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<title>tutoriel</title>
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<!--HEVEA command line is: /usr/bin/hevea -fix tutoriel.tex -->
<!--CUT STYLE article--><!--CUT DEF section 1 --><p>
<br>
<br>
<br>

</p><div class="center">
<span style="font-size:xx-large"><span style="font-size:150%">An </span></span><span style="font-size:xx-large"><span style="font-size:150%"><span style="font-family:monospace">Xcas</span></span></span><span style="font-size:xx-large"><span style="font-size:150%"> Tutorial</span></span>
</div><!--TOC section id="sec1" Contents-->
<h2 id="sec1" class="section">Contents</h2><!--SEC END --><ul class="toc"><li class="li-toc">
<a href="#sec2">1&#XA0;&#XA0;Getting started</a>
<ul class="toc"><li class="li-toc">
<a href="#sec3">1.1&#XA0;&#XA0;Starting <span style="font-family:monospace">Xcas</span></a>
</li><li class="li-toc"><a href="#sec4">1.2&#XA0;&#XA0;<span style="font-family:monospace">XCas</span> as a calculator</a>
</li><li class="li-toc"><a href="#sec5">1.3&#XA0;&#XA0;Functions and variables</a>
</li><li class="li-toc"><a href="#sec6">1.4&#XA0;&#XA0;Simplifying expressions</a>
</li><li class="li-toc"><a href="#sec7">1.5&#XA0;&#XA0;Graphs</a>
</li><li class="li-toc"><a href="#sec8">1.6&#XA0;&#XA0;The Help Index</a>
</li></ul>
</li><li class="li-toc"><a href="#sec9">2&#XA0;&#XA0;The interface</a>
<ul class="toc"><li class="li-toc">
<a href="#sec10">2.1&#XA0;&#XA0;Overview</a>
</li><li class="li-toc"><a href="#sec11">2.2&#XA0;&#XA0;The menu bar</a>
</li><li class="li-toc"><a href="#sec12">2.3&#XA0;&#XA0;Configuration</a>
</li><li class="li-toc"><a href="#sec13">2.4&#XA0;&#XA0;The command line</a>
</li></ul>
</li><li class="li-toc"><a href="#sec14">3&#XA0;&#XA0;Computational objects</a>
<ul class="toc"><li class="li-toc">
<a href="#sec15">3.1&#XA0;&#XA0;Numbers</a>
</li><li class="li-toc"><a href="#sec16">3.2&#XA0;&#XA0;Variables</a>
</li><li class="li-toc"><a href="#sec17">3.3&#XA0;&#XA0;Expressions</a>
</li><li class="li-toc"><a href="#sec18">3.4&#XA0;&#XA0;Functions</a>
</li><li class="li-toc"><a href="#sec19">3.5&#XA0;&#XA0;Lists, sequences and sets</a>
</li><li class="li-toc"><a href="#sec20">3.6&#XA0;&#XA0;Characters and strings</a>
</li><li class="li-toc"><a href="#sec21">3.7&#XA0;&#XA0;Calculation time and memory space</a>
</li></ul>
</li><li class="li-toc"><a href="#sec22">4&#XA0;&#XA0;Analysis with <span style="font-family:monospace">Xcas</span></a>
<ul class="toc"><li class="li-toc">
<a href="#sec23">4.1&#XA0;&#XA0;Derivatives</a>
</li><li class="li-toc"><a href="#sec24">4.2&#XA0;&#XA0;Limits and series</a>
</li><li class="li-toc"><a href="#sec25">4.3&#XA0;&#XA0;Antiderivatives and integrals</a>
</li><li class="li-toc"><a href="#sec26">4.4&#XA0;&#XA0;Solving equations</a>
</li><li class="li-toc"><a href="#sec27">4.5&#XA0;&#XA0;Differential equations</a>
</li></ul>
</li><li class="li-toc"><a href="#sec28">5&#XA0;&#XA0;Algebra with <span style="font-family:monospace">Xcas</span></a>
<ul class="toc"><li class="li-toc">
<a href="#sec29">5.1&#XA0;&#XA0;Integer arithmetic</a>
</li><li class="li-toc"><a href="#sec30">5.2&#XA0;&#XA0;Polynomials and rational functions</a>
</li><li class="li-toc"><a href="#sec31">5.3&#XA0;&#XA0;Trigonometry</a>
</li><li class="li-toc"><a href="#sec32">5.4&#XA0;&#XA0;Vectors and matrices</a>
</li><li class="li-toc"><a href="#sec33">5.5&#XA0;&#XA0;Linear systems</a>
</li><li class="li-toc"><a href="#sec34">5.6&#XA0;&#XA0;Matrix reduction</a>
</li></ul>
</li><li class="li-toc"><a href="#sec35">6&#XA0;&#XA0;Graphs</a>
<ul class="toc"><li class="li-toc">
<a href="#sec36">6.1&#XA0;&#XA0;Curves</a>
</li><li class="li-toc"><a href="#sec37">6.2&#XA0;&#XA0;Plane geometry</a>
</li><li class="li-toc"><a href="#sec38">6.3&#XA0;&#XA0;3D graphical objects</a>
</li></ul>
</li><li class="li-toc"><a href="#sec39">7&#XA0;&#XA0;Programming</a>
<ul class="toc"><li class="li-toc">
<a href="#sec40">7.1&#XA0;&#XA0;The language</a>
</li><li class="li-toc"><a href="#sec41">7.2&#XA0;&#XA0;Some examples</a>
</li><li class="li-toc"><a href="#sec42">7.3&#XA0;&#XA0;Programming style</a>
</li></ul>
</li><li class="li-toc"><a href="#sec43">8&#XA0;&#XA0;Exercises</a>
</li></ul><p><span style="font-family:monospace">Xcas</span> is a free (as in Free Software) computer algebra system.
Although there are other computer algebra systems, both free and
commercial, few if any are as versatile as <span style="font-family:monospace">Xcas</span>,
which is capable of, among other things:
</p><ul class="itemize"><li class="li-itemize">
Symbolic and numeric calculation.
</li><li class="li-itemize">Programming.
</li><li class="li-itemize">Graphing functions.
</li><li class="li-itemize">Working with spreadsheets.
</li><li class="li-itemize">Interactive geometry (both two- and three-dimensional).
</li><li class="li-itemize">Turtle geometry.
</li></ul><p>
This tutorial will briefly introduce you to calculating,
programming and graphing with <span style="font-family:monospace">Xcas</span>. The first section, 
&#X201C;Getting started&#X201D;, will be just enough information to get you
started. The second section will be a brief overview of the
graphic interface and the rest will be more in-depth tutorial.</p><p>For more information, you can refer to the manual or any of the other
sources of information under the <span style="font-family:monospace">Help</span> menu.</p>
<!--TOC section id="sec2" Getting started-->
<h2 id="sec2" class="section">1&#XA0;&#XA0;Getting started</h2><!--SEC END -->
<!--TOC subsection id="sec3" Starting <span style="font-family:monospace">Xcas</span>-->
<h3 id="sec3" class="subsection">1.1&#XA0;&#XA0;Starting <span style="font-family:monospace">Xcas</span></h3><!--SEC END --><p>
<a id="hevea_default0"></a></p><p><span style="font-family:monospace">Xcas</span> is available from 
<a href="http://www-fourier.ujf-grenoble.fr/~parisse/giac_fr.html"><span style="font-family:monospace">http://www-fourier.ujf-grenoble.fr/~parisse/giac_fr.html</span></a>.
<a id="hevea_default1"></a>
This page also has information on installation. Once installed, the
way to start the program depends on the operating system.
</p><ul class="itemize"><li class="li-itemize">
Under Windows, there should be a shortcut <span style="font-family:monospace">xcasen.bat</span> that
you can click on.
</li><li class="li-itemize">Under Linux, you can either find it on a menu provided by the
desktop environment, or enter <span style="font-family:monospace">xcas &amp;</span> in a terminal window.
</li><li class="li-itemize">Under MacOS, you can click on <span style="font-family:monospace">xcas</span> in the Applications menu.
</li></ul><p>
For this tutorial, you will mostly be working with the
command line, which will be a white rectangle next to the number
<span style="font-family:monospace">1</span>.<a id="hevea_default2"></a> There you can enter a command, after which there will be
a window with the result, followed by another command line with the
number <span style="font-family:monospace">2</span>.</p>
<!--TOC subsection id="sec4" <span style="font-family:monospace">XCas</span> as a calculator-->
<h3 id="sec4" class="subsection">1.2&#XA0;&#XA0;<span style="font-family:monospace">XCas</span> as a calculator</h3><!--SEC END --><p>Once you have started <span style="font-family:monospace">Xcas</span>, you can immediately use it as a
calculator. Simply type in the expression that you wish to
use using the standard arithmetic operators; namely <span style="font-family:monospace">+</span> for
addition, <span style="font-family:monospace">-</span> for subtraction, <span style="font-family:monospace">*</span> for multiplication,
<span style="font-family:monospace">/</span> for division and <span style="font-family:monospace">^</span> for exponentiation.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">34+45*12
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
574
</td></tr>
</table><p>
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">2/3 + 98/7
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">44</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">3</td></tr>
</table></td></tr>
</table><p>
The operators have a standard order of operations, and parentheses
can be used for grouping. (Brackets have a different meaning, see
section <a href="#lists">3.5</a>, &#X201C;Lists, sequences and sets&#X201D;.)</p><p>Notice that if you enter integers or other exact 
values,<a id="hevea_default3"></a>
<span style="font-family:monospace">Xcas</span> will give you the exact result. If you enter an
approximate value, such as a number with a decimal point (computers
regard numbers with decimal points as approximate values), then
<span style="font-family:monospace">Xcas</span> will give you an approximate 
result.<a id="hevea_default4"></a> For example, if
you enter
</p><blockquote class="quote"><span style="font-family:monospace">2/3 + 3/2
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">13</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">6</td></tr>
</table></td></tr>
</table><p>
but if you enter
</p><blockquote class="quote"><span style="font-family:monospace">2/3 + 1.5
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2.16666666667
</td></tr>
</table><p>
You can also get a decimal approximation using the <span style="font-family:monospace">evalf</span>
function.<a id="hevea_default5"></a> If you enter
</p><blockquote class="quote"><span style="font-family:monospace">evalf(2/3 + 3/2)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2.16666666667
</td></tr>
</table><p>
By default, <span style="font-family:monospace">Xcas</span> will give you 12 decimal places of accuracy
in its approximations, but this is configurable (see section
<a href="#config">2.3</a>, &#X201C;Configuration&#X201D;).</p><p><span style="font-family:monospace">Xcas</span> also has the standard functions, such as
<span style="font-family:monospace">sin</span>, <span style="font-family:monospace">cos</span>, <span style="font-family:monospace">asin</span> (for the arcsin),
<span style="font-family:monospace">log</span> (for the natural logarithm).
