos(x)^2'
by 1,

     sin(x)^2 + cos(x)^2  :=  1

misses many cases because the sine and cosine may both be multiplied by
an equal factor.  Here's a more successful rule:

     opt(a) sin(x)^2 + opt(a) cos(x)^2  :=  a

   Note that this rule will _not_ match `sin(x)^2 + 6 cos(x)^2' because
one `a' would have "matched" 1 while the other matched 6.

   Calc automatically converts a rule like

     f(x-1, x)  :=  g(x)

into the form

     f(temp, x)  :=  g(x)  :: temp = x-1

(where `temp' stands for a new, invented meta-variable that doesn't
actually have a name).  This modified rule will successfully match
`f(6, 7)', binding `temp' and `x' to 6 and 7, respectively, then
verifying that they differ by one even though `6' does not
superficially look like `x-1'.

   However, Calc does not solve equations to interpret a rule.  The
following rule,

     f(x-1, x+1)  :=  g(x)

will not work.  That is, it will match `f(a - 1 + b, a + 1 + b)' but
not `f(6, 8)'.  Calc always interprets at least one occurrence of a
variable by literal matching.  If the variable appears "isolated" then
Calc is smart enough to use it for literal matching.  But in this last
example, Calc is forced to rewrite the rule to `f(x-1, temp) := g(x) ::
temp = x+1' where the `x-1' term must correspond to an actual
"something-minus-one" in the target formula.

   A successful way to write this would be `f(x, x+2) := g(x+1)'.  You
could make this resemble the original form more closely by using `let'
notation, which is described in the next section:

     f(xm1, x+1)  :=  g(x)  :: let(x := xm1+1)

   Calc does this rewriting or "conditionalizing" for any sub-pattern
which involves only the functions in the following list, operating only
on constants and meta-variables which have already been matched
elsewhere in the pattern.  When matching a function call, Calc is
careful to match arguments which are plain variables before arguments
which are calls to any of the functions below, so that a pattern like
`f(x-1, x)' can be conditionalized even though the isolated `x' comes
after the `x-1'.

     + - * / \ % ^  abs sign  round rounde roundu trunc floor ceil
     max min  re im conj arg

   You can suppress all of the special treatments described in this
section by surrounding a function call with a `plain' marker.  This
marker causes the function call which is its argument to be matched
literally, without regard to commutativity, associativity, negation, or
conditionalization.  When you use `plain', the "deep structure" of the
formula being matched can show through.  For example,

     plain(a - a b)  :=  f(a, b)

will match only literal subtractions.  However, the `plain' marker does
not affect its arguments' arguments.  In this case, commutativity and
associativity is still considered while matching the `a b' sub-pattern,
so the whole pattern will match `x - y x' as well as `x - x y'.  We
could go still further and use

     plain(a - plain(a b))  :=  f(a, b)

which would do a completely strict match for the pattern.

   By contrast, the `quote' marker means that not only the function
name but also the arguments must be literally the same.  The above
pattern will match `x - x y' but

     quote(a - a b)  :=  f(a, b)

will match only the single formula `a - a b'.  Also,

     quote(a - quote(a b))  :=  f(a, b)

will match only `a - quote(a b)'--probably not the desired effect!

   A certain amount of algebra is also done when substituting the
meta-variables on the righthand side of a rule.  For example, in the
rule

     a + f(b)  :=  f(a + b)

matching `f(x) - y' would produce `f((-y) + x)' if taken literally, but
the rewrite mechanism will simplify the righthand side to `f(x - y)'
automatically.  (Of course, the default simplifications would do this
anyway, so this special simplification is only noticeable if you have
turned the default simplifications off.)  This rewriting is done only
when a meta-variable expands to a "negative-looking" expression.  If
this simplification is not desirable, you can use a `plain' marker on
the righthand side:

     a + f(b)  :=  f(plain(a + b))

In this example, we are still allowing the pattern-matcher to use all
the algebra it can muster, but the righthand side will always simplify
to a literal addition like `f((-y) + x)'.

File: calc,  Node: Other Features of Rewrite Rules,  Next: Composing Patterns in Rewrite Rules,  Prev: Algebraic Properties of Rewrite Rules,  Up: Rewrite Rules

12.11.5 Other Features of Rewrite Rules
---------------------------------------

Certain "function names" serve as markers in rewrite rules.  Here is a
complete list of these markers.  First are listed the markers that work
inside a pattern; then come the markers that work in the righthand side
of a rule.

   One kind of marker, `import(x)', takes the place of a whole rule.
Here `x' is the name of a variable containing another rule set; those
rules are "spliced into" the rule set that imports them.  For example,
if `[f(a+b) := f(a) + f(b), f(a b) := a f(b) :: real(a)]' is stored in
variable `linearF', then the rule set `[f(0) := 0, import(linearF)]'
will apply all three rules.  It is possible to modify the imported rules
slightly:  `import(x, v1, x1, v2, x2, ...)' imports the rule set `x'
with all occurrences of `v1', as either a variable name or a function
name, replaced with `x1' and so on.  (If `v1' is used as a function
name, then `x1' must be either a function name itself or a `< >'
nameless function; *note Specifying Operators::.)  For example, `[g(0)
:= 0, import(linearF, f, g)]' applies the linearity rules to the
function `g' instead of `f'.  Imports can be nested, but the
import-with-renaming feature may fail to rename sub-imports properly.

   The special functions allowed in patterns are:

`quote(x)'
     This pattern matches exactly `x'; variable names in `x' are not
     interpreted as meta-variables.  The only flexibility is that
     numbers are compared for numeric equality, so that the pattern
     `f(quote(12))' will match both `f(12)' and `f(12.0)'.  (Numbers
     are always treated this way by the rewrite mechanism: The rule
     `f(x,x) := g(x)' will match `f(12, 12.0)'.  The rewrite may
     produce either `g(12)' or `g(12.0)' as a result in this case.)

`plain(x)'
     Here `x' must be a function call `f(x1,x2,...)'.  This pattern
     matches a call to function `f' with the specified argument
     patterns.  No special knowledge of the properties of the function
     `f' is used in this case; `+' is not commutative or associative.
     Unlike `quote', the arguments `x1,x2,...' are treated as patterns.
     If you wish them to be treated "plainly" as well, you must enclose
     them with more `plain' markers: `plain(plain(-a) + plain(b c))'.

`opt(x,def)'
     Here `x' must be a variable name.  This must appear as an argument
     to a function or an element of a vector; it specifies that the
     argument or element is optional.  As an argument to `+', `-', `*',
     `&&', or `||', or as the second argument to `/' or `^', the value
     DEF may be omitted.  The pattern `x + opt(y)' matches a sum by
     binding one summand to `x' and the other to `y', and it matches
     anything else by binding the whole expression to `x' and zero to
     `y'.  The other operators above work similarly.

     For general miscellaneous functions, the default value `def' must
     be specified.  Optional arguments are dropped starting with the
     rightmost one during matching.  For example, the pattern
     `f(opt(a,0), b, opt(c,b))' will match `f(b)', `f(a,b)', or
     `f(a,b,c)'.  Default values of zero and `b' are supplied in this
     example for the omitted arguments.  Note that the literal variable
     `b' will be the default in the latter case, _not_ the value that
     matched the meta-variable `b'.  In other words, the default DEF is
     effectively quoted.

`condition(x,c)'
     This matches the pattern `x', with the attached condition `c'.  It
     is the same as `x :: c'.

`pand(x,y)'
     This matches anything that matches both pattern `x' and pattern
     `y'.  It is the same as `x &&& y'.  *note Composing Patterns in
     Rewrite Rules::.

`por(x,y)'
     This matches anything that matches either pattern `x' or pattern
     `y'.  It is the same as `x ||| y'.

`pnot(x)'
     This matches anything that does not match pattern `x'.  It is the
     same as `!!! x'.

`cons(h,t)'
     This matches any vector of one or more elements.  The first
     element is matched to `h'; a vector of the remaining elements is
     matched to `t'.  Note that vectors of fixed length can also be
     matched as actual vectors:  The rule `cons(a,cons(b,[])) :=
     cons(a+b,[])' is equivalent to the rule `[a,b] := [a+b]'.

`rcons(t,h)'
     This is like `cons', except that the _last_ element is matched to
     `h', with the remaining elements matched to `t'.

`apply(f,args)'
     This matches any function call.  The name of the function, in the
     form of a variable, is matched to `f'.  The arguments of the
     function, as a vector of zero or more objects, are matched to
     `args'.  Constants, variables, and vectors do _not_ match an
     `apply' pattern.  For example, `apply(f,x)' matches any function
     call, `apply(quote(f),x)' matches any call to the function `f',
     `apply(f,[a,b])' matches any function call with exactly two
     arguments, and `apply(quote(f), cons(a,cons(b,x)))' matches any
     call to the function `f' with two or more arguments.  Another way
     to implement the latter, if the rest of the rule does not need to
     refer to the first two arguments of `f' by name, would be
     `apply(quote(f), x :: vlen(x) >= 2)'.  Here's a more interesting
     sample use of `apply':

          apply(f,[x+n])  :=  n + apply(f,[x])
             :: in(f, [floor,ceil,round,trunc]) :: integer(n)

     Note, however, that this will be slower to match than a rule set
     with four separate rules.  The reason is that Calc sorts the rules
     of a rule set according to top-level function name; if the
     top-level function is `apply', Calc must try the rule for every
     single formula and sub-formula.  If the top-level function in the
     pattern is, say, `floor', then Calc invokes the rule only for
     sub-formulas which are calls to `floor'.

     Formulas normally written with operators like `+' are still
     considered function calls:  `apply(f,x)' matches `a+b' with `f =
     add', `x = [a,b]'.

     You must use `apply' for meta-variables with function names on
     both sides of a rewrite rule:  `apply(f, [x]) := f(x+1)' is _not_
     correct, because it rewrites `spam(6)' into `f(7)'.  The righthand
     side should be `apply(f, [x+1])'.  Also note that you will have to
     use No-Simplify mode (`m O') when entering this rule so that the
     `apply' isn't evaluated immediately to get the new rule `f(x) :=
     f(x+1)'.  Or, use `s e' to enter the rule without going through
     the stack, or enter the rule as `apply(f, [x]) := apply(f, [x+1])
     :: 1'.  *Note Conditional Rewrite Rules::.

`select(x)'
     This is used for applying rules to formulas with selections; *note
     Selections with Rewrite Rules::.

   Special functions for the righthand sides of rules are:

`quote(x)'
     The notation `quote(x)' is changed to `x' when the righthand side
     is used.  As far as the rewrite rule is concerned, `quote' is
     invisible.  However, `quote' has the special property in Calc that
     its argument is not evaluated.  Thus, while it will not work to
     put the rule `t(a) := typeof(a)' on the stack because `typeof(a)'
     is evaluated immediately to produce `t(a) := 100', you can use
     `quote' to protect the righthand side:  `t(a) := quote(typeof(a))'.
     (*Note Conditional Rewrite Rules::, for another trick for
     protecting rules from evaluation.)

`plain(x)'
     Special properties of and simplifications for the function call
     `x' are not used.  One interesting case where `plain' is useful is
     the rule, `q(x) := quote(x)', trying to expand a shorthand
     notation for the `quote' function.  This rule will not work as
     shown; instead of replacing `q(foo)' with `quote(foo)', it will
     replace it with `foo'!  The correct rule would be `q(x) :=
     plain(quote(x))'.

`cons(h,t)'
     Where `t' is a vector, this is converted into an expanded vector
     during rewrite processing.  Note that `cons' is a regular Calc
     function which normally does this anyway; the only way `cons' is
     treated specially by rewrites is that `cons' on the righthand side
     of a rule will be evaluated even if default simplifications have
     been turned off.

`rcons(t,h)'
     Analogous to `cons' except putting `h' at the _end_ of the vector
     `t'.

`apply(f,args)'
     Where `f' is a variable and ARGS is a vector, this is converted to
     a function call.  Once again, note that `apply' is also a regular
     Calc function.

`eval(x)'
     The formula `x' is handled in the usual way, then the default
     simplifications are applied to it even if they have been turned
     off normally.  This allows you to treat any function similarly to
     the way `cons' and `apply' are always treated.  However, there is
     a slight difference:  `cons(2+3, [])' with default simplifications
     off will be converted to `[2+3]', whereas `eval(cons(2+3, []))'
     will be converted to `[5]'.

`evalsimp(x)'
     The formula `x' has meta-variables substituted in the usual way,
     then algebraically simplified as if by the `a s' command.

`evalextsimp(x)'
     The formula `x' has meta-variables substituted in the normal way,
     then "extendedly" simplified as if by the `a e' command.

`select(x)'
     *Note Selections with Rewrite Rules::.

   There are also some special functions you can use in conditions.

`let(v := x)'
     The expression `x' is evaluated with meta-variables substituted.
     The `a s' command's simplifications are _not_ applied by default,
     but `x' can include calls to `evalsimp' or `evalextsimp' as
     described above to invoke higher levels of simplification.  The
     result of `x' is then bound to the meta-variable `v'.  As usual,
     if this meta-variable has already been matched to something else
     the two values must be equal; if the meta-variable is new then it
     is bound to the result of the expression.  This variable can then
     appear in later conditions, and on the righthand side of the rule.
     In fact, `v' may be any pattern in which case the result of
     evaluating `x' is matched to that pattern, binding any
     meta-variables that appear in that pattern.  Note that `let' can
     only appear by itself as a condition, or as one term of an `&&'
     which is a whole condition:  It cannot be inside an `||' term or
     otherwise buried.

