This is ../info/calc, produced by makeinfo version 4.11 from calc.texi.

This file documents Calc, the GNU Emacs calculator.

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

     Permission is granted to copy, distribute and/or modify this
     document under the terms of the GNU Free Documentation License,
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     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 freedom to copy and
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INFO-DIR-SECTION Emacs
START-INFO-DIR-ENTRY
* Calc: (calc).         Advanced desk calculator and mathematical tool.
END-INFO-DIR-ENTRY


File: calc,  Node: Top,  Next: Getting Started,  Prev: (dir),  Up: (dir)

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

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

   This manual 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.)

* 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.

   The manual is divided into three major parts: the "Getting Started"
chapter you are reading now, the Calc tutorial (chapter 2), and the
Calc reference manual (the remaining chapters and appendices).

   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.  In the printed manual, each paragraph
that is referenced in the Key or Function Index is marked in the margin
with its index entry.

   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.  Also, you can jump
directly to the Tutorial by pressing `h t' or `C-x * t', or to the
Summary by pressing `h s' or `C-x * s'.  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 between `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 a letter `l', not a 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----- --%%-Emacs: *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.  `C-x * o' is a handy way to switch
out of Calc momentarily to edit your file; 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 2.1------+----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:

     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 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

   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 uses any non-numeric
characters rather than blank lines to delimit the formula it reads.
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 will
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; Robert J. Chassell, who
suggested the Calc Tutorial and exercises; and Juha Sarlin, who first
worked out how to split Calc into quickly-loading parts.  Bob Weiner
helped immensely with the Lucid Emacs port.

   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 `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.  (If
you have a printed copy of the manual you can use that instead.)  Press
`C-x * c' to turn Calc on or to switch into the Calc window, and press
`C-x * i' to start the Info system or to switch into its window.  Or,
you may prefer to use the tutorial in printed form.

   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.

   *Warning:* Note that `/' has lower precedence than `*', so that
`a/b*c' is interpreted as `a/(b*c)'.  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 u' to "unpack" this
vector into 8 separate stack entries, then `M-8 *' to multiply them
back together.  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::