It also has common constants such as <span style="font-family:monospace">pi</span>, <span style="font-family:monospace">e</span> and <span style="font-family:monospace">i</span>.</p><p>The trigonometric functions assume that angles are measured in
radians. (This can be configured, see section <a href="#config">2.3</a>,
&#X201C;Configuration&#X201D;.) If you enter
</p><blockquote class="quote"><span style="font-family:monospace">sin(pi/4)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >2</td></tr>
</table></td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td></tr>
</table>
<!--TOC subsection id="sec5" Functions and variables-->
<h3 id="sec5" class="subsection">1.3&#XA0;&#XA0;Functions and variables</h3><!--SEC END --><p><span style="font-family:monospace">Xcas</span> can work with expressions and variables as well as 
numbers.<a id="hevea_default6"></a>
A variable in <span style="font-family:monospace">Xcas</span> needs to begin with a letter and can
include numbers and underscores. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">x^2 + x + 2*x + 2
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span><sup>2</sup>&#XA0;+&#XA0;3*<span style="font-style:italic">x</span>&#XA0;+&#XA0;2
</td></tr>
</table><p>
You can give a variable a value with the assignment operator,
<span style="font-family:monospace">:=</span>.<a id="hevea_default7"></a> To assign the variable
<span style="font-family:monospace">myvar</span> the value 5, for example, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">myvar := 5
</span></blockquote><p>
If you later use <span style="font-family:monospace">myvar</span> in an expression, it will be replaced
by <span style="font-family:monospace">5</span>; entering
</p><blockquote class="quote"><span style="font-family:monospace">myvar*x + myvar^2
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
5*<span style="font-style:italic">x</span>&#XA0;+&#XA0;25
</td></tr>
</table><p>The assignment operator can also be used to define 
functions.<a id="hevea_default8"></a> To
define the squaring function, for example, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">sqr(x) := x^2
</span></blockquote><p>
Afterwards, whenever you enter <span style="font-family:monospace">sqr(expression)</span> you will
get the expression squared. For example, entering
</p><blockquote class="quote"><span style="font-family:monospace">sqr(7)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
49
</td></tr>
</table><p>
and entering
</p><blockquote class="quote"><span style="font-family:monospace">sqr(x+1)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(<span style="font-style:italic">x</span>+1)<sup>2</sup>
</td></tr>
</table>
<!--TOC subsection id="sec6" Simplifying expressions-->
<h3 id="sec6" class="subsection">1.4&#XA0;&#XA0;Simplifying expressions</h3><!--SEC END --><p>When you enter an expression into <span style="font-family:monospace">Xcas</span>, some simplifications
will be done automatically. For example, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">a := 3
</span></blockquote><blockquote class="quote"><span style="font-family:monospace">b := 4
</span></blockquote><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">a*b*x + 4*b^2
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
12*<span style="font-style:italic">x</span>&#XA0;+&#XA0;64
</td></tr>
</table><p><span style="font-family:monospace">Xcas</span> has several transformations in case you want an
expression to be simplified beyond the automatic simplifications, or
perhaps transformed in another way. Some examples are:
</p><dl class="description"><dt class="dt-description">
<span style="font-weight:bold"><span style="font-family:monospace">expand</span></span></dt><dd class="dd-description"><a id="hevea_default9"></a>
This will expand integer powers, and more generally distribute
multiplication across addition. For example, if you enter
<blockquote class="quote"><span style="font-family:monospace">expand((x+1)^3)
</span></blockquote>
you will get
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span><sup>3</sup>&#XA0;+&#XA0;3*<span style="font-style:italic">x</span><sup>2</sup>&#XA0;+&#XA0;3*<span style="font-style:italic">x</span>&#XA0;+&#XA0;1
</td></tr>
</table></dd><dt class="dt-description"><span style="font-weight:bold"><span style="font-family:monospace">factor</span></span></dt><dd class="dd-description"><a id="hevea_default10"></a>
This will factor polynomials. For example, if you enter
<blockquote class="quote"><span style="font-family:monospace">factor(x^2 + 3*x + 2)
</span></blockquote>
you will get
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(<span style="font-style:italic">x</span>&#XA0;+&#XA0;1)*(<span style="font-style:italic">x</span>&#XA0;+&#XA0;2)
</td></tr>
</table>
</dd></dl>
<!--TOC subsection id="sec7" Graphs-->
<h3 id="sec7" class="subsection">1.5&#XA0;&#XA0;Graphs</h3><!--SEC END --><p><span style="font-family:monospace">Xcas</span> has several functions for plotting graphs; perhaps the
simplest is the <span style="font-family:monospace">plot</span> function.<a id="hevea_default11"></a> 
The <span style="font-family:monospace">plot</span> function requires two arguments, an expression to be
graphed and a variable. For example, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">plot(sin(x),x)
</span></blockquote><p>
you will get the graph
</p><div class="center">
<img src="tutoriel001.png">
</div><p>
To the right of the graph will be a panel you can use to control
various aspects. By default, the graph will cover values of the
variable from &#X2212;10 to 10; this is of course configurable (see
section <a href="#config">2.3</a>, &#X201C;Configuration&#X201D;). To plot
over a different interval you can also use a second argument of 
<span style="font-style:italic">var</span>=<span style="font-style:italic">min</span>..<span style="font-style:italic">max</span> instead of simply
<span style="font-style:italic">var</span>. For example, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">plot(sin(x),x=-pi..pi)
</span></blockquote><p>
you will get the graph
</p><div class="center">
<img src="tutoriel002.png">
</div>
<!--TOC subsection id="sec8" The Help Index-->
<h3 id="sec8" class="subsection">1.6&#XA0;&#XA0;The Help Index</h3><!--SEC END --><p>You can get a list of all <span style="font-family:monospace">Xcas</span> commands and variables using the
<span style="font-family:monospace">Help</span>&#X25B8;<span style="font-family:monospace">Index</span><a id="hevea_default12"></a> menu item.
This will bring up the following window:
</p><div class="center">
<img src="tutoriel003.png">
</div><p>
Under <span style="font-family:monospace">Index</span> will be a scrollable list of all the commands and
variables. 
If you begin typing to the right of the question mark, you will be
taken to the part of the list beginning with the characters you typed;
for example, if you type <span style="font-family:monospace">evalf</span>, <a id="hevea_default13"></a>
you will get the following:
</p><div class="center">
<img src="tutoriel004.png">
</div><p>
In the upper right-hand pane under <span style="font-family:monospace">Related</span> is a list of
commands related to the chosen command, below that is a list of
synonyms; you can see that the <span style="font-family:monospace">approx</span> command is the same as
the <span style="font-family:monospace">evalf</span> command. Below the line where you typed the
command name is a description of the command; here you
can see that <span style="font-family:monospace">evalf</span> takes an optional second argument
(brackets in the description indicate that an argument is optional)
which can specify the number of digits in the approximation; for
example, 
</p><blockquote class="quote"><span style="font-family:monospace">evalf(pi)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
3.14159265359
</td></tr>
</table><p>
but 
</p><blockquote class="quote"><span style="font-family:monospace">evalf(pi,20)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
3.1415926535897932385
</td></tr>
</table><p>
Below the description is a line where you can enter the arguments for
the command; if you enter values in these boxes then the command with
the chosen arguments will be placed on the <span style="font-family:monospace">Xcas</span> command line.
At the bottom of the help window is a list of examples of the command
being used; if you click on one of these examples it will appear on
the <span style="font-family:monospace">Xcas</span> command line.</p><p>As well as using the menu, you can get to the help 
index<a id="hevea_default14"></a> by using the
tab key while at the <span style="font-family:monospace">Xcas</span> command line. If you have typed
the beginning of a command before using the tab key, then that
beginning will be presented to you in the help index.</p>
<!--TOC section id="sec9" The interface-->
<h2 id="sec9" class="section">2&#XA0;&#XA0;The interface</h2><!--SEC END -->
<!--TOC subsection id="sec10" Overview-->
<h3 id="sec10" class="subsection">2.1&#XA0;&#XA0;Overview</h3><!--SEC END --><p>When you start <span style="font-family:monospace">Xcas</span>, you will be presented with a window which
looks like the following:
</p><div class="center">
<img src="tutoriel005.png">
</div><p>
From top to bottom, there is<a id="hevea_default15"></a>
</p><ul class="itemize"><li class="li-itemize">
A menu bar.<a id="hevea_default16"></a>
</li><li class="li-itemize">A tab indicating the name of the session, or <span style="font-family:monospace">Unnamed</span> if the
session has not been saved. You can run several sessions
simultaneously, in which case each session will get its own tab.
</li><li class="li-itemize">A session management bar, with
<ul class="itemize"><li class="li-itemize">
A <span style="font-family:monospace">?</span> button, which will open the help index.
</li><li class="li-itemize">A <span style="font-family:monospace">save</span> button to save the session.
</li><li class="li-itemize">A configuration button indicating how <span style="font-family:monospace">Xcas</span> is currently
configured. Clicking on this button will open a configuration
window.
</li><li class="li-itemize">A <span style="font-family:monospace">STOP</span> button you can use to interrupt a calculation
which is running on too long.
</li><li class="li-itemize">A <span style="font-family:monospace">Kbd</span> button to bring up an on-screen keyboard which you
can use to help enter your commands. It will come with a control panel
which you can use to display a message window (<span style="font-family:monospace">msg</span>) or
show an extra menu (<span style="font-family:monospace">cmds</span>) at the bottom.
</li><li class="li-itemize">An <span style="font-family:monospace">X</span> button to close the session.
</li></ul>
</li><li class="li-itemize">A numbered command line.