     The alternate, equivalent form `let(v, x)' is also recognized.
     Note that the use of `:=' by `let', while still being
     assignment-like in character, is unrelated to the use of `:=' in
     the main part of a rewrite rule.

     As an example, `f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)'
     replaces `f(a)' with `g' of the inverse of `a', if that inverse
     exists and is constant.  For example, if `a' is a singular matrix
     the operation `1/a' is left unsimplified and `constant(ia)' fails,
     but if `a' is an invertible matrix then the rule succeeds.
     Without `let' there would be no way to express this rule that
     didn't have to invert the matrix twice.  Note that, because the
     meta-variable `ia' is otherwise unbound in this rule, the `let'
     condition itself always "succeeds" because no matter what `1/a'
     evaluates to, it can successfully be bound to `ia'.

     Here's another example, for integrating cosines of linear terms:
     `myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))'.  The
     `lin' function returns a 3-vector if its argument is linear, or
     leaves itself unevaluated if not.  But an unevaluated `lin' call
     will not match the 3-vector on the lefthand side of the `let', so
     this `let' both verifies that `y' is linear, and binds the
     coefficients `a' and `b' for use elsewhere in the rule.  (It would
     have been possible to use `sin(a x + b)/b' for the righthand side
     instead, but using `sin(y)/b' avoids gratuitous rearrangement of
     the argument of the sine.)

     Similarly, here is a rule that implements an inverse-`erf'
     function.  It uses `root' to search for a solution.  If `root'
     succeeds, it will return a vector of two numbers where the first
     number is the desired solution.  If no solution is found, `root'
     remains in symbolic form.  So we use `let' to check that the
     result was indeed a vector.

          ierf(x)  :=  y  :: let([y,z] := root(erf(a) = x, a, .5))

`matches(v,p)'
     The meta-variable V, which must already have been matched to
     something elsewhere in the rule, is compared against pattern P.
     Since `matches' is a standard Calc function, it can appear
     anywhere in a condition.  But if it appears alone or as a term of
     a top-level `&&', then you get the special extra feature that
     meta-variables which are bound to things inside P can be used
     elsewhere in the surrounding rewrite rule.

     The only real difference between `let(p := v)' and `matches(v, p)'
     is that the former evaluates `v' using the default
     simplifications, while the latter does not.

`remember'
     This is actually a variable, not a function.  If `remember'
     appears as a condition in a rule, then when that rule succeeds the
     original expression and rewritten expression are added to the
     front of the rule set that contained the rule.  If the rule set
     was not stored in a variable, `remember' is ignored.  The lefthand
     side is enclosed in `quote' in the added rule if it contains any
     variables.

     For example, the rule `f(n) := n f(n-1) :: remember' applied to
     `f(7)' will add the rule `f(7) := 7 f(6)' to the front of the rule
     set.  The rule set `EvalRules' works slightly differently:  There,
     the evaluation of `f(6)' will complete before the result is added
     to the rule set, in this case as `f(7) := 5040'.  Thus `remember'
     is most useful inside `EvalRules'.

     It is up to you to ensure that the optimization performed by
     `remember' is safe.  For example, the rule `foo(n) := n ::
     evalv(eatfoo) > 0 :: remember' is a bad idea (`evalv' is the
     function equivalent of the `=' command); if the variable `eatfoo'
     ever contains 1, rules like `foo(7) := 7' will be added to the
     rule set and will continue to operate even if `eatfoo' is later
     changed to 0.

`remember(c)'
     Remember the match as described above, but only if condition `c'
     is true.  For example, `remember(n % 4 = 0)' in the above factorial
     rule remembers only every fourth result.  Note that `remember(1)'
     is equivalent to `remember', and `remember(0)' has no effect.

File: calc,  Node: Composing Patterns in Rewrite Rules,  Next: Nested Formulas with Rewrite Rules,  Prev: Other Features of Rewrite Rules,  Up: Rewrite Rules

12.11.6 Composing Patterns in Rewrite Rules
-------------------------------------------

There are three operators, `&&&', `|||', and `!!!', that combine
rewrite patterns to make larger patterns.  The combinations are "and,"
"or," and "not," respectively, and these operators are the pattern
equivalents of `&&', `||' and `!' (which operate on zero-or-nonzero
logical values).

   Note that `&&&', `|||', and `!!!' are left in symbolic form by all
regular Calc features; they have special meaning only in the context of
rewrite rule patterns.

   The pattern `P1 &&& P2' matches anything that matches both P1 and
P2.  One especially useful case is when one of P1 or P2 is a
meta-variable.  For example, here is a rule that operates on error
forms:

     f(x &&& a +/- b, x)  :=  g(x)

   This does the same thing, but is arguably simpler than, the rule

     f(a +/- b, a +/- b)  :=  g(a +/- b)

   Here's another interesting example:

     ends(cons(a, x) &&& rcons(y, b))  :=  [a, b]

which effectively clips out the middle of a vector leaving just the
first and last elements.  This rule will change a one-element vector
`[a]' to `[a, a]'.  The similar rule

     ends(cons(a, rcons(y, b)))  :=  [a, b]

would do the same thing except that it would fail to match a
one-element vector.

   The pattern `P1 ||| P2' matches anything that matches either P1 or
P2.  Calc first tries matching against P1; if that fails, it goes on to
try P2.

   A simple example of `|||' is

     curve(inf ||| -inf)  :=  0

which converts both `curve(inf)' and `curve(-inf)' to zero.

   Here is a larger example:

     log(a, b) ||| (ln(a) :: let(b := e))  :=  mylog(a, b)

   This matches both generalized and natural logarithms in a single
rule.  Note that the `::' term must be enclosed in parentheses because
that operator has lower precedence than `|||' or `:='.

   (In practice this rule would probably include a third alternative,
omitted here for brevity, to take care of `log10'.)

   While Calc generally treats interior conditions exactly the same as
conditions on the outside of a rule, it does guarantee that if all the
variables in the condition are special names like `e', or already bound
in the pattern to which the condition is attached (say, if `a' had
appeared in this condition), then Calc will process this condition
right after matching the pattern to the left of the `::'.  Thus, we
know that `b' will be bound to `e' only if the `ln' branch of the `|||'
was taken.

   Note that this rule was careful to bind the same set of
meta-variables on both sides of the `|||'.  Calc does not check this,
but if you bind a certain meta-variable only in one branch and then use
that meta-variable elsewhere in the rule, results are unpredictable:

     f(a,b) ||| g(b)  :=  h(a,b)

   Here if the pattern matches `g(17)', Calc makes no promises about
the value that will be substituted for `a' on the righthand side.

   The pattern `!!! PAT' matches anything that does not match PAT.  Any
meta-variables that are bound while matching PAT remain unbound outside
of PAT.

   For example,

     f(x &&& !!! a +/- b, !!![])  :=  g(x)

converts `f' whose first argument is anything _except_ an error form,
and whose second argument is not the empty vector, into a similar call
to `g' (but without the second argument).

   If we know that the second argument will be a vector (empty or not),
then an equivalent rule would be:

     f(x, y)  :=  g(x)  :: typeof(x) != 7 :: vlen(y) > 0

where of course 7 is the `typeof' code for error forms.  Another final
condition, that works for any kind of `y', would be `!istrue(y == [])'.
(The `istrue' function returns an explicit 0 if its argument was left
in symbolic form; plain `!(y == [])' or `y != []' would not work to
replace `!!![]' since these would be left unsimplified, and thus cause
the rule to fail, if `y' was something like a variable name.)

   It is possible for a `!!!' to refer to meta-variables bound
elsewhere in the pattern.  For example,

     f(a, !!!a)  :=  g(a)

matches any call to `f' with different arguments, changing this to `g'
with only the first argument.

   If a function call is to be matched and one of the argument patterns
contains a `!!!' somewhere inside it, that argument will be matched
last.  Thus

     f(!!!a, a)  :=  g(a)

will be careful to bind `a' to the second argument of `f' before
testing the first argument.  If Calc had tried to match the first
argument of `f' first, the results would have been disastrous: since
`a' was unbound so far, the pattern `a' would have matched anything at
all, and the pattern `!!!a' therefore would _not_ have matched anything
at all!

File: calc,  Node: Nested Formulas with Rewrite Rules,  Next: Multi-Phase Rewrite Rules,  Prev: Composing Patterns in Rewrite Rules,  Up: Rewrite Rules

12.11.7 Nested Formulas with Rewrite Rules
------------------------------------------

When `a r' (`calc-rewrite') is used, it takes an expression from the
top of the stack and attempts to match any of the specified rules to
any part of the expression, starting with the whole expression and
then, if that fails, trying deeper and deeper sub-expressions.  For
each part of the expression, the rules are tried in the order they
appear in the rules vector.  The first rule to match the first
sub-expression wins; it replaces the matched sub-expression according
to the NEW part of the rule.

   Often, the rule set will match and change the formula several times.
The top-level formula is first matched and substituted repeatedly until
it no longer matches the pattern; then, sub-formulas are tried, and so
on.  Once every part of the formula has gotten its chance, the rewrite
mechanism starts over again with the top-level formula (in case a
substitution of one of its arguments has caused it again to match).
This continues until no further matches can be made anywhere in the
formula.

   It is possible for a rule set to get into an infinite loop.  The
most obvious case, replacing a formula with itself, is not a problem
because a rule is not considered to "succeed" unless the righthand side
actually comes out to something different than the original formula or
sub-formula that was matched.  But if you accidentally had both `ln(a
b) := ln(a) + ln(b)' and the reverse `ln(a) + ln(b) := ln(a b)' in your
rule set, Calc would run forever switching a formula back and forth
between the two forms.

   To avoid disaster, Calc normally stops after 100 changes have been
made to the formula.  This will be enough for most multiple rewrites,
but it will keep an endless loop of rewrites from locking up the
computer forever.  (On most systems, you can also type `C-g' to halt
any Emacs command prematurely.)

   To change this limit, give a positive numeric prefix argument.  In
particular, `M-1 a r' applies only one rewrite at a time, useful when
you are first testing your rule (or just if repeated rewriting is not
what is called for by your application).

   You can also put a "function call" `iterations(N)' in place of a
rule anywhere in your rules vector (but usually at the top).  Then, N
will be used instead of 100 as the default number of iterations for
this rule set.  You can use `iterations(inf)' if you want no iteration
limit by default.  A prefix argument will override the `iterations'
limit in the rule set.

     [ iterations(1),
       f(x) := f(x+1) ]

   More precisely, the limit controls the number of "iterations," where
each iteration is a successful matching of a rule pattern whose
righthand side, after substituting meta-variables and applying the
default simplifications, is different from the original sub-formula
that was matched.

   A prefix argument of zero sets the limit to infinity.  Use with
caution!

   Given a negative numeric prefix argument, `a r' will match and
substitute the top-level expression up to that many times, but will not
attempt to match the rules to any sub-expressions.

   In a formula, `rewrite(EXPR, RULES, N)' does a rewriting operation.
Here EXPR is the expression being rewritten, RULES is the rule, vector
of rules, or variable containing the rules, and N is the optional
iteration limit, which may be a positive integer, a negative integer,
or `inf' or `-inf'.  If N is omitted the `iterations' value from the
rule set is used; if both are omitted, 100 is used.

File: calc,  Node: Multi-Phase Rewrite Rules,  Next: Selections with Rewrite Rules,  Prev: Nested Formulas with Rewrite Rules,  Up: Rewrite Rules

12.11.8 Multi-Phase Rewrite Rules
---------------------------------

It is possible to separate a rewrite rule set into several "phases".
During each phase, certain rules will be enabled while certain others
will be disabled.  A "phase schedule" controls the order in which
phases occur during the rewriting process.

   If a call to the marker function `phase' appears in the rules vector
in place of a rule, all rules following that point will be members of
the phase(s) identified in the arguments to `phase'.  Phases are given
integer numbers.  The markers `phase()' and `phase(all)' both mean the
following rules belong to all phases; this is the default at the start
of the rule set.

   If you do not explicitly schedule the phases, Calc sorts all phase
numbers that appear in the rule set and executes the phases in
ascending order.  For example, the rule set

     [ f0(x) := g0(x),
       phase(1),
       f1(x) := g1(x),
       phase(2),
       f2(x) := g2(x),
       phase(3),
       f3(x) := g3(x),
       phase(1,2),
       f4(x) := g4(x) ]

has three phases, 1 through 3.  Phase 1 consists of the `f0', `f1', and
`f4' rules (in that order).  Phase 2 consists of `f0', `f2', and `f4'.
Phase 3 consists of `f0' and `f3'.

   When Calc rewrites a formula using this rule set, it first rewrites
the formula using only the phase 1 rules until no further changes are
possible.  Then it switches to the phase 2 rule set and continues until
no further changes occur, then finally rewrites with phase 3.  When no
more phase 3 rules apply, rewriting finishes.  (This is assuming `a r'
with a large enough prefix argument to allow the rewriting to run to
completion; the sequence just described stops early if the number of
iterations specified in the prefix argument, 100 by default, is
reached.)

   During each phase, Calc descends through the nested levels of the
formula as described previously.  (*Note Nested Formulas with Rewrite
Rules::.)  Rewriting starts at the top of the formula, then works its
way down to the parts, then goes back to the top and works down again.
The phase 2 rules do not begin until no phase 1 rules apply anywhere in
the formula.

   A `schedule' marker appearing in the rule set (anywhere, but
conventionally at the top) changes the default schedule of phases.  In
the simplest case, `schedule' has a sequence of phase numbers for
arguments; each phase number is invoked in turn until the arguments to
`schedule' are exhausted.  Thus adding `schedule(3,2,1)' at the top of
the above rule set would reverse the order of the phases;
`schedule(1,2,3)' would have no effect since this is the default
schedule; and `schedule(1,2,1,3)' would give phase 1 a second chance
after phase 2 has completed, before moving on to phase 3.

   Any argument to `schedule' can instead be a vector of phase numbers
(or even of sub-vectors).  Then the sub-sequence of phases described by
the vector are tried repeatedly until no change occurs in any phase in
the sequence.  For example, `schedule([1, 2], 3)' tries phase 1, then
phase 2, then, if either phase made any changes to the formula, repeats
these two phases until they can make no further progress.  Finally, it
goes on to phase 3 for finishing touches.

   Also, items in `schedule' can be variable names as well as numbers.
A variable name is interpreted as the name of a function to call on the
whole formula.  For example, `schedule(1, simplify)' says to apply the
phase-1 rules (presumably, all of them), then to call `simplify' which
is the function name equivalent of `a s'.  Likewise, `schedule([1,
simplify])' says to alternate between phase 1 and `a s' until no
further changes occur.

   Phases can be used purely to improve efficiency; if it is known that
a certain group of rules will apply only at the beginning of rewriting,
and a certain other group will apply only at the end, then rewriting
will be faster if these groups are identified as separate phases.  Once
the phase 1 rules are done, Calc can put them aside and no longer spend
any time on them while it works on phase 2.

   There are also some problems that can only be solved with several
rewrite phases.  For a real-world example of a multi-phase rule set,
examine the set `FitRules', which is used by the curve-fitting command
to convert a model expression to linear form.  *Note Curve Fitting
Details::.  This set is divided into four phases.  The first phase
rewrites certain kinds of expressions to be more easily linearizable,
but less computationally efficient.  After the linear components have
been picked out, the final phase includes the opposite rewrites to put
each component back into an efficient form.  If both sets of rules were
included in one big phase, Calc could get into an infinite loop going
back and forth between the two forms.

   Elsewhere in `FitRules', the components are first isolated, then
recombined where possible to reduce the complexity of the linear fit,
then finally packaged one component at a time into vectors.  If the
packaging rules were allowed to begin before the recombining rules were
finished, some components might be put away into vectors before they
had a chance to recombine.  By putting these rules in two separate
phases, this problem is neatly avoided.

File: calc,  Node: Selections with Rewrite Rules,  Next: Matching Commands,  Prev: Multi-Phase Rewrite Rules,  Up: Rewrite Rules

12.11.9 Selections with Rewrite Rules
-------------------------------------

If a sub-formula of the current formula is selected (as by `j s'; *note
Selecting Subformulas::), the `a r' (`calc-rewrite') command applies
only to that sub-formula.  Together with a negative prefix argument,
you can use this fact to apply a rewrite to one specific part of a
formula without affecting any other parts.

   The `j r' (`calc-rewrite-selection') command allows more
sophisticated operations on selections.  This command prompts for the
rules in the same way as `a r', but it then applies those rules to the
whole formula in question even though a sub-formula of it has been
selected.  However, the selected sub-formula will first have been
surrounded by a `select( )' function call.  (Calc's evaluator does not
understand the function name `select'; this is only a tag used by the
`j r' command.)

   For example, suppose the formula on the stack is `2 (a + b)^2' and
the sub-formula `a + b' is selected.  This formula will be rewritten to
`2 select(a + b)^2' and then the rewrite rules will be applied in the
usual way.  The rewrite rules can include references to `select' to
tell where in the pattern the selected sub-formula should appear.

   If there is still exactly one `select( )' function call in the
formula after rewriting is done, it indicates which part of the formula
should be selected afterwards.  Otherwise, the formula will be
unselected.

   You can make `j r' act much like `a r' by enclosing both parts of
the rewrite rule with `select()'.  However, `j r' allows you to use the
current selection in more flexible ways.  Suppose you wished to make a
rule which removed the exponent from the selected term; the rule
`select(a)^x := select(a)' would work.  In the above example, it would
rewrite `2 select(a + b)^2' to `2 select(a + b)'.  This would then be
returned to the stack as `2 (a + b)' with the `a + b' selected.

   The `j r' command uses one iteration by default, unlike `a r' which
defaults to 100 iterations.  A numeric prefix argument affects `j r' in
the same way as `a r'.  *Note Nested Formulas with Rewrite Rules::.

   As with other selection commands, `j r' operates on the stack entry
that contains the cursor.  (If the cursor is on the top-of-stack `.'
marker, it works as if the cursor were on the formula at stack level 1.)

   If you don't specify a set of rules, the rules are taken from the
top of the stack, just as with `a r'.  In this case, the cursor must
indicate stack entry 2 or above as the formula to be rewritten
(otherwise the same formula would be used as both the target and the
rewrite rules).

   If the indicated formula has no selection, the cursor position within
the formula temporarily selects a sub-formula for the purposes of this
command.  If the cursor is not on any sub-formula (e.g., it is in the
line-number area to the left of the formula), the `select( )' markers
are ignored by the rewrite mechanism and the rules are allowed to apply
anywhere in the formula.

   As a special feature, the normal `a r' command also ignores `select(
)' calls in rewrite rules.  For example, if you used the above rule
`select(a)^x := select(a)' with `a r', it would apply the rule as if it
were `a^x := a'.  Thus, you can write general purpose rules with
`select( )' hints inside them so that they will "do the right thing" in
both `a r' and `j r', both with and without selections.

File: calc,  Node: Matching Commands,  Next: Automatic Rewrites,  Prev: Selections with Rewrite Rules,  Up: Rewrite Rules

12.11.10 Matching Commands
--------------------------

The `a m' (`calc-match') [`match'] function takes a vector of formulas
and a rewrite-rule-style pattern, and produces a vector of all formulas
which match the pattern.  The command prompts you to enter the pattern;
as for `a r', you can enter a single pattern (i.e., a formula with
meta-variables), or a vector of patterns, or a variable which contains
patterns, or you can give a blank response in which case the patterns
are taken from the top of the stack.  The pattern set will be compiled
once and saved if it is stored in a variable.  If there are several
patterns in the set, vector elements are kept if they match any of the
patterns.

   For example, `match(a+b, [x, x+y, x-y, 7, x+y+z])' will return
`[x+y, x-y, x+y+z]'.

   The `import' mechanism is not available for pattern sets.

   The `a m' command can also be used to extract all vector elements
which satisfy any condition:  The pattern `x :: x>0' will select all
the positive vector elements.

   With the Inverse flag [`matchnot'], this command extracts all vector
elements which do _not_ match the given pattern.

   There is also a function `matches(X, P)' which evaluates to 1 if
expression X matches pattern P, or to 0 otherwise.  This is sometimes
useful for including into the conditional clauses of other rewrite
rules.

   The function `vmatches' is just like `matches', except that if the
match succeeds it returns a vector of assignments to the meta-variables
instead of the number 1.  For example, `vmatches(f(1,2), f(a,b))'
returns `[a := 1, b := 2]'.  If the match fails, the function returns
the number 0.

File: calc,  Node: Automatic Rewrites,  Next: Debugging Rewrites,  Prev: Matching Commands,  Up: Rewrite Rules

12.11.11 Automatic Rewrites
---------------------------

It is possible to get Calc to apply a set of rewrite rules on all
results, effectively adding to the built-in set of default
simplifications.  To do this, simply store your rule set in the
variable `EvalRules'.  There is a convenient `s E' command for editing
`EvalRules'; *note Operations on Variables::.

   For example, suppose you want `sin(a + b)' to be expanded out to
`sin(b) cos(a) + cos(b) sin(a)' wherever it appears, and similarly for
`cos(a + b)'.  The corresponding rewrite rule set would be,

     [ sin(a + b)  :=  cos(a) sin(b) + sin(a) cos(b),
       cos(a + b)  :=  cos(a) cos(b) - sin(a) sin(b) ]

   To apply these manually, you could put them in a variable called
`trigexp' and then use `a r trigexp' every time you wanted to expand
trig functions.  But if instead you store them in the variable
`EvalRules', they will automatically be applied to all sines and
cosines of sums.  Then, with `2 x' and `45' on the stack, typing `+ S'
will (assuming Degrees mode) result in `0.7071 sin(2 x) + 0.7071 cos(2
x)' automatically.

   As each level of a formula is evaluated, the rules from `EvalRules'
are applied before the default simplifications.  Rewriting continues
until no further `EvalRules' apply.  Note that this is different from
the usual order of application of rewrite rules:  `EvalRules' works
from the bottom up, simplifying the arguments to a function before the
function itself, while `a r' applies rules from the top down.

   Because the `EvalRules' are tried first, you can use them to
override the normal behavior of any built-in Calc function.

   It is important not to write a rule that will get into an infinite
loop.  For example, the rule set `[f(0) := 1, f(n) := n f(n-1)]'
appears to be a good definition of a factorial function, but it is
unsafe.  Imagine what happens if `f(2.5)' is simplified.  Calc will
continue to subtract 1 from this argument forever without reaching
zero.  A safer second rule would be `f(n) := n f(n-1) :: n>0'.  Another
dangerous rule is `g(x, y) := g(y, x)'.  Rewriting `g(2, 4)', this
would bounce back and forth between that and `g(4, 2)' forever.  If an
infinite loop in `EvalRules' occurs, Emacs will eventually stop with a
"Computation got stuck or ran too long" message.

   Another subtle difference between `EvalRules' and regular rewrites
concerns rules that rewrite a formula into an identical formula.  For
example, `f(n) := f(floor(n))' "fails to match" when `n' is already an
integer.  But in `EvalRules' this case is detected only if the
righthand side literally becomes the original formula before any
further simplification.  This means that `f(n) := f(floor(n))' will get
into an infinite loop if it occurs in `EvalRules'.  Calc will replace
`f(6)' with `f(floor(6))', which is different from `f(6)', so it will
consider the rule to have matched and will continue simplifying that
formula; first the argument is simplified to get `f(6)', then the rule
matches again to get `f(floor(6))' again, ad infinitum.  A much safer
rule would check its argument first, say, with `f(n) := f(floor(n)) ::
!dint(n)'.

   (What really happens is that the rewrite mechanism substitutes the
meta-variables in the righthand side of a rule, compares to see if the
result is the same as the original formula and fails if so, then uses
the default simplifications to simplify the result and compares again
(and again fails if the formula has simplified back to its original
form).  The only special wrinkle for the `EvalRules' is that the same
rules will come back into play when the default simplifications are
used.  What Calc wants to do is build `f(floor(6))', see that this is
different from the original formula, simplify to `f(6)', see that this
is the same as the original formula, and thus halt the rewriting.  But
while simplifying, `f(6)' will again trigger the same `EvalRules' rule
and Calc will get into a loop inside the rewrite mechanism itself.)

   The `phase', `schedule', and `iterations' markers do not work in
`EvalRules'.  If the rule set is divided into phases, only the phase 1
rules are applied, and the schedule is ignored.  The rules are always
repeated as many times as possible.

   The `EvalRules' are applied to all function calls in a formula, but
not to numbers (and other number-like objects like error forms), nor to
vectors or individual variable names.  (Though they will apply to
_components_ of vectors and error forms when appropriate.)  You might
try to make a variable `phihat' which automatically expands to its
definition without the need to press `=' by writing the rule
`quote(phihat) := (1-sqrt(5))/2', but unfortunately this rule will not
work as part of `EvalRules'.

   Finally, another limitation is that Calc sometimes calls its built-in
functions directly rather than going through the default
simplifications.  When it does this, `EvalRules' will not be able to
override those functions.  For example, when you take the absolute
value of the complex number `(2, 3)', Calc computes `sqrt(2*2 + 3*3)'
by calling the multiplication, addition, and square root functions
directly rather than applying the default simplifications to this
formula.  So an `EvalRules' rule that (perversely) rewrites `sqrt(13)
:= 6' would not apply.  (However, if you put Calc into Symbolic mode so
that `sqrt(13)' will be left in symbolic form by the built-in square
root function, your rule will be able to apply.  But if the complex
number were `(3,4)', so that `sqrt(25)' must be calculated, then
Symbolic mode will not help because `sqrt(25)' can be evaluated exactly
to 5.)

   One subtle restriction that normally only manifests itself with
`EvalRules' is that while a given rewrite rule is in the process of
being checked, that same rule cannot be recursively applied.  Calc
effectively removes the rule from its rule set while checking the rule,
then puts it back once the match succeeds or fails.  (The technical
reason for this is that compiled pattern programs are not reentrant.)
For example, consider the rule `foo(x) := x :: foo(x/2) > 0' attempting
to match `foo(8)'.  This rule will be inactive while the condition
`foo(4) > 0' is checked, even though it might be an integral part of
evaluating that condition.  Note that this is not a problem for the
more usual recursive type of rule, such as `foo(x) := foo(x/2)',
because there the rule has succeeded and been reactivated by the time
the righthand side is evaluated.

   If `EvalRules' has no stored value (its default state), or if
anything but a vector is stored in it, then it is ignored.

   Even though Calc's rewrite mechanism is designed to compare rewrite
rules to formulas as quickly as possible, storing rules in `EvalRules'
may make Calc run substantially slower.  This is particularly true of
rules where the top-level call is a commonly used function, or is not
fixed.  The rule `f(n) := n f(n-1) :: n>0' will only activate the
rewrite mechanism for calls to the function `f', but `lg(n) + lg(m) :=
lg(n m)' will check every `+' operator.

     apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])

may seem more "efficient" than two separate rules for `ln' and `log10',
but actually it is vastly less efficient because rules with `apply' as
the top-level pattern must be tested against _every_ function call that
is simplified.