</li></ul>
<!--TOC subsection id="sec11" The menu bar-->
<h3 id="sec11" class="subsection">2.2&#XA0;&#XA0;The menu bar</h3><!--SEC END --><p>
<a id="hevea_default17"></a>
The menu items have submenus, and sometimes sub-submenus. When
indicating a submenu item, it will be separated from the menu item with
&#X25B8;; for example, you can save a session with
<span style="font-family:monospace">File</span>&#X25B8;<span style="font-family:monospace">Save</span>, which is the <span style="font-family:monospace">Save</span>
item in the <span style="font-family:monospace">File</span> menu.</p><p>The menu bar contains the usual menus for graphic programs. It also
contains all of the <span style="font-family:monospace">Xcas</span> commands grouped by themes. For
some commands, if you choose it from a menu then the command will be
put on the command line. If the message window is open (which you can
open with the
<span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Show</span>&#X25B8;<span style="font-family:monospace">msg</span>
menu item), a brief description of the command will appear in that
window. For other commands, for example the <span style="font-family:monospace">Graphic</span>
commands, you will get a dialog box which lets you specify the
arguments; afterwards, the command with the arguments will be placed
on the command line.</p><p>The <span style="font-family:monospace">Help</span> menu has links to the Help index, various manuals,
as well as the online forum. </p>
<!--TOC subsection id="sec12" Configuration-->
<h3 id="sec12" class="subsection">2.3&#XA0;&#XA0;Configuration</h3><!--SEC END --><p>
<a id="config"></a></p><p>The <span style="font-family:monospace">Cfg</span> menu has various items that allow you to configure
various aspects of <span style="font-family:monospace">Xcas</span>. This tutorial will
refer to the <span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Cas configuration</span> and 
<span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Graph configuration</span> menu items. The
<span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Cas configuration</span> menu item will
bring up the same configuration page as clicking on the status bar.
These items will bring up windows with various entry fields and
check boxes; after you make any changes, you can click the
<span style="font-family:monospace">Apply</span> button to apply them and the <span style="font-family:monospace">Save</span> button to
save them for future sessions.</p><p>The <span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Cas configuration</span> menu item will
bring up a window with options that determine how <span style="font-family:monospace">Xcas</span> computes.
This includes some things mentioned in this tutorial, such as:
</p><ul class="itemize"><li class="li-itemize">
<span style="font-weight:bold">Digits.</span><a id="hevea_default18"></a>
This entry field determines the number of significant digits used in
calculations. This resets the value of the variable <span style="font-family:monospace">Digits</span>,
which you can also reset from the command line.
</li><li class="li-itemize"><span style="font-weight:bold">epsilon.</span><a id="hevea_default19"></a>
This entry field determines how close the fraction returned by
<span style="font-family:monospace">exact</span> will be to the input. This resets the value of the
variable <span style="font-family:monospace">epsilon</span>, which you can also reset from the command
line.
</li><li class="li-itemize"><span style="font-weight:bold">radian.</span><a id="hevea_default20"></a>
This checkbox determines whether angles are measured in radians or
degrees.
</li><li class="li-itemize"><span style="font-weight:bold">Complex.</span><a id="hevea_default21"></a>
This checkbox determines whether computations will find complex
solutions to equations.
</li><li class="li-itemize"><span style="font-weight:bold">All_trig_sol.</span><a id="hevea_default22"></a>
This checkbox determines whether <span style="font-family:monospace">solve</span> will find the
primary solutions to trigonometric equations or all solutions.
</li></ul><p>The <span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Graph configuration</span> menu item
will bring up a window with options that determine how <span style="font-family:monospace">Xcas</span>
draws graphs. This includes the default ranges for the axes; the
<span style="font-style:italic">x</span>-axis will go from <span style="font-family:monospace">X-</span> to <span style="font-family:monospace">X+</span>, the <span style="font-style:italic">y</span>-axis will go
from <span style="font-family:monospace">Y-</span> to <span style="font-family:monospace">Y+</span> and the <span style="font-style:italic">z</span>-axis will go from
<span style="font-family:monospace">Z-</span> to <span style="font-family:monospace">Z+</span>.</p>
<!--TOC subsection id="sec13" The command line-->
<h3 id="sec13" class="subsection">2.4&#XA0;&#XA0;The command line</h3><!--SEC END --><p>
<a id="hevea_default23"></a></p><p>You can run a command by typing it into the command
line and pressing <span style="font-family:monospace">Enter</span>. If you want to enter more than one
command on a line, you can separate them with semicolons. If you want
to suppress the output of a command, you can end it with a
colon-semicolon (<span style="font-family:monospace">:;</span>).</p><p>If you have enough commands, there will be a scroll bar on the right
which you can use to scroll through different command line levels. The
<span style="font-family:monospace">Edit</span> menu will allow you to merge levels, group levels and
add comments.</p><p>All commands are kept in memory. You can scroll through previous
commands with <span style="font-family:monospace">Ctrl+</span> arrow keys, and modify them if you want.</p>
<!--TOC section id="sec14" Computational objects-->
<h2 id="sec14" class="section">3&#XA0;&#XA0;Computational objects</h2><!--SEC END -->
<!--TOC subsection id="sec15" Numbers-->
<h3 id="sec15" class="subsection">3.1&#XA0;&#XA0;Numbers</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Operations</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">+</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >addition</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">-</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >subtraction </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">*</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >multiplication </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">/</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >division</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">^</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >power </td></tr>
</table>
</div><p>
<a id="hevea_default24"></a>
<a id="hevea_default25"></a>
<a id="hevea_default26"></a>
<a id="hevea_default27"></a>
<a id="hevea_default28"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Conversions</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">evalf</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximate a number</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">exact</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >find an exact number close to the given number</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">epsilon</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >determine how close a fraction has to be to
a floating point number to be returned by
		 <span style="font-family:monospace">exact</span></td></tr>
</table>
</div><p>
<a id="hevea_default29"></a>
<a id="hevea_default30"></a>
<a id="hevea_default31"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Constants</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">pi</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >&#X3C0;&#X2243; 3.14159265359 </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">e</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-style:italic">e</span> &#X2243; 2.71828182846 </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">i</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-style:italic">i</span>=&#X221A;<span style="text-decoration:overline">&#X2212;1</span> </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">infinity</span> or <span style="font-family:monospace">inf</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >&#X221E; </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">+infinity</span> or <span style="font-family:monospace">+inf</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >+&#X221E; </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">-infinity</span> or <span style="font-family:monospace">-inf</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >&#X2212;&#X221E; </td></tr>
</table>
</div><p>
<a id="hevea_default32"></a>
<a id="hevea_default33"></a>
<a id="hevea_default34"></a>
<a id="hevea_default35"></a>
<a id="hevea_default36"></a></p><p>There are two types of numbers in <span style="font-family:monospace">Xcas</span>, approximate and exact.</p><p>Computer programs like <span style="font-family:monospace">Xcas</span> regard floating point numbers,
which are numbers displayed with decimal points, as 
approximations.<a id="hevea_default37"></a> Other
numbers will be regarded as exact.<a id="hevea_default38"></a> For example, the number
<span style="font-family:monospace">2</span> is exactly 2, while <span style="font-family:monospace">2.0</span> represents a number that
equals 2 to within the current precision, which by default is about 12
significant digits (see section <a href="#config">2.3</a>, &#X201C;Configuration&#X201D;).
Approximate numbers can be entered by
typing in a number with a decimal point or in scientific notation
(which is a decimal number followed by <span style="font-family:monospace">e</span> and then an integer,
where the integer represents the power of 10). So <span style="font-family:monospace">2000.0</span>,
<span style="font-family:monospace">2e3</span> and <span style="font-family:monospace">2.0e3</span> all represent the same approximate
number.</p><p>Exact numbers are integers, symbolic constants (like <span style="font-style:italic">e</span> and &#X3C0;),
and numeric expressions which only involve exact 
numbers.<a id="hevea_default39"></a> For example,
sin(1) will be exact, and so won&#X2019;t be given a decimal
approximation; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">sin(1)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
sin(1)
</td></tr>
</table><p>
However, sin(1.0) involves the approximate number <span style="font-family:monospace">1.0</span>, and
so will be regarded as approximate itself. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">sin(1.0)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
0.841470984808
</td></tr>
</table><p>As with many computer languages, if you enter an integer beginning
with the digit 0, the <span style="font-family:monospace">Xcas</span> will regard it as an integer
base 8;<a id="hevea_default40"></a> if you enter
</p><blockquote class="quote"><span style="font-family:monospace">011
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
9
</td></tr>
</table><p>
since 11 is the base 8 representation of the decimal number 9.
Similarly, if you write <span style="font-family:monospace">0x</span> at the beginning of an integer,
<span style="font-family:monospace">Xcas</span> will regard it as a hexadecimal (base 16) 
integer.<a id="hevea_default41"></a> If you enter
</p><blockquote class="quote"><span style="font-family:monospace">0x11
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
17
</td></tr>
</table><p>
since 11 is the base 16 representation of the decimal number 17.</p><p>The symbolic constants that are built in to <span style="font-family:monospace">Xcas</span> are
<span style="font-family:monospace">pi</span>, <span style="font-family:monospace">e</span>, <span style="font-family:monospace">i</span>, <span style="font-family:monospace">infinity</span>,
<span style="font-family:monospace">+infinity</span> and <span style="font-family:monospace">-infinity</span>.
Note that <span style="font-family:monospace">Xcas</span> distinguishes between <span style="font-family:monospace">+infinity</span>,
<span style="font-family:monospace">-infinity</span> and <span style="font-family:monospace">infinity</span>, which is unsigned infinity.
The distinction can be noted in the following calculations:
</p><blockquote class="quote"><span style="font-family:monospace">1/0
</span></blockquote><p>
will result in <span style="font-family:monospace">infinity</span>,
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X221E;
</td></tr>
</table><p>
while
</p><blockquote class="quote"><span style="font-family:monospace">(1/0)^2
</span></blockquote><p>
will result in <span style="font-family:monospace">+infinity</span>,
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
+&#X221E;
</td></tr>
</table><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">-(1/0)^2
</span></blockquote><p>
will result in <span style="font-family:monospace">-infinity</span>,
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X2212;&#X221E;
</td></tr>
</table><p>
These variables cannot be reassigned; in particular, the variable
<span style="font-family:monospace">i</span> can&#X2019;t be used as a loop index.</p><p><span style="font-family:monospace">Xcas</span> can handle integers of arbitrary length; if, for
example, you enter
</p><blockquote class="quote"><span style="font-family:monospace">500!