   Suppose you want `sin(a + b)' to be expanded out not all the time,
but only when `a s' is used to simplify the formula.  The variable
`AlgSimpRules' holds rules for this purpose.  The `a s' command will
apply `EvalRules' and `AlgSimpRules' to the formula, as well as all of
its built-in simplifications.

   Most of the special limitations for `EvalRules' don't apply to
`AlgSimpRules'.  Calc simply does an `a r AlgSimpRules' command with an
infinite repeat count as the first step of `a s'.  It then applies its
own built-in simplifications throughout the formula, and then repeats
these two steps (along with applying the default simplifications) until
no further changes are possible.

   There are also `ExtSimpRules' and `UnitSimpRules' variables that are
used by `a e' and `u s', respectively; these commands also apply
`EvalRules' and `AlgSimpRules'.  The variable `IntegSimpRules' contains
simplification rules that are used only during integration by `a i'.

File: calc,  Node: Debugging Rewrites,  Next: Examples of Rewrite Rules,  Prev: Automatic Rewrites,  Up: Rewrite Rules

12.11.12 Debugging Rewrites
---------------------------

If a buffer named `*Trace*' exists, the rewrite mechanism will record
some useful information there as it operates.  The original formula is
written there, as is the result of each successful rewrite, and the
final result of the rewriting.  All phase changes are also noted.

   Calc always appends to `*Trace*'.  You must empty this buffer
yourself periodically if it is in danger of growing unwieldy.

   Note that the rewriting mechanism is substantially slower when the
`*Trace*' buffer exists, even if the buffer is not visible on the
screen.  Once you are done, you will probably want to kill this buffer
(with `C-x k *Trace* <RET>').  If you leave it in existence and forget
about it, all your future rewrite commands will be needlessly slow.

File: calc,  Node: Examples of Rewrite Rules,  Prev: Debugging Rewrites,  Up: Rewrite Rules

12.11.13 Examples of Rewrite Rules
----------------------------------

Returning to the example of substituting the pattern `sin(x)^2 +
cos(x)^2' with 1, we saw that the rule `opt(a) sin(x)^2 + opt(a)
cos(x)^2 := a' does a good job of finding suitable cases.  Another
solution would be to use the rule `cos(x)^2 := 1 - sin(x)^2', followed
by algebraic simplification if necessary.  This rule will be the most
effective way to do the job, but at the expense of making some changes
that you might not desire.

   Another algebraic rewrite rule is `exp(x+y) := exp(x) exp(y)'.  To
make this work with the `j r' command so that it can be easily targeted
to a particular exponential in a large formula, you might wish to write
the rule as `select(exp(x+y)) := select(exp(x) exp(y))'.  The `select'
markers will be ignored by the regular `a r' command (*note Selections
with Rewrite Rules::).

   A surprisingly useful rewrite rule is `a/(b-c) := a*(b+c)/(b^2-c^2)'.
This will simplify the formula whenever `b' and/or `c' can be made
simpler by squaring.  For example, applying this rule to `2 / (sqrt(2)
+ 3)' yields `6:7 - 2:7 sqrt(2)' (assuming Symbolic mode has been
enabled to keep the square root from being evaluated to a
floating-point approximation).  This rule is also useful when working
with symbolic complex numbers, e.g., `(a + b i) / (c + d i)'.

   As another example, we could define our own "triangular numbers"
function with the rules `[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]'.
Enter this vector and store it in a variable:  `s t trirules'.  Now,
given a suitable formula like `tri(5)' on the stack, type `a r trirules'
to apply these rules repeatedly.  After six applications, `a r' will
stop with 15 on the stack.  Once these rules are debugged, it would
probably be most useful to add them to `EvalRules' so that Calc will
evaluate the new `tri' function automatically.  We could then use `Z K'
on the keyboard macro `' tri($) <RET>' to make a command that applies
`tri' to the value on the top of the stack.  *Note Programming::.

   The following rule set, contributed by Francois Pinard, implements
"quaternions", a generalization of the concept of complex numbers.
Quaternions have four components, and are here represented by function
calls `quat(W, [X, Y, Z])' with "real part" W and the three "imaginary"
parts collected into a vector.  Various arithmetical operations on
quaternions are supported.  To use these rules, either add them to
`EvalRules', or create a command based on `a r' for simplifying
quaternion formulas.  A convenient way to enter quaternions would be a
command defined by a keyboard macro containing: `' quat($$$$, [$$$, $$,
$]) <RET>'.

     [ quat(w, x, y, z) := quat(w, [x, y, z]),
       quat(w, [0, 0, 0]) := w,
       abs(quat(w, v)) := hypot(w, v),
       -quat(w, v) := quat(-w, -v),
       r + quat(w, v) := quat(r + w, v) :: real(r),
       r - quat(w, v) := quat(r - w, -v) :: real(r),
       quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
       r * quat(w, v) := quat(r * w, r * v) :: real(r),
       plain(quat(w1, v1) * quat(w2, v2))
          := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
       quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
       z / quat(w, v) := z * quatinv(quat(w, v)),
       quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
       quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
       quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
                    :: integer(k) :: k > 0 :: k % 2 = 0,
       quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
                    :: integer(k) :: k > 2,
       quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]

   Quaternions, like matrices, have non-commutative multiplication.  In
other words, `q1 * q2 = q2 * q1' is not necessarily true if `q1' and
`q2' are `quat' forms.  The `quat*quat' rule above uses `plain' to
prevent Calc from rearranging the product.  It may also be wise to add
the line `[quat(), matrix]' to the `Decls' matrix, to ensure that
Calc's other algebraic operations will not rearrange a quaternion
product.  *Note Declarations::.

   These rules also accept a four-argument `quat' form, converting it
to the preferred form in the first rule.  If you would rather see
results in the four-argument form, just append the two items `phase(2),
quat(w, [x, y, z]) := quat(w, x, y, z)' to the end of the rule set.
(But remember that multi-phase rule sets don't work in `EvalRules'.)

File: calc,  Node: Units,  Next: Store and Recall,  Prev: Algebra,  Up: Top

13 Operating on Units
*********************

One special interpretation of algebraic formulas is as numbers with
units.  For example, the formula `5 m / s^2' can be read "five meters
per second squared."  The commands in this chapter help you manipulate
units expressions in this form.  Units-related commands begin with the
`u' prefix key.

* Menu:

* Basic Operations on Units::
* The Units Table::
* Predefined Units::
* User-Defined Units::

File: calc,  Node: Basic Operations on Units,  Next: The Units Table,  Prev: Units,  Up: Units

13.1 Basic Operations on Units
==============================

A "units expression" is a formula which is basically a number
multiplied and/or divided by one or more "unit names", which may
optionally be raised to integer powers.  Actually, the value part need
not be a number; any product or quotient involving unit names is a units
expression.  Many of the units commands will also accept any formula,
where the command applies to all units expressions which appear in the
formula.

   A unit name is a variable whose name appears in the "unit table", or
a variable whose name is a prefix character like `k' (for "kilo") or
`u' (for "micro") followed by a name in the unit table.  A substantial
table of built-in units is provided with Calc; *note Predefined
Units::.  You can also define your own unit names; *note User-Defined
Units::.

   Note that if the value part of a units expression is exactly `1', it
will be removed by the Calculator's automatic algebra routines:  The
formula `1 mm' is "simplified" to `mm'.  This is only a display
anomaly, however; `mm' will work just fine as a representation of one
millimeter.

   You may find that Algebraic mode (*note Algebraic Entry::) makes
working with units expressions easier.  Otherwise, you will have to
remember to hit the apostrophe key every time you wish to enter units.

   The `u s' (`calc-simplify-units') [`usimplify'] command simplifies a
units expression.  It uses `a s' (`calc-simplify') to simplify the
expression first as a regular algebraic formula; it then looks for
features that can be further simplified by converting one object's units
to be compatible with another's.  For example, `5 m + 23 mm' will
simplify to `5.023 m'.  When different but compatible units are added,
the righthand term's units are converted to match those of the lefthand
term.  *Note Simplification Modes::, for a way to have this done
automatically at all times.

   Units simplification also handles quotients of two units with the
same dimensionality, as in `2 in s/L cm' to `5.08 s/L'; fractional
powers of unit expressions, as in `sqrt(9 mm^2)' to `3 mm' and `sqrt(9
acre)' to a quantity in meters; and `floor', `ceil', `round', `rounde',
`roundu', `trunc', `float', `frac', `abs', and `clean' applied to units
expressions, in which case the operation in question is applied only to
the numeric part of the expression.  Finally, trigonometric functions
of quantities with units of angle are evaluated, regardless of the
current angular mode.

   The `u c' (`calc-convert-units') command converts a units expression
to new, compatible units.  For example, given the units expression `55
mph', typing `u c m/s <RET>' produces `24.5872 m/s'.  If you have
previously converted a units expression with the same type of units (in
this case, distance over time), you will be offered the previous choice
of new units as a default.  Continuing the above example, entering the
units expression `100 km/hr' and typing `u c <RET>' (without specifying
new units) produces `27.7777777778 m/s'.

   While many of Calc's conversion factors are exact, some are
necessarily approximate.  If Calc is in fraction mode (*note Fraction
Mode::), then unit conversions will try to give exact, rational
conversions, but it isn't always possible.  Given `55 mph' in fraction
mode, typing `u c m/s <RET>' produces  `15367:625 m/s', for example,
while typing `u c au/yr <RET>' produces `5.18665819999e-3 au/yr'.

   If the units you request are inconsistent with the original units,
the number will be converted into your units times whatever "remainder"
units are left over.  For example, converting `55 mph' into acres
produces `6.08e-3 acre / m s'.  (Recall that multiplication binds more
strongly than division in Calc formulas, so the units here are acres
per meter-second.)  Remainder units are expressed in terms of
"fundamental" units like `m' and `s', regardless of the input units.

   One special exception is that if you specify a single unit name, and
a compatible unit appears somewhere in the units expression, then that
compatible unit will be converted to the new unit and the remaining
units in the expression will be left alone.  For example, given the
input `980 cm/s^2', the command `u c ms' will change the `s' to `ms' to
get `9.8e-4 cm/ms^2'.  The "remainder unit" `cm' is left alone rather
than being changed to the base unit `m'.

   You can use explicit unit conversion instead of the `u s' command to
gain more control over the units of the result of an expression.  For
example, given `5 m + 23 mm', you can type `u c m' or `u c mm' to
express the result in either meters or millimeters.  (For that matter,
you could type `u c fath' to express the result in fathoms, if you
preferred!)

   In place of a specific set of units, you can also enter one of the
units system names `si', `mks' (equivalent), or `cgs'.  For example, `u
c si <RET>' converts the expression into International System of Units
(SI) base units.  Also, `u c base' converts to Calc's base units, which
are the same as `si' units except that `base' uses `g' as the
fundamental unit of mass whereas `si' uses `kg'.

   The `u c' command also accepts "composite units", which are
expressed as the sum of several compatible unit names.  For example,
converting `30.5 in' to units `mi+ft+in' (miles, feet, and inches)
produces `2 ft + 6.5 in'.  Calc first sorts the unit names into order
of decreasing relative size.  It then accounts for as much of the input
quantity as it can using an integer number times the largest unit, then
moves on to the next smaller unit, and so on.  Only the smallest unit
may have a non-integer amount attached in the result.  A few standard
unit names exist for common combinations, such as `mfi' for `mi+ft+in',
and `tpo' for `ton+lb+oz'.  Composite units are expanded as if by `a
x', so that `(ft+in)/hr' is first converted to `ft/hr+in/hr'.

   If the value on the stack does not contain any units, `u c' will
prompt first for the old units which this value should be considered to
have, then for the new units.  Assuming the old and new units you give
are consistent with each other, the result also will not contain any
units.  For example, `u c cm <RET> in <RET>' converts the number 2 on
the stack to 5.08.

   The `u b' (`calc-base-units') command is shorthand for `u c base';
it converts the units expression on the top of the stack into `base'
units.  If `u s' does not simplify a units expression as far as you
would like, try `u b'.

   The `u c' and `u b' commands treat temperature units (like `degC'
and `K') as relative temperatures.  For example, `u c' converts `10
degC' to `18 degF': A change of 10 degrees Celsius corresponds to a
change of 18 degrees Fahrenheit.

   The `u t' (`calc-convert-temperature') command converts absolute
temperatures.  The value on the stack must be a simple units expression
with units of temperature only.  This command would convert `10 degC'
to `50 degF', the equivalent temperature on the Fahrenheit scale.

   The `u r' (`calc-remove-units') command removes units from the
formula at the top of the stack.  The `u x' (`calc-extract-units')
command extracts only the units portion of a formula.  These commands
essentially replace every term of the formula that does or doesn't
(respectively) look like a unit name by the constant 1, then resimplify
the formula.

   The `u a' (`calc-autorange-units') command turns on and off a mode
in which unit prefixes like `k' ("kilo") are automatically applied to
keep the numeric part of a units expression in a reasonable range.
This mode affects `u s' and all units conversion commands except `u b'.
For example, with autoranging on, `12345 Hz' will be simplified to
`12.345 kHz'.  Autoranging is useful for some kinds of units (like `Hz'
and `m'), but is probably undesirable for non-metric units like `ft'
and `tbsp'.  (Composite units are more appropriate for those; see
above.)