</span></blockquote><p>
you will be given all 1135 digits of the factorial of 500.</p><p>When <span style="font-family:monospace">Xcas</span> combines two numbers, the result will be exact
unless one of the numbers is approximate, in which case the result
will be approximate. For example, entering
</p><blockquote class="quote"><span style="font-family:monospace">3/2 + 1
</span></blockquote><p>
will return the exact value
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">5</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td></tr>
</table><p>
while entering
</p><blockquote class="quote"><span style="font-family:monospace">1.5 + 1
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2.5
</td></tr>
</table><p>The <span style="font-family:monospace">evalf</span><a id="hevea_default42"></a> function will
transform a number to an approximate value. While entering
</p><blockquote class="quote"><span style="font-family:monospace">sqrt(2)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >2</td></tr>
</table></td></tr>
</table><p>
entering
</p><blockquote class="quote"><span style="font-family:monospace">evalf(sqrt(2))
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1.41421356237
</td></tr>
</table><p>
which is the square root of two to the default precision, in this case
12 digits. The <span style="font-family:monospace">evalf</span> function can also take a second
argument which you can use to specify how many digits of precision
that you want; for example if you want to know the square root of two
to 50 digits, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">evalf(sqrt(2),50)
</span></blockquote><p>
and get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1.4142135623730950488016887242096980785696718753770
</td></tr>
</table><p>The <span style="font-family:monospace">exact</span><a id="hevea_default43"></a> function will turn 
an approximate value into a nearby
exact value. Specifically, given an approximate value <span style="font-style:italic">x</span>,
<span style="font-family:monospace">exact(</span><span style="font-style:italic">x</span><span style="font-family:monospace">)</span> will be a rational number <span style="font-style:italic">r</span> with
|<span style="font-style:italic">x</span> &#X2212; <span style="font-style:italic">r</span>| &lt; &#X454;, where &#X454; is the value of the variable
<span style="font-family:monospace">epsilon</span>, which has a default value of 10<sup>&#X2212;12</sup>. This value
is configurable (see section <a href="#config">2.3</a>, &#X201C;Configuration&#X201D;).</p>
<!--TOC subsection id="sec16" Variables-->
<h3 id="sec16" class="subsection">3.2&#XA0;&#XA0;Variables</h3><!--SEC END --><p>
<a id="hevea_default44"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Variables</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">:=</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >assignment </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">subst</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >give a variable a value for a single instance</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">assume</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >put assumptions on variables </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">and</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >combine assumptions</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">or</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >combine assumptions</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">purge</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >remove values and assumptions attached to variables </td></tr>
</table>
</div><p>
<a id="hevea_default45"></a>
<a id="hevea_default46"></a>
<a id="hevea_default47"></a>
<a id="hevea_default48"></a>
<a id="hevea_default49"></a>
<a id="hevea_default50"></a></p><p>A variable in <span style="font-family:monospace">Xcas</span> begins with a letter and can contain
letters, numbers and underscores. </p><p>A variable can be given a value with the assignment operator <span style="font-family:monospace">:=</span>.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">a := 3
</span></blockquote><p>
then <span style="font-family:monospace">a</span> will be replaced by <span style="font-family:monospace">3</span> in all later
calculations. If you later enter
</p><blockquote class="quote"><span style="font-family:monospace">4*a^2
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
36
</td></tr>
</table><p>
The <span style="font-family:monospace">purge</span> command will unassign a variable; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">purge(a)
</span></blockquote><p>
and then
</p><blockquote class="quote"><span style="font-family:monospace">4*a^2
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
4&#XB7;&#XA0;<span style="font-style:italic">a</span><sup>2</sup>
</td></tr>
</table><p>The assignment operator <span style="font-family:monospace">:=</span> is one of three types of
equalities used in <span style="font-family:monospace">Xcas</span>. They are
</p><ul class="itemize"><li class="li-itemize">
The assignment operator, <span style="font-family:monospace">:=</span>,<a id="hevea_default51"></a> 
which is used to assign values.
</li><li class="li-itemize">The Boolean equality, <span style="font-family:monospace">==</span>,<a id="hevea_default52"></a> 
which tells you whether two
quantities are equal to each other or not. If you enter
<span style="font-style:italic">A</span><span style="font-family:monospace">==</span><span style="font-style:italic">B</span>, then you will get either
<span style="font-family:monospace">true</span> or <span style="font-family:monospace">false</span> as a result. The predefined
constants <span style="font-family:monospace">true</span> and <span style="font-family:monospace">True</span> are equal to 1, the
predefined constants <span style="font-family:monospace">false</span> and <span style="font-family:monospace">False</span> are equal to 0.
<a id="hevea_default53"></a>
<a id="hevea_default54"></a>
<a id="hevea_default55"></a>
<a id="hevea_default56"></a>
<a id="hevea_default57"></a>
<a id="hevea_default58"></a> 
</li><li class="li-itemize">The equal sign <span style="font-family:monospace">=</span><a id="hevea_default59"></a> is used to define an 
equation. In this case, the equation will be the expression.
<a id="hevea_default60"></a>
<a id="hevea_default61"></a>
</li></ul><p>If you want to replace a variable by a value for a single expression,
you can use the <span style="font-family:monospace">subst</span> command. This command takes an
expression and an equation <span style="font-style:italic">var</span> <span style="font-family:monospace">=</span> <span style="font-style:italic">value</span> as a
second argument. If <span style="font-family:monospace">a</span> is an unassigned variable, for example, then
entering
</p><blockquote class="quote"><span style="font-family:monospace">subst(a^2 + 2, a=3)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
11
</td></tr>
</table><p>
Afterwards, <span style="font-family:monospace">a</span> will still be unassigned.</p><p>Even without giving a variable a value, you can still tell
<span style="font-family:monospace">Xcas</span> some of its properties with the <span style="font-family:monospace">assume</span> command.
For example, for a real number <span style="font-style:italic">a</span>, the expression &#X221A;<span style="text-decoration:overline"><span style="font-style:italic">a</span></span><sup><span style="text-decoration:overline">2</span></sup>
simplifies to |<span style="font-style:italic">a</span>|, since <span style="font-style:italic">a</span> could be positive or negative. If you
enter 
</p><blockquote class="quote"><span style="font-family:monospace">sqrt(a^2)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
|<span style="font-style:italic">a</span>|
</td></tr>
</table><p>
If you enter 
</p><blockquote class="quote"><span style="font-family:monospace">assume(a&lt;0)
</span></blockquote><p>
beforehand, then <span style="font-family:monospace">Xcas</span> will work under the assumption that
<span style="font-family:monospace">a</span> is negative, and so entering
</p><blockquote class="quote"><span style="font-family:monospace">sqrt(a^2)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X2212;<span style="font-style:italic">a</span>
</td></tr>
</table><p>
As well as assuming that a variable satifies an equation or
inequality, you can use the keywords <span style="font-family:monospace">and</span> and <span style="font-family:monospace">or</span> to
assume that a variable satisifies more than one inequality. Some
assumptions on a variable require a second argument; for example, to
assume that <span style="font-style:italic">a</span> is an integer you can enter
</p><blockquote class="quote"><span style="font-family:monospace">assume(a,integer)
</span></blockquote><p><a id="hevea_default62"></a>
Afterwards
</p><blockquote class="quote"><span style="font-family:monospace">sin(a*pi)
</span></blockquote><p>
will result in 
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
0
</td></tr>
</table><p>
The <span style="font-family:monospace">purge</span> command will remove any assumptions about a
variable as well as any assigned values.</p>
<!--TOC subsection id="sec17" Expressions-->
<h3 id="sec17" class="subsection">3.3&#XA0;&#XA0;Expressions</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Conversions</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">expand</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >expand powers and distribute multiplication </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">normal</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >reduce to lowest terms</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ratnormal</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >reduce to lowest terms</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">factor</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >factor</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">simplify</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >reduce an expression to simpler form</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tsimplify</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >reduce and expression to simpler form</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">convert</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert an expression to a different type</td></tr>
</table>
</div><p>
<a id="hevea_default63"></a>
<a id="hevea_default64"></a>
<a id="hevea_default65"></a>
<a id="hevea_default66"></a>
<a id="hevea_default67"></a>
<a id="hevea_default68"></a>
<a id="hevea_default69"></a></p><p>An expression is a combination of numbers and variables combined by
arithmetic operators. For example, <span style="font-family:monospace">x^2 + 2*x + c</span> is an
expression. </p><p>When you enter an expression, <span style="font-family:monospace">Xcas</span> will perform some
automatic simplifications,<a id="hevea_default70"></a> such as
</p><ul class="itemize"><li class="li-itemize">
Any variables that have been assigned are replaced by their values.
</li><li class="li-itemize">Operations on numbers are performed.
</li><li class="li-itemize">Trivial simplifications, such as <span style="font-style:italic">x</span>+0=<span style="font-style:italic">x</span> and <span style="font-style:italic">x</span>&#XB7; 0 = 0, are made.
</li><li class="li-itemize">Some trigonometric forms are rewritten; for example, <span style="font-family:monospace">cos(-x)</span> is
replaced by <span style="font-family:monospace">cos(x)</span> and <span style="font-family:monospace">cos(pi/4)</span> is replaced by &#X221A;<span style="text-decoration:overline">2</span>/2.
</li></ul><p>Other simplifications are not done automatically, since it isn&#X2019;t
always clear what sort of simplifications the user might want, and
besides non-trivial simplifications are time-consuming.
The most used commands for simplifying and transforming commands are:
</p><dl class="description"><dt class="dt-description">
<span style="font-weight:bold"><span style="font-family:monospace">expand</span></span></dt><dd class="dd-description">
This will expand integer powers and more generally distribute
multiplication across addition. For example, if you enter
<blockquote class="quote"><span style="font-family:monospace">expand((x+1)^3)
</span></blockquote>
you will get
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span><sup>3</sup>&#XA0;+&#XA0;3*<span style="font-style:italic">x</span><sup>2</sup>&#XA0;+&#XA0;3*<span style="font-style:italic">x</span>&#XA0;+&#XA0;1
</td></tr>
</table></dd><dt class="dt-description"><span style="font-weight:bold"><span style="font-family:monospace">normal</span></span><span style="font-weight:bold"> and </span><span style="font-weight:bold"><span style="font-family:monospace">ratnormal</span></span></dt><dd class="dd-description">
These commands will reduce a rational function to lowest terms. For
example, if you enter
<blockquote class="quote"><span style="font-family:monospace">normal((x^3-1)/(x^2-1))
</span></blockquote>
then <span style="font-family:monospace">Xcas</span> will cancel a common factor of <span style="font-style:italic">x</span>&#X2212;1 from the top
and bottom and return
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup>+<span style="font-style:italic">x</span>+1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>+1</td></tr>
</table></td></tr>
</table>
<span style="font-family:monospace">ratnormal</span> will have the same behavior on this expression.
The difference between the two commands is that <span style="font-family:monospace">ratnormal</span>
does not take into account reductions with algebraic numbers, while
<span style="font-family:monospace">normal</span> does. If you enter
<blockquote class="quote"><span style="font-family:monospace">ratnormal((x^2-2)/(x-sqrt(2)))
</span></blockquote>
you will get
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup>&#X2212;2</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">x</span>&#X2212;</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >2</td></tr>
</table></td></tr>
</table></td></tr>
</table></td></tr>
</table>
but if you enter
<blockquote class="quote"><span style="font-family:monospace">normal((x^2-2)/(x-sqrt(2)))
</span></blockquote>
you will get
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span>+</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >2</td></tr>
</table></td></tr>
</table><p>Neither of these commands will take into account relationships
between transcendental functions such as <span style="font-family:monospace">sin</span> and <span style="font-family:monospace">cos</span>.</p></dd><dt class="dt-description"><span style="font-weight:bold"><span style="font-family:monospace">factor</span></span></dt><dd class="dd-description">
This will factor polynomials and reduce rational expressions. This
is a little slower than <span style="font-family:monospace">normal</span> and <span style="font-family:monospace">ratnormal</span> and
different in that it will give the result in factored form. For
example, if you enter 
<blockquote class="quote"><span style="font-family:monospace">factor(x^2 + 3*x + 2)
</span></blockquote>
you will get
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(<span style="font-style:italic">x</span>&#XA0;+&#XA0;1)*(<span style="font-style:italic">x</span>&#XA0;+&#XA0;2)
</td></tr>
</table></dd><dt class="dt-description"><span style="font-weight:bold"><span style="font-family:monospace">simplify</span></span></dt><dd class="dd-description">
This command will try to reduce an expression to algebraically
independent variables, then it will apply <span style="font-family:monospace">normal</span>.
Simplifications requiring algebraic extensions (such as roots) may
require two calls to <span style="font-family:monospace">simplify</span> and possibly adding some
assumptions with <span style="font-family:monospace">assume</span>. </dd><dt class="dt-description"><span style="font-weight:bold"><span style="font-family:monospace">tsimplify</span></span></dt><dd class="dd-description">
Like <span style="font-family:monospace">simplify</span>, this will try to reduce an expression to
algebraically independent variables, but will not apply
<span style="font-family:monospace">normal</span> afterwards.
</dd></dl><p>The <span style="font-family:monospace">convert</span> command will rewrite expressions to different
formats; the first argument will be the expression and the second
argument will indicate the format to convert the expression to. For
example, you can convert <span style="font-style:italic">e</span><sup><span style="font-style:italic">i</span> &#X3B8;</sup> to sines and 
cosines<a id="hevea_default71"></a> with
</p><blockquote class="quote"><span style="font-family:monospace">convert(exp(i*theta),sincos)
</span></blockquote><p>
the result will be
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
cos(&#X3B8;)&#XA0;+&#XA0;<span style="font-style:italic">i</span>*sin(&#X3B8;)
</td></tr>
</table><p>
You can use <span style="font-family:monospace">convert</span> to find the partial fraction
decomposition of a rational expression with a second argument of
<span style="font-family:monospace">partfrac</span>;<a id="hevea_default72"></a> 
for example, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">convert((x-1)/(x^2 - x -2), partfrac)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">2</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>+1)*3</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>&#X2212;2)*3</td></tr>
</table></td></tr>
</table>
<!--TOC subsection id="sec18" Functions-->
<h3 id="sec18" class="subsection">3.4&#XA0;&#XA0;Functions</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Common functions</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">abs</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >absolute value</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sign</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >sign (-1,0,+1)</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">max</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >maximum</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">min</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >minimum</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">round</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >round to the nearest integer </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">floor</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >greatest integer less than or equal to</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">frac</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >fractional part</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ceil</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >least integer greater than or equal to</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">re</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >real part</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">im</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >imaginary part</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">abs</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >absolute value</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">arg</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >argument</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">conj</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >conjugate</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">coordinates</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the coordinates of a point</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">factorial</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >factorial</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">!</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >factorial</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sqrt</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >square root</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">exp</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >exponential</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">log</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >natural logarithm</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ln</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >natural logarithm</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">log10</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >logarithm base 10</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sin</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >sine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cos</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >cosine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tan</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >tangent</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >cotangent</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">asin</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >arcsine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">acos</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >arccosine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">atan</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >arctangent</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sinh</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >hyperbolic sine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cosh</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >hyperbolic cosine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tanh</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >hyperbolic tangent</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">asinh</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >inverse hyperbolic sine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">acosh</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >inverse hyperbolic cosine</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">atanh</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >inverse hyperbolic tangent</td></tr>
</table>
</div><p>
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<a id="hevea_default129"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Create functions</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">:=</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >assign an expression to a function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">-&gt;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >define a function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">unapply</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >turn an expression into a function</td></tr>
</table>
</div><p>
<a id="hevea_default130"></a>
<a id="hevea_default131"></a></p><p><span style="font-family:monospace">Xcas</span> has many built in functions; you can get a complete list
with the help index. You can also define your own functions with the
assignment (<span style="font-family:monospace">:=</span>) operator.
To define a function <span style="font-style:italic">f</span> given by <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>) = <span style="font-style:italic">x</span>*exp(<span style="font-style:italic">x</span>), for example, you
can enter 
</p><blockquote class="quote"><span style="font-family:monospace">f(x) := x*exp(x)
</span></blockquote><p>
Note that in this case the name of the function is <span style="font-style:italic">f</span>; <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>) is the
value of the function evaluated at <span style="font-style:italic">x</span>. The function is a rule which
takes an input <span style="font-family:monospace">x</span> and returns <span style="font-family:monospace">x*exp(x)</span>. This rule
can be written without giving it a name as <span style="font-family:monospace">x -&gt; x*exp(x)</span>. In
fact, another way you can define the function <span style="font-style:italic">f</span> as above is
</p><blockquote class="quote"><span style="font-family:monospace">f := x -&gt;x*exp(x)
</span></blockquote><p>
In either case, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">f(2)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2*exp(2)
</td></tr>
</table><p>You can similarly define functions of more than one variable.
For example, to convert polar coordinates to rectangular coordinates,
you could define
</p><blockquote class="quote"><span style="font-family:monospace">p(r,theta) := (r*cos(theta), r*sin(theta))
</span></blockquote><p>
or equivalently
</p><blockquote class="quote"><span style="font-family:monospace">p := (r, theta) -&gt; (r*cos(theta),r*sin(theta))
</span></blockquote><p>The <span style="font-family:monospace">unapply</span> command will transform an expression into a
function. It takes as arguments an expression and a variable, it will
return the function defined by the expression. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">unapply(x*exp(x),x)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span>&#XA0;&#X2212;&gt;<span style="font-style:italic">x</span>*exp(<span style="font-style:italic">x</span>)
</td></tr>
</table><p>
The <span style="font-family:monospace">unapply</span> command will return the function written in terms
of built in functions; for example, for the function <span style="font-style:italic">f</span> defined
above, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">unapply(f(x),x)
</span></blockquote><p>
you will also get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span>&#XA0;&#X2212;&gt;<span style="font-style:italic">x</span>*exp(<span style="font-style:italic">x</span>)
</td></tr>
</table><p>You can define a function in terms of a function that you previously
defined, but it&#X2019;s probably better to define any new functions in
terms of built-in functions. For example, if you define
</p><blockquote class="quote"><span style="font-family:monospace">f(x) := exp(x)*sin(x)
</span></blockquote><p>
you can define a new function
</p><blockquote class="quote"><span style="font-family:monospace">g(x) := x*f(x)
</span></blockquote><p>
but it might be better to write
</p><blockquote class="quote"><span style="font-family:monospace">g(x) := x*exp(x)*sin(x)
</span></blockquote><p>
Perhaps a better alternative is to use <span style="font-family:monospace">unapply</span>; you can
define <span style="font-family:monospace">g</span> by
<span style="font-family:monospace">g := unapply(x*f(x),x)</span></p><p>In some cases, it will be necessary to use <span style="font-family:monospace">unapply</span> to define
a function. For example (see section <a href="#deriv">4.1</a>, &#X201C;Derivatives&#X201D;),
the <span style="font-family:monospace">diff</span> command will
find the derivative of an expression; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">diff(x*sin(x),x)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
sin(<span style="font-style:italic">x</span>)&#XA0;+&#XA0;<span style="font-style:italic">x</span>&#XA0;*&#XA0;cos(<span style="font-style:italic">x</span>)
</td></tr>
</table><p>
However, you cannot simply define a function
<span style="font-family:monospace">g(x) := diff(x*sin(x),x)</span>
if you tried to do this, then evaluating <span style="font-family:monospace">g(0)</span> for example
would give you <span style="font-family:monospace">diff(0*sin(0),0)</span>, which is not what you want.
Instead, you could define <span style="font-family:monospace">g</span> by
</p><blockquote class="quote"><span style="font-family:monospace">g := unapply(diff(x*sin(x),x)
</span></blockquote><p>Another case where you need to use <span style="font-family:monospace">unapply</span> to define a
function is when you have a function of two variables and you want to
use it to define a function of one variable, where the other variable
is a parameter. For example, consider the polar coordinate function
</p><blockquote class="quote"><span style="font-family:monospace">p(r,theta) := (r*cos(theta), r*sin(theta))
</span></blockquote><p>
If you want to use this to define <span style="font-family:monospace">C(r)</span> as a function of
&#X3B8; for any value of <span style="font-style:italic">r</span>, you cannot simply define it as
</p><blockquote class="quote"><span style="font-family:monospace">C(r) := p(r,theta)
</span></blockquote><p>
Doing this will define <span style="font-family:monospace">C(r)</span> as an expression involving
&#X3B8;, not a function of &#X3B8;. Entering
</p><blockquote class="quote"><span style="font-family:monospace">C(1)(pi/4)
</span></blockquote><p>
would be the same as
</p><blockquote class="quote"><span style="font-family:monospace">(cos(theta),sin(theta))(pi/4)
</span></blockquote><p>
which is not what you want. To define <span style="font-family:monospace">C(r)</span>, you
would have to use <span style="font-family:monospace">unapply</span>:
</p><blockquote class="quote"><span style="font-family:monospace">C(r) := unapply(p(r,theta),theta)
</span></blockquote><p>The necessity of using <span style="font-family:monospace">unapply</span> in these cases is because when
you define a function, the right hand side of the assignment is 
not evaluated. For example, if you try to define the squaring
function by
</p><blockquote class="quote"><span style="font-family:monospace">sq := x^2
</span></blockquote><blockquote class="quote"><span style="font-family:monospace">f(x) := sq
</span></blockquote><p>
it will not work; if you enter <span style="font-family:monospace">f(5)</span>, for example, it will
get the value <span style="font-family:monospace">sq</span>, which will then be replaced by its value.