   Autoranging always applies the prefix to the leftmost unit name.
Calc chooses the largest prefix that causes the number to be greater
than or equal to 1.0.  Thus an increasing sequence of adjusted times
would be `1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks'.  Generally the
rule of thumb is that the number will be adjusted to be in the interval
`[1 .. 1000)', although there are several exceptions to this rule.
First, if the unit has a power then this is not possible; `0.1 s^2'
simplifies to `100000 ms^2'.  Second, the "centi-" prefix is allowed to
form `cm' (centimeters), but will not apply to other units.  The
"deci-," "deka-," and "hecto-" prefixes are never used.  Thus the
allowable interval is `[1 .. 10)' for millimeters and `[1 .. 100)' for
centimeters.  Finally, a prefix will not be added to a unit if the
resulting name is also the actual name of another unit; `1e-15 t' would
normally be considered a "femto-ton," but it is written as `1000 at'
(1000 atto-tons) instead because `ft' would be confused with feet.

File: calc,  Node: The Units Table,  Next: Predefined Units,  Prev: Basic Operations on Units,  Up: Units

13.2 The Units Table
====================

The `u v' (`calc-enter-units-table') command displays the units table
in another buffer called `*Units Table*'.  Each entry in this table
gives the unit name as it would appear in an expression, the definition
of the unit in terms of simpler units, and a full name or description of
the unit.  Fundamental units are defined as themselves; these are the
units produced by the `u b' command.  The fundamental units are meters,
seconds, grams, kelvins, amperes, candelas, moles, radians, and
steradians.

   The Units Table buffer also displays the Unit Prefix Table.  Note
that two prefixes, "kilo" and "hecto," accept either upper- or
lower-case prefix letters.  `Meg' is also accepted as a synonym for the
`M' prefix.  Whenever a unit name can be interpreted as either a
built-in name or a prefix followed by another built-in name, the former
interpretation wins.  For example, `2 pt' means two pints, not two
pico-tons.

   The Units Table buffer, once created, is not rebuilt unless you
define new units.  To force the buffer to be rebuilt, give any numeric
prefix argument to `u v'.

   The `u V' (`calc-view-units-table') command is like `u v' except
that the cursor is not moved into the Units Table buffer.  You can type
`u V' again to remove the Units Table from the display.  To return from
the Units Table buffer after a `u v', type `C-x * c' again or use the
regular Emacs `C-x o' (`other-window') command.  You can also kill the
buffer with `C-x k' if you wish; the actual units table is safely
stored inside the Calculator.

   The `u g' (`calc-get-unit-definition') command retrieves a unit's
defining expression and pushes it onto the Calculator stack.  For
example, `u g in' will produce the expression `2.54 cm'.  This is the
same definition for the unit that would appear in the Units Table
buffer.  Note that this command works only for actual unit names; `u g
km' will report that no such unit exists, for example, because `km' is
really the unit `m' with a `k' ("kilo") prefix.  To see a definition of
a unit in terms of base units, it is easier to push the unit name on
the stack and then reduce it to base units with `u b'.

   The `u e' (`calc-explain-units') command displays an English
description of the units of the expression on the stack.  For example,
for the expression `62 km^2 g / s^2 mol K', the description is
"Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin)."  This
command uses the English descriptions that appear in the righthand
column of the Units Table.

File: calc,  Node: Predefined Units,  Next: User-Defined Units,  Prev: The Units Table,  Up: Units

13.3 Predefined Units
=====================

The definitions of many units have changed over the years.  For example,
the meter was originally defined in 1791 as one ten-millionth of the
distance from the equator to the north pole.  In order to be more
precise, the definition was adjusted several times, and now a meter is
defined as the distance that light will travel in a vacuum in
1/299792458 of a second; consequently, the speed of light in a vacuum
is exactly 299792458 m/s.  Many other units have been redefined in
terms of fundamental physical processes; a second, for example, is
currently defined as 9192631770 periods of a certain radiation related
to the cesium-133 atom.  The only SI unit that is not based on a
fundamental physical process (although there are efforts to change
this) is the kilogram, which was originally defined as the mass of one
liter of water, but is now defined as the mass of the International
Prototype Kilogram (IPK), a cylinder of platinum-iridium kept at the
Bureau International des Poids et Mesures in Se`vres, France.  (There
are several copies of the IPK throughout the world.)  The British
imperial units, once defined in terms of physical objects, were
redefined in 1963 in terms of SI units.  The US customary units, which
were the same as British units until the British imperial system was
created in 1824, were also defined in terms of the SI units in 1893.
Because of these redefinitions, conversions between metric, British
Imperial, and US customary units can often be done precisely.

   Since the exact definitions of many kinds of units have evolved over
the years, and since certain countries sometimes have local differences
in their definitions, it is a good idea to examine Calc's definition of
a unit before depending on its exact value.  For example, there are
three different units for gallons, corresponding to the US (`gal'),
Canadian (`galC'), and British (`galUK') definitions.  Also, note that
`oz' is a standard ounce of mass, `ozt' is a Troy ounce, and `ozfl' is
a fluid ounce.

   The temperature units corresponding to degrees Kelvin and Centigrade
(Celsius) are the same in this table, since most units commands treat
temperatures as being relative.  The `calc-convert-temperature' command
has special rules for handling the different absolute magnitudes of the
various temperature scales.

   The unit of volume "liters" can be referred to by either the
lower-case `l' or the upper-case `L'.

   The unit `A' stands for Amperes; the name `Ang' is used for
Angstroms.

   The unit `pt' stands for pints; the name `point' stands for a
typographical point, defined by `72 point = 1 in'.  This is slightly
different than the point defined by the American Typefounder's
Association in 1886, but the point used by Calc has become standard
largely due to its use by the PostScript page description language.
There is also `texpt', which stands for a printer's point as defined by
the TeX typesetting system:  `72.27 texpt = 1 in'.  Other units used by
TeX are available; they are `texpc' (a pica), `texbp' (a "big point",
equal to a standard point which is larger than the point used by TeX),
`texdd' (a Didot point), `texcc' (a Cicero) and `texsp' (a scaled TeX
point, all dimensions representable in TeX are multiples of this value).

   The unit `e' stands for the elementary (electron) unit of charge;
because algebra command could mistake this for the special constant
`e', Calc provides the alternate unit name `ech' which is preferable to
`e'.

   The name `g' stands for one gram of mass; there is also `gf', one
gram of force.  (Likewise for `lb', pounds, and `lbf'.)  Meanwhile, one
"`g'" of acceleration is denoted `ga'.

   The unit `ton' is a U.S. ton of `2000 lb', and `t' is a metric ton
of `1000 kg'.

   The names `s' (or `sec') and `min' refer to units of time; `arcsec'
and `arcmin' are units of angle.

   Some "units" are really physical constants; for example, `c'
represents the speed of light, and `h' represents Planck's constant.
You can use these just like other units: converting `.5 c' to `m/s'
expresses one-half the speed of light in meters per second.  You can
also use this merely as a handy reference; the `u g' command gets the
definition of one of these constants in its normal terms, and `u b'
expresses the definition in base units.

   Two units, `pi' and `alpha' (the fine structure constant,
approximately 1/137) are dimensionless.  The units simplification
commands simply treat these names as equivalent to their corresponding
values.  However you can, for example, use `u c' to convert a pure
number into multiples of the fine structure constant, or `u b' to
convert this back into a pure number.  (When `u c' prompts for the "old
units," just enter a blank line to signify that the value really is
unitless.)

File: calc,  Node: User-Defined Units,  Prev: Predefined Units,  Up: Units

13.4 User-Defined Units
=======================

Calc provides ways to get quick access to your selected "favorite"
units, as well as ways to define your own new units.

   To select your favorite units, store a vector of unit names or
expressions in the Calc variable `Units'.  The `u 1' through `u 9'
commands (`calc-quick-units') provide access to these units.  If the
value on the top of the stack is a plain number (with no units
attached), then `u 1' gives it the specified units.  (Basically, it
multiplies the number by the first item in the `Units' vector.)  If the
number on the stack _does_ have units, then `u 1' converts that number
to the new units.  For example, suppose the vector `[in, ft]' is stored
in `Units'.  Then `30 u 1' will create the expression `30 in', and `u
2' will convert that expression to `2.5 ft'.

   The `u 0' command accesses the tenth element of `Units'.  Only ten
quick units may be defined at a time.  If the `Units' variable has no
stored value (the default), or if its value is not a vector, then the
quick-units commands will not function.  The `s U' command is a
convenient way to edit the `Units' variable; *note Operations on
Variables::.

   The `u d' (`calc-define-unit') command records the units expression
on the top of the stack as the definition for a new, user-defined unit.
For example, putting `16.5 ft' on the stack and typing `u d rod'
defines the new unit `rod' to be equivalent to 16.5 feet.  The unit
conversion and simplification commands will now treat `rod' just like
any other unit of length.  You will also be prompted for an optional
English description of the unit, which will appear in the Units Table.
If you wish the definition of this unit to be displayed in a special
way in the Units Table buffer (such as with an asterisk to indicate an
approximate value), then you can call this command with an argument,
`C-u u d'; you will then also be prompted for a string that will be
used to display the definition.

   The `u u' (`calc-undefine-unit') command removes a user-defined
unit.  It is not possible to remove one of the predefined units,
however.

   If you define a unit with an existing unit name, your new definition
will replace the original definition of that unit.  If the unit was a
predefined unit, the old definition will not be replaced, only
"shadowed."  The built-in definition will reappear if you later use `u
u' to remove the shadowing definition.

   To create a new fundamental unit, use either 1 or the unit name
itself as the defining expression.  Otherwise the expression can
involve any other units that you like (except for composite units like
`mfi').  You can create a new composite unit with a sum of other units
as the defining expression.  The next unit operation like `u c' or `u v'
will rebuild the internal unit table incorporating your modifications.
Note that erroneous definitions (such as two units defined in terms of
each other) will not be detected until the unit table is next rebuilt;
`u v' is a convenient way to force this to happen.

   Temperature units are treated specially inside the Calculator; it is
not possible to create user-defined temperature units.

   The `u p' (`calc-permanent-units') command stores the user-defined
units in your Calc init file (the file given by the variable
`calc-settings-file', typically `~/.calc.el'), so that the units will
still be available in subsequent Emacs sessions.  If there was already
a set of user-defined units in your Calc init file, it is replaced by
the new set.  (*Note General Mode Commands::, for a way to tell Calc to
use a different file for the Calc init file.)

File: calc,  Node: Store and Recall,  Next: Graphics,  Prev: Units,  Up: Top

14 Storing and Recalling
************************

Calculator variables are really just Lisp variables that contain numbers
or formulas in a form that Calc can understand.  The commands in this
section allow you to manipulate variables conveniently.  Commands
related to variables use the `s' prefix key.

* Menu:

* Storing Variables::
* Recalling Variables::
* Operations on Variables::
* Let Command::
* Evaluates-To Operator::

File: calc,  Node: Storing Variables,  Next: Recalling Variables,  Prev: Store and Recall,  Up: Store and Recall

14.1 Storing Variables
======================

The `s s' (`calc-store') command stores the value at the top of the
stack into a specified variable.  It prompts you to enter the name of
the variable.  If you press a single digit, the value is stored
immediately in one of the "quick" variables `q0' through `q9'.  Or you
can enter any variable name.

   The `s s' command leaves the stored value on the stack.  There is
also an `s t' (`calc-store-into') command, which removes a value from
the stack and stores it in a variable.

   If the top of stack value is an equation `a = 7' or assignment `a :=
7' with a variable on the lefthand side, then Calc will assign that
variable with that value by default, i.e., if you type `s s <RET>' or
`s t <RET>'.  In this example, the value 7 would be stored in the
variable `a'.  (If you do type a variable name at the prompt, the
top-of-stack value is stored in its entirety, even if it is an
equation:  `s s b <RET>' with `a := 7' on the stack stores `a := 7' in
`b'.)

   In fact, the top of stack value can be a vector of equations or
assignments with different variables on their lefthand sides; the
default will be to store all the variables with their corresponding
righthand sides simultaneously.

   It is also possible to type an equation or assignment directly at
the prompt for the `s s' or `s t' command:  `s s foo = 7'.  In this
case the expression to the right of the `=' or `:=' symbol is evaluated
as if by the `=' command, and that value is stored in the variable.  No
value is taken from the stack; `s s' and `s t' are equivalent when used
in this way.

   The prefix keys `s' and `t' may be followed immediately by a digit;
`s 9' is equivalent to `s s 9', and `t 9' is equivalent to `s t 9'.
(The `t' prefix is otherwise used for trail and time/date commands.)

   There are also several "arithmetic store" commands.  For example, `s
+' removes a value from the stack and adds it to the specified
variable.  The other arithmetic stores are `s -', `s *', `s /', `s ^',
and `s |' (vector concatenation), plus `s n' and `s &' which negate or
invert the value in a variable, and `s [' and `s ]' which decrease or
increase a variable by one.