You will end up getting <span style="font-family:monospace">x^2</span> and not <span style="font-family:monospace">5^2</span>. You
should either define the function <span style="font-family:monospace">f</span> by
</p><blockquote class="quote"><span style="font-family:monospace">f(x) := x^2
</span></blockquote><p>
or perhaps
</p><blockquote class="quote"><span style="font-family:monospace">f := unapply(sq,x)
</span></blockquote><p>Functions (not just expressions) can be added and multiplied. To
define a function which is the sine function times the exponential,
instead of defining <span style="font-family:monospace">f(x)</span> as the expression
<span style="font-family:monospace">sin(x)*exp(x)</span>, you could simply enter
</p><blockquote class="quote"><span style="font-family:monospace">f := sin*exp
</span></blockquote><p>
Functions can also be composed with the <span style="font-family:monospace">@</span> symbol. For example, if
you define functions <span style="font-family:monospace">f</span> and <span style="font-family:monospace">g</span> by
</p><blockquote class="quote"><span style="font-family:monospace">f(x) := x^2 + 1
</span></blockquote><blockquote class="quote"><span style="font-family:monospace">g(x) := sin(x)
</span></blockquote><p>
then 
</p><blockquote class="quote"><span style="font-family:monospace">f @ g
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span>&#XA0;&#X2212;&gt;(sin(<span style="font-style:italic">x</span>))<sup>2</sup>&#XA0;+&#XA0;1
</td></tr>
</table><p>
You can use the <span style="font-family:monospace">@</span> operator to compose a function with itself;
<span style="font-family:monospace">f@f(x)</span> is the same as <span style="font-family:monospace">f(f(x))</span>, but if you want to
compose a function with itself several times, you can use the
<span style="font-family:monospace">@@</span> operator. Entering <span style="font-family:monospace">f @@ </span><span style="font-style:italic">n</span> for a positive
integer <span style="font-style:italic">n</span> will give you the composition of <span style="font-style:italic">f</span> with itself <span style="font-style:italic">n</span>
times; for example, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">sin @@ 3
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span>&#XA0;&#X2212;&gt;&#XA0;sin(sin(sin(<span style="font-style:italic">x</span>)))
</td></tr>
</table>
<!--TOC subsection id="sec19" Lists, sequences and sets-->
<h3 id="sec19" class="subsection">3.5&#XA0;&#XA0;Lists, sequences and sets</h3><!--SEC END --><p>
<a id="lists"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Sequences and lists</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">(&#XA0;&#XA0;)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >sequence delimiters</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">[&#XA0;&#XA0;]</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >list delimiters</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">%{&#XA0;&#XA0;%}</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >set delimiters</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">NULL</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >empty sequence</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">E$(k=n..m)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >create a sequence</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">seq(E,k=n..m)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >create a sequence</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">[E$(k=n..m)]</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >create a list</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">makelist(f,k,n,m,p)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >create a list</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">append</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >append an element to a list</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">op(li)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a list to a sequence</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">nop(se)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a sequence to a list</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">nops(li)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the number of elements</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">size(li)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the number of elements</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">mid(li)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >extract a subsequence</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sum</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the sum of the elements</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">product</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the product of the elements</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cumSum</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the cumulative sums</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">apply(f,li)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >apply a function to the list elements</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">map(li,f)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >apply a function to the list elements</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">map(li,f,matrix)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >apply a function to the elements of a matrix</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">poly2symb</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a polynomial expression to a polynomial
list</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">symb2poly</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a polynomial list to a polynomial
expression</td></tr>
</table>
</div><p>
<a id="hevea_default132"></a>
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<a id="hevea_default134"></a>
<a id="hevea_default135"></a>
<a id="hevea_default136"></a>
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<a id="hevea_default150"></a></p><p><span style="font-family:monospace">Xcas</span> can combine objects in several different ways.
</p><dl class="description"><dt class="dt-description">
<span style="font-weight:bold">sequences</span></dt><dd class="dd-description"><a id="hevea_default151"></a>
A sequence is simply several items between parentheses, separated by
commas. For example, <span style="font-family:monospace">(1,2,x,4)</span> is a sequence.
(The parentheses can be omitted, but it&#X2019;s a good idea to use them.) 
Sequences are flat, meaning an element in a sequence cannot be
another sequence.
The empty sequence is denoted <span style="font-family:monospace">NULL</span>.
</dd><dt class="dt-description"><span style="font-weight:bold">lists</span></dt><dd class="dd-description"><a id="hevea_default152"></a>
A list consists of several items between square brackets,
separated by commas. For example, <span style="font-family:monospace">[1,2,x,4]</span> is a
list. A list can contain other lists as elements. Matrices, which
will be discussed later, are lists of lists. The empty list is
denoted <span style="font-family:monospace">[]</span>.
</dd><dt class="dt-description"><span style="font-weight:bold">sets</span></dt><dd class="dd-description"><a id="hevea_default153"></a>
A set consists of several items between <span style="font-family:monospace">%{</span> and
<span style="font-family:monospace">%}</span>, separated by commas. For example,
<span style="font-family:monospace">%{1,2,3%}</span> is a set. In a set, order doesn&#X2019;t matter and
each item only counts once. The sets <span style="font-family:monospace">%{1,2,3%}</span>
<span style="font-family:monospace">%{3,2,1%}</span> and <span style="font-family:monospace">%{1,2,2,3%}</span> are all the same
set.
</dd><dt class="dt-description"><span style="font-weight:bold">tables</span></dt><dd class="dd-description"><a id="hevea_default154"></a>
Tables are described later.
</dd></dl><p>
A sequence can be turned into a list or a set by putting it between
the appropriate delimiters. For example, if you define a sequence
</p><blockquote class="quote"><span style="font-family:monospace">se := (1,2,4,2)
</span></blockquote><p>
then if you enter
</p><blockquote class="quote"><span style="font-family:monospace">[se]
</span></blockquote><p>
you will get the list
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[1,2,4,2]
</td></tr>
</table><p>
You can turn a set or list into a sequence with the <span style="font-family:monospace">op</span>
command; if you define a set
</p><blockquote class="quote"><span style="font-family:monospace">st := %{1,2,3%}
</span></blockquote><p>
and then enter
</p><blockquote class="quote"><span style="font-family:monospace">op(st)
</span></blockquote><p>
you will get the sequence
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(1,2,3)
</td></tr>
</table><p>
You can find the number of elements in a sequence, list or set with
the <span style="font-family:monospace">size</span> command; with <span style="font-family:monospace">st</span> as above,
</p><blockquote class="quote"><span style="font-family:monospace">size(st)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
3
</td></tr>
</table><p>Sequences can be built using one of the iteration commands
<span style="font-family:monospace">seq</span> or <span style="font-family:monospace">$</span>. The <span style="font-family:monospace">seq</span> command takes an
expression as the first argument, the second argument will be a
variable followed by a range in the form
<span style="font-style:italic">variable</span><span style="font-family:monospace">=</span><span style="font-style:italic">beginning
value</span><span style="font-family:monospace">..</span><span style="font-style:italic">ending value</span>. The resulting sequence will be
the values of the expression with the variable replaced by the
sequence of values. For example,
</p><blockquote class="quote"><span style="font-family:monospace">seq(k^2,k=-2..2)
</span></blockquote><p>
will result in the sequence
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(4,1,0,1,4)
</td></tr>
</table><p>
The <span style="font-family:monospace">$</span> operator is an infix version of <span style="font-family:monospace">seq</span>. If you
enter
</p><blockquote class="quote"><span style="font-family:monospace">k^2$k=-2..2
</span></blockquote><p>
you will get, as above,
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(4,1,0,1,4)
</td></tr>
</table><p>A list can be built by putting a sequence in brackets; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">[k^3,k=1..3]
</span></blockquote><p>
you will get the list
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[1,8,27]
</td></tr>
</table><p>
You can also create a list with the <span style="font-family:monospace">makelist</span> command. It
takes three arguments; a function (not an expression), an initial
value for the variable and an ending value for the variable. If you
enter
</p><blockquote class="quote"><span style="font-family:monospace">makelist(x -&gt; x^2,-2,2)
</span></blockquote><p>
you will get the list
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[4,1,0,1,4]
</td></tr>
</table><p>
There is an optional fourth argument, which will be the step size.</p><p>You can add an element to the end of a list with the <span style="font-family:monospace">append</span>
command. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">append([1,5],3)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[1,5,3]
</td></tr>
</table><p>The elements of sequences and lists are indexed, beginning with the
index 0. You can get an element by following the sequence or
list with the index number in square brackets; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">ls := [A,B,C,D,E,F]
</span></blockquote><p>
then
</p><blockquote class="quote"><span style="font-family:monospace">ls[1]
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">B</span>
</td></tr>
</table><p>
You can get a subsequence<a id="hevea_default155"></a> (or sublist<a id="hevea_default156"></a>) 
by putting an interval (a
beginning value and an ending values separated by two dots) in brackets.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">ls[2..4]
</span></blockquote><p>
you will get 
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[<span style="font-style:italic">C</span>,<span style="font-style:italic">D</span>,<span style="font-style:italic">E</span>]
</td></tr>
</table><p>The <span style="font-family:monospace">mid</span><a id="hevea_default157"></a><a id="hevea_default158"></a> command is another
way to get a subsequence or sublist. 