   All the arithmetic stores accept the Inverse prefix to reverse the
order of the operands.  If `v' represents the contents of the variable,
and `a' is the value drawn from the stack, then regular `s -' assigns
`v := v - a', but `I s -' assigns `v := a - v'.  While `I s *' might
seem pointless, it is useful if matrix multiplication is involved.
Actually, all the arithmetic stores use formulas designed to behave
usefully both forwards and backwards:

     s +        v := v + a          v := a + v
     s -        v := v - a          v := a - v
     s *        v := v * a          v := a * v
     s /        v := v / a          v := a / v
     s ^        v := v ^ a          v := a ^ v
     s |        v := v | a          v := a | v
     s n        v := v / (-1)       v := (-1) / v
     s &        v := v ^ (-1)       v := (-1) ^ v
     s [        v := v - 1          v := 1 - v
     s ]        v := v - (-1)       v := (-1) - v

   In the last four cases, a numeric prefix argument will be used in
place of the number one.  (For example, `M-2 s ]' increases a variable
by 2, and `M-2 I s ]' replaces a variable by minus-two minus the
variable.

   The first six arithmetic stores can also be typed `s t +', `s t -',
etc.  The commands `s s +', `s s -', and so on are analogous arithmetic
stores that don't remove the value `a' from the stack.

   All arithmetic stores report the new value of the variable in the
Trail for your information.  They signal an error if the variable
previously had no stored value.  If default simplifications have been
turned off, the arithmetic stores temporarily turn them on for numeric
arguments only (i.e., they temporarily do an `m N' command).  *Note
Simplification Modes::.  Large vectors put in the trail by these
commands always use abbreviated (`t .') mode.

   The `s m' command is a general way to adjust a variable's value
using any Calc function.  It is a "mapping" command analogous to `V M',
`V R', etc.  *Note Reducing and Mapping::, to see how to specify a
function for a mapping command.  Basically, all you do is type the Calc
command key that would invoke that function normally.  For example, `s
m n' applies the `n' key to negate the contents of the variable, so `s
m n' is equivalent to `s n'.  Also, `s m Q' takes the square root of
the value stored in a variable, `s m v v' uses `v v' to reverse the
vector stored in the variable, and `s m H I S' takes the hyperbolic
arcsine of the variable contents.

   If the mapping function takes two or more arguments, the additional
arguments are taken from the stack; the old value of the variable is
provided as the first argument.  Thus `s m -' with `a' on the stack
computes `v - a', just like `s -'.  With the Inverse prefix, the
variable's original value becomes the _last_ argument instead of the
first.  Thus `I s m -' is also equivalent to `I s -'.

   The `s x' (`calc-store-exchange') command exchanges the value of a
variable with the value on the top of the stack.  Naturally, the
variable must already have a stored value for this to work.

   You can type an equation or assignment at the `s x' prompt.  The
command `s x a=6' takes no values from the stack; instead, it pushes
the old value of `a' on the stack and stores `a = 6'.

   Until you store something in them, most variables are "void," that
is, they contain no value at all.  If they appear in an algebraic
formula they will be left alone even if you press `=' (`calc-evaluate').
The `s u' (`calc-unstore') command returns a variable to the void state.

   The `s c' (`calc-copy-variable') command copies the stored value of
one variable to another.  One way it differs from a simple `s r'
followed by an `s t' (aside from saving keystrokes) is that the value
never goes on the stack and thus is never rounded, evaluated, or
simplified in any way; it is not even rounded down to the current
precision.

   The only variables with predefined values are the "special constants"
`pi', `e', `i', `phi', and `gamma'.  You are free to unstore these
variables or to store new values into them if you like, although some
of the algebraic-manipulation functions may assume these variables
represent their standard values.  Calc displays a warning if you change
the value of one of these variables, or of one of the other special
variables `inf', `uinf', and `nan' (which are normally void).

   Note that `pi' doesn't actually have 3.14159265359 stored in it, but
rather a special magic value that evaluates to `pi' at the current
precision.  Likewise `e', `i', and `phi' evaluate according to the
current precision or polar mode.  If you recall a value from `pi' and
store it back, this magic property will be lost.  The magic property is
preserved, however, when a variable is copied with `s c'.

   If one of the "special constants" is redefined (or undefined) so that
it no longer has its magic property, the property can be restored with
`s k' (`calc-copy-special-constant').  This command will prompt for a
special constant and a variable to store it in, and so a special
constant can be stored in any variable.  Here, the special constant that
you enter doesn't depend on the value of the corresponding variable;
`pi' will represent 3.14159... regardless of what is currently stored
in the Calc variable `pi'.  If one of the other special variables,
`inf', `uinf' or `nan', is given a value, its original behavior can be
restored by voiding it with `s u'.

File: calc,  Node: Recalling Variables,  Next: Operations on Variables,  Prev: Storing Variables,  Up: Store and Recall

14.2 Recalling Variables
========================

The most straightforward way to extract the stored value from a variable
is to use the `s r' (`calc-recall') command.  This command prompts for
a variable name (similarly to `calc-store'), looks up the value of the
specified variable, and pushes that value onto the stack.  It is an
error to try to recall a void variable.

   It is also possible to recall the value from a variable by
evaluating a formula containing that variable.  For example, `' a <RET>
=' is the same as `s r a <RET>' except that if the variable is void, the
former will simply leave the formula `a' on the stack whereas the
latter will produce an error message.

   The `r' prefix may be followed by a digit, so that `r 9' is
equivalent to `s r 9'.

File: calc,  Node: Operations on Variables,  Next: Let Command,  Prev: Recalling Variables,  Up: Store and Recall

14.3 Other Operations on Variables
==================================

The `s e' (`calc-edit-variable') command edits the stored value of a
variable without ever putting that value on the stack or simplifying or
evaluating the value.  It prompts for the name of the variable to edit.
If the variable has no stored value, the editing buffer will start out
empty.  If the editing buffer is empty when you press `C-c C-c' to
finish, the variable will be made void.  *Note Editing Stack Entries::,
for a general description of editing.

   The `s e' command is especially useful for creating and editing
rewrite rules which are stored in variables.  Sometimes these rules
contain formulas which must not be evaluated until the rules are
actually used.  (For example, they may refer to `deriv(x,y)', where `x'
will someday become some expression involving `y'; if you let Calc
evaluate the rule while you are defining it, Calc will replace
`deriv(x,y)' with 0 because the formula `x' does not itself refer to
`y'.)  By contrast, recalling the variable, editing with ``', and
storing will evaluate the variable's value as a side effect of putting
the value on the stack.

   There are several special-purpose variable-editing commands that use
the `s' prefix followed by a shifted letter:

`s A'
     Edit `AlgSimpRules'.  *Note Algebraic Simplifications::.

`s D'
     Edit `Decls'.  *Note Declarations::.

`s E'
     Edit `EvalRules'.  *Note Default Simplifications::.

`s F'
     Edit `FitRules'.  *Note Curve Fitting::.

`s G'
     Edit `GenCount'.  *Note Solving Equations::.

`s H'
     Edit `Holidays'.  *Note Business Days::.

`s I'
     Edit `IntegLimit'.  *Note Calculus::.

`s L'
     Edit `LineStyles'.  *Note Graphics::.

`s P'
     Edit `PointStyles'.  *Note Graphics::.

`s R'
     Edit `PlotRejects'.  *Note Graphics::.

`s T'
     Edit `TimeZone'.  *Note Time Zones::.

`s U'
     Edit `Units'.  *Note User-Defined Units::.

`s X'
     Edit `ExtSimpRules'.  *Note Unsafe Simplifications::.

   These commands are just versions of `s e' that use fixed variable
names rather than prompting for the variable name.

   The `s p' (`calc-permanent-variable') command saves a variable's
value permanently in your Calc init file (the file given by the
variable `calc-settings-file', typically `~/.calc.el'), so that its
value will still be available in future Emacs sessions.  You can
re-execute `s p' later on to update the saved value, but the only way
to remove a saved variable is to edit your calc init file by hand.
(*Note General Mode Commands::, for a way to tell Calc to use a
different file for the Calc init file.)

   If you do not specify the name of a variable to save (i.e., `s p
<RET>'), all Calc variables with defined values are saved except for
the special constants `pi', `e', `i', `phi', and `gamma'; the variables
`TimeZone' and `PlotRejects'; `FitRules', `DistribRules', and other
built-in rewrite rules; and `PlotDataN' variables generated by the
graphics commands.  (You can still save these variables by explicitly
naming them in an `s p' command.)

   The `s i' (`calc-insert-variables') command writes the values of all
Calc variables into a specified buffer.  The variables are written with
the prefix `var-' in the form of Lisp `setq' commands which store the
values in string form.  You can place these commands in your Calc init
file (or `.emacs') if you wish, though in this case it would be easier
to use `s p <RET>'.  (Note that `s i' omits the same set of variables
as `s p <RET>'; the difference is that `s i' will store the variables
in any buffer, and it also stores in a more human-readable format.)

File: calc,  Node: Let Command,  Next: Evaluates-To Operator,  Prev: Operations on Variables,  Up: Store and Recall

14.4 The Let Command
====================

If you have an expression like `a+b^2' on the stack and you wish to
compute its value where `b=3', you can simply store 3 in `b' and then
press `=' to reevaluate the formula.  This has the side-effect of
leaving the stored value of 3 in `b' for future operations.

   The `s l' (`calc-let') command evaluates a formula under a
_temporary_ assignment of a variable.  It stores the value on the top
of the stack into the specified variable, then evaluates the
second-to-top stack entry, then restores the original value (or lack of
one) in the variable.  Thus after `' a+b^2 <RET> 3 s l b <RET>', the
stack will contain the formula `a + 9'.  The subsequent command
`5 s l a <RET>' will replace this formula with the number 14.  The
variables `a' and `b' are not permanently affected in any way by these
commands.

   The value on the top of the stack may be an equation or assignment,
or a vector of equations or assignments, in which case the default will
be analogous to the case of `s t <RET>'.  *Note Storing Variables::.

   Also, you can answer the variable-name prompt with an equation or
assignment:  `s l b=3 <RET>' is the same as storing 3 on the stack and
typing `s l b <RET>'.

   The `a b' (`calc-substitute') command is another way to substitute a
variable with a value in a formula.  It does an actual substitution
rather than temporarily assigning the variable and evaluating.  For
example, letting `n=2' in `f(n pi)' with `a b' will produce `f(2 pi)',
whereas `s l' would give `f(6.28)' since the evaluation step will also
evaluate `pi'.

File: calc,  Node: Evaluates-To Operator,  Prev: Let Command,  Up: Store and Recall

14.5 The Evaluates-To Operator
==============================

The special algebraic symbol `=>' is known as the "evaluates-to
operator".  (It will show up as an `evalto' function call in other
language modes like Pascal and LaTeX.)  This is a binary operator, that
is, it has a lefthand and a righthand argument, although it can be
entered with the righthand argument omitted.

   A formula like `A => B' is evaluated by Calc as follows:  First, A
is not simplified or modified in any way.  The previous value of
argument B is thrown away; the formula A is then copied and evaluated
as if by the `=' command according to all current modes and stored
variable values, and the result is installed as the new value of B.

   For example, suppose you enter the algebraic formula `2 + 3 => 17'.
The number 17 is ignored, and the lefthand argument is left in its
unevaluated form; the result is the formula `2 + 3 => 5'.

   You can enter an `=>' formula either directly using algebraic entry
(in which case the righthand side may be omitted since it is going to
be replaced right away anyhow), or by using the `s =' (`calc-evalto')
command, which takes A from the stack and replaces it with `A => B'.

   Calc keeps track of all `=>' operators on the stack, and recomputes
them whenever anything changes that might affect their values, i.e., a
mode setting or variable value.  This occurs only if the `=>' operator
is at the top level of the formula, or if it is part of a top-level
vector.  In other words, pushing `2 + (a => 17)' will change the 17 to
the actual value of `a' when you enter the formula, but the result will
not be dynamically updated when `a' is changed later because the `=>'
operator is buried inside a sum.  However, a vector of `=>' operators
will be recomputed, since it is convenient to push a vector like `[a
=>, b =>, c =>]' on the stack to make a concise display of all the
variables in your problem.  (Another way to do this would be to use
`[a, b, c] =>', which provides a slightly different format of display.
You can use whichever you find easiest to read.)

   The `m C' (`calc-auto-recompute') command allows you to turn this
automatic recomputation on or off.  If you turn recomputation off, you
must explicitly recompute an `=>' operator on the stack in one of the
usual ways, such as by pressing `='.  Turning recomputation off
temporarily can save a lot of time if you will be changing several
modes or variables before you look at the `=>' entries again.

   Most commands are not especially useful with `=>' operators as
arguments.  For example, given `x + 2 => 17', it won't work to type `1
+' to get `x + 3 => 18'.  If you want to operate on the lefthand side
of the `=>' operator on the top of the stack, type `j 1' (that's the
digit "one") to select the lefthand side, execute your commands, then
type `j u' to unselect.

   All current modes apply when an `=>' operator is computed, including
the current simplification mode.  Recall that the formula `x + y + x'
is not handled by Calc's default simplifications, but the `a s' command
will reduce it to the simpler form `y + 2 x'.  You can also type `m A'
to enable an Algebraic Simplification mode in which the equivalent of
`a s' is used on all of Calc's results.  If you enter `x + y + x =>'
normally, the result will be `x + y + x => x + y + x'.  If you change to
Algebraic Simplification mode, the result will be `x + y + x => y + 2
x'.  However, just pressing `a s' once will have no effect on `x + y +
x => x + y + x', because the righthand side depends only on the
lefthand side and the current mode settings, and the lefthand side is
not affected by commands like `a s'.