Given a sequence or list, a beginning index and a length, then
<span style="font-family:monospace">mid</span> will return the subsequence of the sequence beginning at
the given index of the given length. With <span style="font-family:monospace">ls</span> as above, if
you enter
</p><blockquote class="quote"><span style="font-family:monospace">mid(ls,2,3)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[<span style="font-style:italic">C</span>,<span style="font-style:italic">D</span>,<span style="font-style:italic">E</span>]
</td></tr>
</table><p>
If the length is left off, then the subsequence will go to the end of
the given sequence; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">mid(ls,2)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[<span style="font-style:italic">C</span>,<span style="font-style:italic">D</span>,<span style="font-style:italic">E</span>,<span style="font-style:italic">F</span>]
</td></tr>
</table><p>You can change the element in a particular position with the
<span style="font-family:monospace">:=</span> operator; for example, to change the second element in
<span style="font-family:monospace">ls</span>, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">ls[1] := 7
</span></blockquote><p>
The value of <span style="font-family:monospace">ls</span> will then be
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[<span style="font-style:italic">A</span>,7,<span style="font-style:italic">C</span>,<span style="font-style:italic">D</span>,<span style="font-style:italic">E</span>,<span style="font-style:italic">F</span>]
</td></tr>
</table><p>If a variable <span style="font-style:italic">var</span> is not a list or sequence and you assign a
value to <span style="font-style:italic">var</span><span style="font-family:monospace">[</span><span style="font-style:italic">n</span><span style="font-family:monospace">]</span>, then <span style="font-style:italic">var</span> becomes a
table. A table is like a list, but the indices don&#X2019;t have to be
integers. If you define
</p><blockquote class="quote"><span style="font-family:monospace">newls := []
</span></blockquote><p>
and then set
</p><blockquote class="quote"><span style="font-family:monospace">newls[2] := 5
</span></blockquote><p>
then since <span style="font-family:monospace">newls</span> was previous a list, it will now be equal to
the list
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[0,0,5]
</td></tr>
</table><p>
If <span style="font-family:monospace">nols</span> is an undefined variable and you set
</p><blockquote class="quote"><span style="font-family:monospace">nols[2] := 5
</span></blockquote><p>
then <span style="font-family:monospace">nols</span> will be a table,<a id="hevea_default159"></a>
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-family:monospace">table</span>(2=5)
</td></tr>
</table><p>When changing an element of a list (or sequence or table) using
<span style="font-family:monospace">:=</span>, the entire list is copied. This can be inefficient. To
save copy time and modify the list element in place, you can use
<span style="font-family:monospace">=&lt;</span>. If you have
</p><blockquote class="quote"><span style="font-family:monospace">ls := [a,b,c]
</span></blockquote><p>
and then enter
</p><blockquote class="quote"><span style="font-family:monospace">ls[2] =&lt; 3
</span></blockquote><p>
then <span style="font-family:monospace">ls</span> will be equal to
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[<span style="font-style:italic">a</span>,<span style="font-style:italic">b</span>,3]
</td></tr>
</table><p>Polynomials are typically given by expressions, but they can also be
given by a list of the coefficients in decreasing order, delimited
with <span style="font-family:monospace">poly1[</span> and <span style="font-family:monospace">]</span>. The <span style="font-family:monospace">symb2poly</span> will
transform a polynomial written as an expression to the list form of
the polynomial. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">symb2poly(2*x^3 - 4*x + 1)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
poly1[2,0,&#X2212;4,1]
</td></tr>
</table><p>
The <span style="font-family:monospace">poly2symb</span> will transform in the other direction; if you
enter
</p><blockquote class="quote"><span style="font-family:monospace">poly2symb(poly1[2,0,-4,1])
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2*<span style="font-style:italic">x</span><sup>3</sup>&#XA0;&#X2212;&#XA0;4*<span style="font-style:italic">x</span>&#XA0;+&#XA0;1
</td></tr>
</table><p>
There is also a way to represent a multivariable polynomial with
lists; see the manual for more information.</p>
<!--TOC subsection id="sec20" Characters and strings-->
<h3 id="sec20" class="subsection">3.6&#XA0;&#XA0;Characters and strings</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">String commands</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">asc</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a string to a list of ASCII codes </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">char</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a list of ASCII codes to a string</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">size</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the number of characters </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">concat</span> or <span style="font-family:monospace">+</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >concatenation </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">mid</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >substring</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">head</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >first character </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tail</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the string without the first character</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">string</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a number or expression to a string </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">expr</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >convert a string to a number or expression </td></tr>
</table>
</div><p>
<a id="hevea_default160"></a>
<a id="hevea_default161"></a>
<a id="hevea_default162"></a>
<a id="hevea_default163"></a>
<a id="hevea_default164"></a>
<a id="hevea_default165"></a>
<a id="hevea_default166"></a>
<a id="hevea_default167"></a>
<a id="hevea_default168"></a></p><p>A string is simply text enclosed within quotation marks.
You can find out how many characters are in a string with the
<span style="font-family:monospace">size</span> command; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">size("this string")
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
11
</td></tr>
</table><p>A character is simply a string with length 1.<a id="hevea_default169"></a> 
The <span style="font-family:monospace">char</span>
command will take an ASCII code (or a list of ASCII codes) and return
the character or string determined by the codes. For example, the
letter &#X201C;a&#X201D; has ASCII code 65, so
</p><blockquote class="quote"><span style="font-family:monospace">char(65)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">A</span>
</td></tr>
</table><p>
The <span style="font-family:monospace">asc</span> command will turn a string into the list of ASCII
codes; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">asc("A")
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[65]
</td></tr>
</table><p>The characters in a string are indexed starting with <span style="font-family:monospace">0</span>.
To get the first character, for example, you can enter a string, or
the name of a string, followed by <span style="font-family:monospace">[0]</span>. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">str := "abcde"
</span></blockquote><blockquote class="quote"><span style="font-family:monospace">str[0]
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">a</span>
</td></tr>
</table><p>
You can choose a substring from a string by putting the beginning and
ending indices in the brackets, separated by two periods <span style="font-family:monospace">..</span>.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">str[1..3]
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">bcd</span>
</td></tr>
</table><p>An alternate way of getting the first character from a string is with
the <span style="font-family:monospace">head</span> command. With <span style="font-family:monospace">str</span> as above, 
</p><blockquote class="quote"><span style="font-family:monospace">head(str)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">a</span>
</td></tr>
</table><p>
The <span style="font-family:monospace">tail</span> command will produce the remaining characters;
</p><blockquote class="quote"><span style="font-family:monospace">tail(str)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">bcde</span>
</td></tr>
</table><p>Strings can be combined with the <span style="font-family:monospace">concat</span> command or the infix
<span style="font-family:monospace">+</span> operator. Both 
</p><blockquote class="quote"><span style="font-family:monospace">concat("abc","def")
</span></blockquote><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">"abc" + "def"
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">abcdef</span>
</td></tr>
</table><p>If a string represents a number, then the <span style="font-family:monospace">expr</span> command will
convert the string to the number. For example,
</p><blockquote class="quote"><span style="font-family:monospace">expr("123")
</span></blockquote><p>
will return the number 123. More generally, <span style="font-family:monospace">expr</span> will
convert a string representing an expression or command into the
corresponding expression or command. The <span style="font-family:monospace">string</span> command
works in the opposite direction; it will take an expression and
convert it to a string.</p>
<!--TOC subsection id="sec21" Calculation time and memory space-->
<h3 id="sec21" class="subsection">3.7&#XA0;&#XA0;Calculation time and memory space</h3><!--SEC END --><p>One major issue with symbolic calculations is the complexity of the
intermediate calculations. This complexity takes the form of the
amount of time required for the calculations and the amount of
computer memory needed. The algorithms used by <span style="font-family:monospace">Xcas</span> are
efficient, but not necessarily optimal. The <span style="font-family:monospace">time</span> command
will tell you how long a calculation takes. For very quick
calculations, <span style="font-family:monospace">Xcas</span> will execute it several times and return
the average for a more accurate result. The amount of memory used by
<span style="font-family:monospace">Xcas</span> is shown in the status line of the Unix version of
<span style="font-family:monospace">Xcas</span>. </p><p>If a command that you are timing takes more than a few seconds, you
could have made an input error and you may have to interrupt the
command (with the red <span style="font-family:monospace">STOP</span> button on the status line, for
example). It is a good idea to make a backup of your session beforehand.</p>
<!--TOC section id="sec22" Analysis with <span style="font-family:monospace">Xcas</span>-->
<h2 id="sec22" class="section">4&#XA0;&#XA0;Analysis with <span style="font-family:monospace">Xcas</span></h2><!--SEC END -->
<!--TOC subsection id="sec23" Derivatives-->
<h3 id="sec23" class="subsection">4.1&#XA0;&#XA0;Derivatives</h3><!--SEC END --><p>
<a id="deriv"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Derivatives</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">diff(ex,t)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the derivative of an expression with respect to t</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">function_diff(f)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the derivative of a function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">diff(ex,x$n,y$m)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >partial derivatives</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">grad</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >gradient</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">divergence</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >divergence</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">curl</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >curl</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">laplacian</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >laplacian</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">hessian</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >hessian matrix</td></tr>
</table>
</div><p>
<a id="hevea_default170"></a>
<a id="hevea_default171"></a>
<a id="hevea_default172"></a>
<a id="hevea_default173"></a>
<a id="hevea_default174"></a>
<a id="hevea_default175"></a>
<a id="hevea_default176"></a>
<a id="hevea_default177"></a>
<a id="hevea_default178"></a>
<a id="hevea_default179"></a></p><p>The <span style="font-family:monospace">diff</span> function will find the derivative of an expression
and returns the derivative as an expression. If you have a function
<span style="font-style:italic">f</span>, you can find the derivative by entering
</p><blockquote class="quote"><span style="font-family:monospace">diff(f(x),x)
</span></blockquote><p>
Note that the result will itself be an expression; do not define the
deritivave function by <span style="font-family:monospace">fprime(x) := diff(f(x),x)</span>. If you
want to define the derivative as a function, you can use
<span style="font-family:monospace">unapply</span>:<a id="hevea_default180"></a>
</p><blockquote class="quote"><span style="font-family:monospace">fprime := unapply(diff(f(x),x),x)
</span></blockquote><p>
Alternatively, you can use <span style="font-family:monospace">function_diff</span>, which takes a
function (not an expression) as input and returns the derivative
function;
</p><blockquote class="quote"><span style="font-family:monospace">fprime := function_diff(f)
</span></blockquote><p>The <span style="font-family:monospace">diff</span> function can take a sequence of variables as the
second argument, and so can calculate successive partial derivatives.