   The "let" command (`s l') has an interesting interaction with the
`=>' operator.  The `s l' command evaluates the second-to-top stack
entry with the top stack entry supplying a temporary value for a given
variable.  As you might expect, if that stack entry is an `=>' operator
its righthand side will temporarily show this value for the variable.
In fact, all `=>'s on the stack will be updated if they refer to that
variable.  But this change is temporary in the sense that the next
command that causes Calc to look at those stack entries will make them
revert to the old variable value.

     2:  a => a             2:  a => 17         2:  a => a
     1:  a + 1 => a + 1     1:  a + 1 => 18     1:  a + 1 => a + 1
         .                      .                   .

                                17 s l a <RET>        p 8 <RET>

   Here the `p 8' command changes the current precision, thus causing
the `=>' forms to be recomputed after the influence of the "let" is
gone.  The `d <SPC>' command (`calc-refresh') is a handy way to force
the `=>' operators on the stack to be recomputed without any other side
effects.

   Embedded mode also uses `=>' operators.  In Embedded mode, the
lefthand side of an `=>' operator can refer to variables assigned
elsewhere in the file by `:=' operators.  The assignment operator `a :=
17' does not actually do anything by itself.  But Embedded mode
recognizes it and marks it as a sort of file-local definition of the
variable.  You can enter `:=' operators in Algebraic mode, or by using
the `s :' (`calc-assign') [`assign'] command which takes a variable and
value from the stack and replaces them with an assignment.

   *Note TeX and LaTeX Language Modes::, for the way `=>' appears in
TeX language output.  The "eqn" mode gives similar treatment to `=>'.

File: calc,  Node: Graphics,  Next: Kill and Yank,  Prev: Store and Recall,  Up: Top

15 Graphics
***********

The commands for graphing data begin with the `g' prefix key.  Calc
uses GNUPLOT 2.0 or later to do graphics.  These commands will only work
if GNUPLOT is available on your system.  (While GNUPLOT sounds like a
relative of GNU Emacs, it is actually completely unrelated.  However,
it is free software.   It can be obtained from
`http://www.gnuplot.info'.)

   If you have GNUPLOT installed on your system but Calc is unable to
find it, you may need to set the `calc-gnuplot-name' variable in your
Calc init file or `.emacs'.  You may also need to set some Lisp
variables to show Calc how to run GNUPLOT on your system; these are
described under `g D' and `g O' below.  If you are using the X window
system, Calc will configure GNUPLOT for you automatically.  If you have
GNUPLOT 3.0 or later and you are not using X, Calc will configure
GNUPLOT to display graphs using simple character graphics that will
work on any terminal.

* Menu:

* Basic Graphics::
* Three Dimensional Graphics::
* Managing Curves::
* Graphics Options::
* Devices::

File: calc,  Node: Basic Graphics,  Next: Three Dimensional Graphics,  Prev: Graphics,  Up: Graphics

15.1 Basic Graphics
===================

The easiest graphics command is `g f' (`calc-graph-fast').  This
command takes two vectors of equal length from the stack.  The vector
at the top of the stack represents the "y" values of the various data
points.  The vector in the second-to-top position represents the
corresponding "x" values.  This command runs GNUPLOT (if it has not
already been started by previous graphing commands) and displays the
set of data points.  The points will be connected by lines, and there
will also be some kind of symbol to indicate the points themselves.

   The "x" entry may instead be an interval form, in which case suitable
"x" values are interpolated between the minimum and maximum values of
the interval (whether the interval is open or closed is ignored).

   The "x" entry may also be a number, in which case Calc uses the
sequence of "x" values `x', `x+1', `x+2', etc.  (Generally the number 0
or 1 would be used for `x' in this case.)

   The "y" entry may be any formula instead of a vector.  Calc
effectively uses `N' (`calc-eval-num') to evaluate variables in the
formula; the result of this must be a formula in a single (unassigned)
variable.  The formula is plotted with this variable taking on the
various "x" values.  Graphs of formulas by default use lines without
symbols at the computed data points.  Note that if neither "x" nor "y"
is a vector, Calc guesses at a reasonable number of data points to use.
See the `g N' command below.  (The "x" values must be either a vector
or an interval if "y" is a formula.)

   If "y" is (or evaluates to) a formula of the form `xy(X, Y)' then
the result is a parametric plot.  The two arguments of the fictitious
`xy' function are used as the "x" and "y" coordinates of the curve,
respectively.  In this case the "x" vector or interval you specified is
not directly visible in the graph.  For example, if "x" is the interval
`[0..360]' and "y" is the formula `xy(sin(t), cos(t))', the resulting
graph will be a circle.

   Also, "x" and "y" may each be variable names, in which case Calc
looks for suitable vectors, intervals, or formulas stored in those
variables.

   The "x" and "y" values for the data points (as pulled from the
vectors, calculated from the formulas, or interpolated from the
intervals) should be real numbers (integers, fractions, or floats).
One exception to this is that the "y" entry can consist of a vector of
numbers combined with error forms, in which case the points will be
plotted with the appropriate error bars.  Other than this, if either
the "x" value or the "y" value of a given data point is not a real
number, that data point will be omitted from the graph.  The points on
either side of the invalid point will _not_ be connected by a line.

   See the documentation for `g a' below for a description of the way
numeric prefix arguments affect `g f'.

   If you store an empty vector in the variable `PlotRejects' (i.e., `[
] s t PlotRejects'), Calc will append information to this vector for
every data point which was rejected because its "x" or "y" values were
not real numbers.  The result will be a matrix where each row holds the
curve number, data point number, "x" value, and "y" value for a
rejected data point.  *Note Evaluates-To Operator::, for a handy way to
keep tabs on the current value of `PlotRejects'.  *Note Operations on
Variables::, for the `s R' command which is another easy way to examine
`PlotRejects'.

   To clear the graphics display, type `g c' (`calc-graph-clear').  If
the GNUPLOT output device is an X window, the window will go away.
Effects on other kinds of output devices will vary.  You don't need to
use `g c' if you don't want to--if you give another `g f' or `g p'
command later on, it will reuse the existing graphics window if there
is one.

File: calc,  Node: Three Dimensional Graphics,  Next: Managing Curves,  Prev: Basic Graphics,  Up: Graphics

15.2 Three-Dimensional Graphics
===============================

The `g F' (`calc-graph-fast-3d') command makes a three-dimensional
graph.  It works only if you have GNUPLOT 3.0 or later; with GNUPLOT
2.0, you will see a GNUPLOT error message if you try this command.

   The `g F' command takes three values from the stack, called "x",
"y", and "z", respectively.  As was the case for 2D graphs, there are
several options for these values.

   In the first case, "x" and "y" are each vectors (not necessarily of
the same length); either or both may instead be interval forms.  The
"z" value must be a matrix with the same number of rows as elements in
"x", and the same number of columns as elements in "y".  The result is
a surface plot where `z_ij' is the height of the point at coordinate
`(x_i, y_j)' on the surface.  The 3D graph will be displayed from a
certain default viewpoint; you can change this viewpoint by adding a
`set view' to the `*Gnuplot Commands*' buffer as described later.  See
the GNUPLOT documentation for a description of the `set view' command.

   Each point in the matrix will be displayed as a dot in the graph,
and these points will be connected by a grid of lines ("isolines").

   In the second case, "x", "y", and "z" are all vectors of equal
length.  The resulting graph displays a 3D line instead of a surface,
where the coordinates of points along the line are successive triplets
of values from the input vectors.

   In the third case, "x" and "y" are vectors or interval forms, and
"z" is any formula involving two variables (not counting variables with
assigned values).  These variables are sorted into alphabetical order;
the first takes on values from "x" and the second takes on values from
"y" to form a matrix of results that are graphed as a 3D surface.

   If the "z" formula evaluates to a call to the fictitious function
`xyz(X, Y, Z)', then the result is a "parametric surface."  In this
case, the axes of the graph are taken from the X and Y values in these
calls, and the "x" and "y" values from the input vectors or intervals
are used only to specify the range of inputs to the formula.  For
example, plotting `[0..360], [0..180], xyz(sin(x)*sin(y),
cos(x)*sin(y), cos(y))' will draw a sphere.  (Since the default
resolution for 3D plots is 5 steps in each of "x" and "y", this will
draw a very crude sphere.  You could use the `g N' command, described
below, to increase this resolution, or specify the "x" and "y" values as
vectors with more than 5 elements.

   It is also possible to have a function in a regular `g f' plot
evaluate to an `xyz' call.  Since `g f' plots a line, not a surface,
the result will be a 3D parametric line.  For example, `[[0..720],
xyz(sin(x), cos(x), x)]' will plot two turns of a helix (a
three-dimensional spiral).

   As for `g f', each of "x", "y", and "z" may instead be variables
containing the relevant data.

File: calc,  Node: Managing Curves,  Next: Graphics Options,  Prev: Three Dimensional Graphics,  Up: Graphics

15.3 Managing Curves
====================

The `g f' command is really shorthand for the following commands: `C-u
g d  g a  g p'.  Likewise, `g F' is shorthand for `C-u g d  g A  g p'.
You can gain more control over your graph by using these commands
directly.

   The `g a' (`calc-graph-add') command adds the "curve" represented by
the two values on the top of the stack to the current graph.  You can
have any number of curves in the same graph.  When you give the `g p'
command, all the curves will be drawn superimposed on the same axes.

   The `g a' command (and many others that affect the current graph)
will cause a special buffer, `*Gnuplot Commands*', to be displayed in
another window.  This buffer is a template of the commands that will be
sent to GNUPLOT when it is time to draw the graph.  The first `g a'
command adds a `plot' command to this buffer.  Succeeding `g a'
commands add extra curves onto that `plot' command.  Other
graph-related commands put other GNUPLOT commands into this buffer.  In
normal usage you never need to work with this buffer directly, but you
can if you wish.  The only constraint is that there must be only one
`plot' command, and it must be the last command in the buffer.  If you
want to save and later restore a complete graph configuration, you can
use regular Emacs commands to save and restore the contents of the
`*Gnuplot Commands*' buffer.

   If the values on the stack are not variable names, `g a' will invent
variable names for them (of the form `PlotDataN') and store the values
in those variables.  The "x" and "y" variables are what go into the
`plot' command in the template.  If you add a curve that uses a certain
variable and then later change that variable, you can replot the graph
without having to delete and re-add the curve.  That's because the
variable name, not the vector, interval or formula itself, is what was
added by `g a'.

   A numeric prefix argument on `g a' or `g f' changes the way stack
entries are interpreted as curves.  With a positive prefix argument
`n', the top `n' stack entries are "y" values for `n' different curves
which share a common "x" value in the `n+1'st stack entry.  (Thus `g a'
with no prefix argument is equivalent to `C-u 1 g a'.)

   A prefix of zero or plain `C-u' means to take two stack entries, "x"
and "y" as usual, but to interpret "y" as a vector of "y" values for
several curves that share a common "x".

   A negative prefix argument tells Calc to read `n' vectors from the
stack; each vector `[x, y]' describes an independent curve.  This is
the only form of `g a' that creates several curves at once that don't
have common "x" values.  (Of course, the range of "x" values covered by
all the curves ought to be roughly the same if they are to look nice on
the same graph.)

   For example, to plot `sin(n x)' for integers `n' from 1 to 5, you
could use `v x' to create a vector of integers (`n'), then `V M '' or
`V M $' to map `sin(n x)' across this vector.  The resulting vector of
formulas is suitable for use as the "y" argument to a `C-u g a' or `C-u
g f' command.

   The `g A' (`calc-graph-add-3d') command adds a 3D curve to the
graph.  It is not valid to intermix 2D and 3D curves in a single graph.
This command takes three arguments, "x", "y", and "z", from the stack.
With a positive prefix `n', it takes `n+2' arguments (common "x" and
"y", plus `n' separate "z"s).  With a zero prefix, it takes three stack
entries but the "z" entry is a vector of curve values.  With a negative
prefix `-n', it takes `n' vectors of the form `[x, y, z]'.  The `g A'
command works by adding a `splot' (surface-plot) command to the
`*Gnuplot Commands*' buffer.

   (Although `g a' adds a 2D `plot' command to the `*Gnuplot Commands*'
buffer, Calc changes this to `splot' before sending it to GNUPLOT if it
notices that the data points are evaluating to `xyz' calls.  It will
not work to mix 2D and 3D `g a' curves in a single graph, although Calc
does not currently check for this.)

   The `g d' (`calc-graph-delete') command deletes the most recently
added curve from the graph.  It has no effect if there are no curves in
the graph.  With a numeric prefix argument of any kind, it deletes all
of the curves from the graph.

   The `g H' (`calc-graph-hide') command "hides" or "unhides" the most
recently added curve.  A hidden curve will not appear in the actual
plot, but information about it such as its name and line and point
styles will be retained.

   The `g j' (`calc-graph-juggle') command moves the curve at the end
of the list (the "most recently added curve") to the front of the list.
The next-most-recent curve is thus exposed for `g d' or similar
commands to use.  With `g j' you can work with any curve in the graph
even though curve-related commands only affect the last curve in the
list.

   The `g p' (`calc-graph-plot') command uses GNUPLOT to draw the graph
described in the `*Gnuplot Commands*' buffer.  Any GNUPLOT parameters
which are not defined by commands in this buffer are reset to their
default values.  The variables named in the `plot' command are written
to a temporary data file and the variable names are then replaced by
the file name in the template.  The resulting plotting commands are fed
to the GNUPLOT program.  See the documentation for the GNUPLOT program
for more specific information.  All temporary files are removed when
Emacs or GNUPLOT exits.

   If you give a formula for "y", Calc will remember all the values that
it calculates for the formula so that later plots can reuse these
values.  Calc throws out these saved values when you change any
circumstances that may affect the data, such as switching from Degrees
to Radians mode, or changing the value of a parameter in the formula.
You can force Calc to recompute the data from scratch by giving a
negative numeric prefix argument to `g p'.

   Calc uses a fairly rough step size when graphing formulas over
intervals.  This is to ensure quick response.  You can "refine" a plot
by giving a positive numeric prefix argument to `g p'.  Calc goes
through the data points it has computed and saved from previous plots
of the function, and computes and inserts a new data point midway
between each of the existing points.  You can refine a plot any number
of times, but beware that the amount of calculation involved doubles
each time.

   Calc does not remember computed values for 3D graphs.  This means the
numerix prefix argument, if any, to `g p' is effectively ignored if the
current graph is three-dimensional.

   The `g P' (`calc-graph-print') command is like `g p', except that it
sends the output to a printer instead of to the screen.  More
precisely, `g p' looks for `set terminal' or `set output' commands in
the `*Gnuplot Commands*' buffer; lacking these it uses the default
settings.  However, `g P' ignores `set terminal' and `set output'
commands and uses a different set of default values.  All of these
values are controlled by the `g D' and `g O' commands discussed below.
Provided everything is set up properly, `g p' will plot to the screen
unless you have specified otherwise and `g P' will always plot to the
printer.

File: calc,  Node: Graphics Options,  Next: Devices,  Prev: Managing Curves,  Up: Graphics

15.4 Graphics Options
=====================

The `g g' (`calc-graph-grid') command turns the "grid" on and off.  It
is off by default; tick marks appear only at the edges of the graph.
With the grid turned on, dotted lines appear across the graph at each
tick mark.  Note that this command only changes the setting in
`*Gnuplot Commands*'; to see the effects of the change you must give
another `g p' command.

   The `g b' (`calc-graph-border') command turns the border (the box
that surrounds the graph) on and off.  It is on by default.  This
command will only work with GNUPLOT 3.0 and later versions.

   The `g k' (`calc-graph-key') command turns the "key" on and off.
The key is a chart in the corner of the graph that shows the
correspondence between curves and line styles.  It is off by default,
and is only really useful if you have several curves on the same graph.

   The `g N' (`calc-graph-num-points') command allows you to select the
number of data points in the graph.  This only affects curves where
neither "x" nor "y" is specified as a vector.  Enter a blank line to
revert to the default value (initially 15).  With no prefix argument,
this command affects only the current graph.  With a positive prefix
argument this command changes or, if you enter a blank line, displays
the default number of points used for all graphs created by `g a' that
don't specify the resolution explicitly.  With a negative prefix
argument, this command changes or displays the default value (initially
5) used for 3D graphs created by `g A'.  Note that a 3D setting of 5
means that a total of `5^2 = 25' points will be computed for the
surface.

   Data values in the graph of a function are normally computed to a
precision of five digits, regardless of the current precision at the
time. This is usually more than adequate, but there are cases where it
will not be.  For example, plotting `1 + x' with `x' in the interval
`[0 .. 1e-6]' will round all the data points down to 1.0!  Putting the
command `set precision N' in the `*Gnuplot Commands*' buffer will cause
the data to be computed at precision N instead of 5.  Since this is
such a rare case, there is no keystroke-based command to set the
precision.

   The `g h' (`calc-graph-header') command sets the title for the
graph.  This will show up centered above the graph.  The default title
is blank (no title).

   The `g n' (`calc-graph-name') command sets the title of an
individual curve.  Like the other curve-manipulating commands, it
affects the most recently added curve, i.e., the last curve on the list
in the `*Gnuplot Commands*' buffer.  To set the title of the other
curves you must first juggle them to the end of the list with `g j', or
edit the `*Gnuplot Commands*' buffer by hand.  Curve titles appear in
the key; if the key is turned off they are not used.

   The `g t' (`calc-graph-title-x') and `g T' (`calc-graph-title-y')
commands set the titles on the "x" and "y" axes, respectively.  These
titles appear next to the tick marks on the left and bottom edges of
the graph, respectively.  Calc does not have commands to control the
tick marks themselves, but you can edit them into the `*Gnuplot
Commands*' buffer if you wish.  See the GNUPLOT documentation for
details.

   The `g r' (`calc-graph-range-x') and `g R' (`calc-graph-range-y')
commands set the range of values on the "x" and "y" axes, respectively.
You are prompted to enter a suitable range.  This should be either a
pair of numbers of the form, `MIN:MAX', or a blank line to revert to the
default behavior of setting the range based on the range of values in
the data, or `$' to take the range from the top of the stack.  Ranges
on the stack can be represented as either interval forms or vectors:
`[MIN .. MAX]' or `[MIN, MAX]'.

   The `g l' (`calc-graph-log-x') and `g L' (`calc-graph-log-y')
commands allow you to set either or both of the axes of the graph to be
logarithmic instead of linear.

   For 3D plots, `g C-t', `g C-r', and `g C-l' (those are letters with
the Control key held down) are the corresponding commands for the "z"
axis.

   The `g z' (`calc-graph-zero-x') and `g Z' (`calc-graph-zero-y')
commands control whether a dotted line is drawn to indicate the "x"
and/or "y" zero axes.  (These are the same dotted lines that would be
drawn there anyway if you used `g g' to turn the "grid" feature on.)
Zero-axis lines are on by default, and may be turned off only in
GNUPLOT 3.0 and later versions.  They are not available for 3D plots.

   The `g s' (`calc-graph-line-style') command turns the connecting
lines on or off for the most recently added curve, and optionally
selects the style of lines to be used for that curve.  Plain `g s'
simply toggles the lines on and off.  With a numeric prefix argument,
`g s' turns lines on and sets a particular line style.  Line style
numbers start at one and their meanings vary depending on the output
device.  GNUPLOT guarantees that there will be at least six different
line styles available for any device.

   The `g S' (`calc-graph-point-style') command similarly turns the
symbols at the data points on or off, or sets the point style.  If you
turn both lines and points off, the data points will show as tiny dots.
If the "y" values being plotted contain error forms and the connecting
lines are turned off, then this command will also turn the error bars
on or off.

   Another way to specify curve styles is with the `LineStyles' and
`PointStyles' variables.  These variables initially have no stored
values, but if you store a vector of integers in one of these variables,
the `g a' and `g f' commands will use those style numbers instead of
the defaults for new curves that are added to the graph.  An entry
should be a positive integer for a specific style, or 0 to let the
style be chosen automatically, or -1 to turn off lines or points
altogether.  If there are more curves than elements in the vector, the
last few curves will continue to have the default styles.  Of course,
you can later use `g s' and `g S' to change any of these styles.

   For example, `'[2 -1 3] <RET> s t LineStyles' causes the first curve
to have lines in style number 2, the second curve to have no connecting
lines, and the third curve to have lines in style 3.  Point styles will
still be assigned automatically, but you could store another vector in
`PointStyles' to define them, too.

File: calc,  Node: Devices,  Prev: Graphics Options,  Up: Graphics

15.5 Graphical Devices
======================

The `g D' (`calc-graph-device') command sets the device name (or
"terminal name" in GNUPLOT lingo) to be used by `g p' commands on this
graph.  It does not affect the permanent default device name.  If you
enter a blank name, the device name reverts to the default.  Enter `?'
to see a list of supported devices.

   With a positive numeric prefix argument, `g D' instead sets the
default device name, used by all plots in the future which do not
override it with a plain `g D' command.  If you enter a blank line this
command shows you the current default.  The special name `default'
signifies that Calc should choose `x11' if the X window system is in
use (as indicated by the presence of a `DISPLAY' environment variable),
or otherwise `dumb' under GNUPLOT 3.0 and later, or `postscript' under
GNUPLOT 2.0.  This is the initial default value.

   The `dumb' device is an interface to "dumb terminals," i.e.,
terminals with no special graphics facilities.  It writes a crude
picture of the graph composed of characters like `-' and `|' to a
buffer called `*Gnuplot Trail*', which Calc then displays.  The graph
is made the same size as the Emacs screen, which on most dumb terminals
will be 80x24 characters.  The graph is displayed in an Emacs
"recursive edit"; type `q' or `C-c C-c' to exit the recursive edit and
return to Calc.  Note that the `dumb' device is present only in GNUPLOT
3.0 and later versions.

   The word `dumb' may be followed by two numbers separated by spaces.
These are the desired width and height of the graph in characters.
Also, the device name `big' is like `dumb' but creates a graph four
times the width and height of the Emacs screen.  You will then have to
scroll around to view the entire graph.  In the `*Gnuplot Trail*'
buffer, <SPC>, <DEL>, `<', and `>' are defined to scroll by one
screenful in each of the four directions.

   With a negative numeric prefix argument, `g D' sets or displays the
device name used by `g P' (`calc-graph-print').  This is initially
`postscript'.  If you don't have a PostScript printer, you may decide
once again to use `dumb' to create a plot on any text-only printer.

   The `g O' (`calc-graph-output') command sets the name of the output
file used by GNUPLOT.  For some devices, notably `x11', there is no
output file and this information is not used.  Many other "devices" are
really file formats like `postscript'; in these cases the output in the
desired format goes into the file you name with `g O'.  Type `g O
stdout <RET>' to set GNUPLOT to write to its standard output stream,
i.e., to `*Gnuplot Trail*'.  This is the default setting.

   Another special output name is `tty', which means that GNUPLOT is
going to write graphics commands directly to its standard output, which
you wish Emacs to pass through to your terminal.  Tektronix graphics
terminals, among other devices, operate this way.  Calc does this by
telling GNUPLOT to write to a temporary file, then running a sub-shell
executing the command `cat tempfile >/dev/tty'.  On typical Unix
systems, this will copy the temporary file directly to the terminal,
bypassing Emacs entirely.  You will have to type `C-l' to Emacs
afterwards to refresh the screen.

   Once again, `g O' with a positive or negative prefix argument sets
the default or printer output file names, respectively.  In each case
you can specify `auto', which causes Calc to invent a temporary file
name for each `g p' (or `g P') command.  This temporary file will be
deleted once it has been displayed or printed.  If the output file name
is not `auto', the file is not automatically deleted.

   The default and printer devices and output files can be saved
permanently by the `m m' (`calc-save-modes') command.  The default
number of data points (see `g N') and the X geometry (see `g X') are
also saved.  Other graph information is _not_ saved; you can save a
graph's configuration simply by saving the contents of the `*Gnuplot
Commands*' buffer.

   You may wish to configure the default and printer devices and output
files for the whole system.  The relevant Lisp variables are
`calc-gnuplot-default-device' and `-output', and
`calc-gnuplot-print-device' and `-output'.  The output file names must
be either strings as described above, or Lisp expressions which are
evaluated on the fly to get the output file names.

   Other important Lisp variables are `calc-gnuplot-plot-command' and
`calc-gnuplot-print-command', which give the system commands to display
or print the output of GNUPLOT, respectively.  These may be `nil' if no
command is necessary, or strings which can include `%s' to signify the
name of the file to be displayed or printed.  Or, these variables may
contain Lisp expressions which are evaluated to display or print the
output.  These variables are customizable (*note Customizing Calc::).

   The `g x' (`calc-graph-display') command lets you specify on which X
window system display your graphs should be drawn.  Enter a blank line
to see the current display name.  This command has no effect unless the
current device is `x11'.

   The `g X' (`calc-graph-geometry') command is a similar command for
specifying the position and size of the X window.  The normal value is
`default', which generally means your window manager will let you place
the window interactively.  Entering `800x500+0+0' would create an
800-by-500 pixel window in the upper-left corner of the screen.

   The buffer called `*Gnuplot Trail*' holds a transcript of the
session with GNUPLOT.  This shows the commands Calc has "typed" to
GNUPLOT and the responses it has received.  Calc tries to notice when an
error message has appeared here and display the buffer for you when
this happens.  You can check this buffer yourself if you suspect
something has gone wrong.

   The `g C' (`calc-graph-command') command prompts you to enter any
line of text, then simply sends that line to the current GNUPLOT
process.  The `*Gnuplot Trail*' buffer looks deceptively like a Shell
buffer but you can't type commands in it yourself.  Instead, you must
use `g C' for this purpose.

   The `g v' (`calc-graph-view-commands') and `g V'
(`calc-graph-view-trail') commands display the `*Gnuplot Commands*' and
`*Gnuplot Trail*' buffers, respectively, in another window.  This
happens automatically when Calc thinks there is something you will want
to see in either of these buffers.  If you type `g v' or `g V' when the
relevant buffer is already displayed, the buffer is hidden again.

   One reason to use `g v' is to add your own commands to the `*Gnuplot
Commands*' buffer.  Press `g v', then use `C-x o' to switch into that
window.  For example, GNUPLOT has `set label' and `set arrow' commands
that allow you to annotate your plots.  Since Calc doesn't understand
these commands, you have to add them to the `*Gnuplot Commands*' buffer
yourself, then use `g p' to replot using these new commands.  Note that
your commands must appear _before_ the `plot' command.  To get help on
any GNUPLOT feature, type, e.g., `g C help set label'.  You may have to
type `g C <RET>' a few times to clear the "press return for more" or
"subtopic of ..." requests.  Note that Calc always sends commands (like
`set nolabel') to reset all plotting parameters to the defaults before
each plot, so to delete a label all you need to do is delete the `set
label' line you added (or comment it out with `#') and then replot with
`g p'.

   You can use `g q' (`calc-graph-quit') to kill the GNUPLOT process
that is running.  The next graphing command you give will