Given
</p><blockquote class="quote"><span style="font-family:monospace">E := sin(x*y)
</span></blockquote><p>
then
</p><blockquote class="quote"><span style="font-family:monospace">diff(E,x)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">y</span>*cos(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>)
</td></tr>
</table><blockquote class="quote"><span style="font-family:monospace">diff(E,y)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span>*cos(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>)
</td></tr>
</table><blockquote class="quote"><span style="font-family:monospace">diff(E,x,y)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X2212;<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>*sin(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>)&#XA0;+&#XA0;cos(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>)
</td></tr>
</table><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">diff(E,x $ 2)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X2212;<span style="font-style:italic">y</span><sup>2</sup>*sin(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>)
</td></tr>
</table><p>If the second argument to <span style="font-family:monospace">diff</span> is a list, then a list of
derivatives is returned. For example, to find the gradient of
<span style="font-family:monospace">E</span>, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">diff(E,[x,y])
</span></blockquote><p>
and get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[<span style="font-style:italic">y</span>*cos(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>),<span style="font-style:italic">x</span>*cos(<span style="font-style:italic">x</span>*<span style="font-style:italic">y</span>)]
</td></tr>
</table><p>
There is also a special <span style="font-family:monospace">grad</span> command for this, as well as
commands for other types of special derivatives.</p>
<!--TOC subsection id="sec24" Limits and series-->
<h3 id="sec24" class="subsection">4.2&#XA0;&#XA0;Limits and series</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Limits and series</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">limit</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the limit of an expression</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">taylor</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Taylor series</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">series</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Taylor series</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">order_size</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >used in the remainder term of a series expansion</td></tr>
</table>
</div><p>
<a id="hevea_default181"></a>
<a id="hevea_default182"></a>
<a id="hevea_default183"></a>
<a id="hevea_default184"></a>
<a id="hevea_default185"></a></p><p>The <span style="font-family:monospace">limit</span> function will take an expression, a variable and a
point and return the limit at the point. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">limit(sin(x)/x,x,0)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1
</td></tr>
</table><p>
<span style="font-family:monospace">Xcas</span> can also find limits at plus and minus infinity;
</p><blockquote class="quote"><span style="font-family:monospace">limit(sin(x)/x,x,+infinity)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
0
</td></tr>
</table><p>
as well as limits of infinity;
</p><blockquote class="quote"><span style="font-family:monospace">limit(1/x,x,0)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X221E;
</td></tr>
</table><p>
which recall is unsigned infinity.
An optional fourth argument can be used to find one-sided limits; if
the fourth argument is <span style="font-family:monospace">1</span> it will be a right-handed limit and
if the argument is <span style="font-family:monospace">-1</span> it will be a left-handed limit.
Entering
</p><blockquote class="quote"><span style="font-family:monospace">limit(1/x,x,0,-1)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X2212;&#X221E;
</td></tr>
</table><p>Given an expression and a variable, the <span style="font-family:monospace">taylor</span> function will
find the Taylor series of the expression. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">taylor(sin(x)/x,x)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">6</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>4</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">120</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;<span style="font-style:italic">x</span><sup>6</sup>*<span style="font-family:monospace">order_size</span>(<span style="font-style:italic">x</span>)
</td></tr>
</table><p>
The <span style="font-family:monospace">series</span> function works the same as the <span style="font-family:monospace">taylor</span>
function.</p><p>By default, <span style="font-family:monospace">taylor</span> will find the terms up to the fifth
degree. The <span style="font-family:monospace">order_size(x)</span> represents a factor for which 
for all <span style="font-style:italic">a</span>&gt;0, the term <span style="font-style:italic">x</span><sup><span style="font-style:italic">a</span></sup><span style="font-family:monospace">order_size</span>(<span style="font-style:italic">x</span>) will approach 0
as <span style="font-style:italic">x</span> approaches 0. </p><p>The series returned by <span style="font-family:monospace">taylor</span> will also
be centered about 0 by default; if you want to center it around the number
<span style="font-family:monospace">a</span>, you can replace <span style="font-family:monospace">x</span> by <span style="font-family:monospace">x=a</span>;
</p><blockquote class="quote"><span style="font-family:monospace">taylor(exp(x),x=1)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
exp(1)+exp(1)*(<span style="font-style:italic">x</span>&#X2212;1)+exp(1)*(<span style="font-style:italic">x</span>&#X2212;1)<sup>2</sup>/2+&#XA0;exp(1)*(<span style="font-style:italic">x</span>&#X2212;1)<sup>3</sup>/6+
</td></tr>
</table><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
exp(1)*(<span style="font-style:italic">x</span>&#X2212;1)<sup>4</sup>/24+exp(1)*(<span style="font-style:italic">x</span>&#X2212;1)<sup>5</sup>/120+(<span style="font-style:italic">x</span>&#X2212;1)<sup>6</sup>*<span style="font-family:monospace">order_size</span>(<span style="font-style:italic">x</span>&#X2212;1)
</td></tr>
</table><p>
You can also give the center of the series with a third argument.
To find the terms to a different you can add an extra argument
giving the order;
</p><blockquote class="quote"><span style="font-family:monospace">taylor(sin(x)/x,x=0,3)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">6</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;<span style="font-style:italic">x</span><sup>4</sup>*<span style="font-family:monospace">order_size</span>(<span style="font-style:italic">x</span>)
</td></tr>
</table><p>
Note that in this case you must explicitly give the center of the
series, even if it is 0.</p><p>To find the Taylor polynomial, you can add an extra argument of
<span style="font-family:monospace">polynom</span>;
</p><blockquote class="quote"><span style="font-family:monospace">taylor(sin(x)/x,x,polynom)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">6</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>4</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">120</td></tr>
</table></td></tr>
</table>
<!--TOC subsection id="sec25" Antiderivatives and integrals-->
<h3 id="sec25" class="subsection">4.3&#XA0;&#XA0;Antiderivatives and integrals</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Integrals</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">int</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >antiderivatives and exact integrals</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">romberg</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximation of integrals</td></tr>
</table>
</div><p>
<a id="hevea_default186"></a>
<a id="hevea_default187"></a>
<a id="hevea_default188"></a>
<a id="hevea_default189"></a></p><p>The <span style="font-family:monospace">int</span> function will find an antiderivative of an
expression. By default, it will assume that the variable is <span style="font-style:italic">x</span>, to
use another variable you can give it as an argument.
</p><blockquote class="quote"><span style="font-family:monospace">int(x*sin(x))
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
sin(<span style="font-style:italic">x</span>)&#XA0;&#X2212;&#XA0;<span style="font-style:italic">x</span>*cos(<span style="font-style:italic">x</span>)
</td></tr>
</table><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">int(t*sin(t),t)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
sin(<span style="font-style:italic">t</span>)&#XA0;&#X2212;&#XA0;<span style="font-style:italic">t</span>*cos(<span style="font-style:italic">t</span>)
</td></tr>
</table><p>To compute a definite integral, you can give the limits of integration
as arguments after the variable; to integrate <span style="font-style:italic">x</span>*sin(<span style="font-style:italic">x</span>) from <span style="font-style:italic">x</span>=0 to
<span style="font-style:italic">x</span>=&#X3C0;, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">int(x*sin(x),x,0,pi)
</span></blockquote><p>
and get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X3C0;
</td></tr>
</table><p>
The limits of integration are allowed to be expressions; this can be
useful when computing a multiple integral over a non-rectangular
region. For example, you can integrate <span style="font-style:italic">x</span> <span style="font-style:italic">y</span> over the triangle 0 &#X2264;
<span style="font-style:italic">x</span> &#X2264; 1, 0 &#X2264; <span style="font-style:italic">y</span> &#X2264; <span style="font-style:italic">x</span> with
</p><blockquote class="quote"><span style="font-family:monospace">int(int(x*y,y,0,x),x,0,1)
</span></blockquote><p>
resulting in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">8</td></tr>
</table></td></tr>
</table><p>The <span style="font-family:monospace">romberg</span> function will approximate the value of a definite
integral, for cases when the exact value can&#X2019;t be computed or you
don&#X2019;t want to compute it. For example,
</p><blockquote class="quote"><span style="font-family:monospace">romberg(exp(-x^2),x,0,10)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
0.886226925452
</td></tr>
</table>
<!--TOC subsection id="sec26" Solving equations-->
<h3 id="sec26" class="subsection">4.4&#XA0;&#XA0;Solving equations</h3><!--SEC END --><p>
<a id="solve"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Solving equations</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">solve(eq,x)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >exact solutions of an equation</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">solve([eq1,eq2],[x,y])</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >exact solutions of a system of equations</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">fsolve(eq,x)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximate solution of an equation</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">fsolve([eq1,eq2],[x,y])</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximate solution of a system of equations</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">linsolve</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >solve a linear system</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">proot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximate roots of a polynomial</td></tr>
</table>
</div><p>
<a id="hevea_default190"></a>
<a id="hevea_default191"></a>
<a id="hevea_default192"></a>
<a id="hevea_default193"></a></p><p>Solving equations is important, but it is often impossible to find
exact solutions. <span style="font-family:monospace">Xcas</span> has the ability to find exact
solutions in some cases and to approximate solutions.</p><p>The <span style="font-family:monospace">solve</span> function will attempt to find the exact solution of
an equation that you give it. If you enter an expression that isn&#X2019;t
an equation, it will try to solve for the expression equal to zero.
By default, the variable will be <span style="font-style:italic">x</span>, but you can give a different
variable as a second argument. If you enter 
</p><blockquote class="quote"><span style="font-family:monospace">solve(x^3 -2*x^2 + 1=0, x)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell">&#X23A1;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A3;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X2212;(</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td></tr>
</table></td><td class="dcell">)+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">,1,</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td></tr>
</table></td><td class="dcell">+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">&#X23A4;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A6;</td></tr>
</table><p>
By default, <span style="font-family:monospace">solve</span> will only try to find real solutions; if
you enter
</p><blockquote class="quote"><span style="font-family:monospace">solve(x^3+1=0,x)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[&#X2212;1]
</td></tr>
</table><p>
You can configure <span style="font-family:monospace">Xcas</span> to find complex solutions (see
section <a href="#config">2.3</a>, &#X201C;Configuration&#X201D;). If you do that,
then entering
</p><blockquote class="quote"><span style="font-family:monospace">solve(x^3+1=0,x)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell">&#X23A1;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A3;</td><td class="dcell">&#X2212;1,</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X2212;</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >3</td></tr>
</table></td><td class="dcell">*<span style="font-style:italic">i</span>+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">,</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >3</td></tr>
</table></td><td class="dcell">*<span style="font-style:italic">i</span>+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">&#X23A4;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A6;</td></tr>
</table><p>For linear and quadratic functions, <span style="font-family:monospace">solve</span> will always return
the exact solution. For higher degree polynomials, <span style="font-family:monospace">solve</span>
will try some approaches, but may return intermediate results or
approximate solutions. (It doesn&#X2019;t use the Cardan and Ferrari
formulas for polynomials of degrees 3 and 4, since the solutions would
then not be easily managable.)</p><p>For trigonometric equations, the primary solutions are returned. For
example,
</p><blockquote class="quote"><span style="font-family:monospace">solve(cos(x) + sin(x) = 0, x)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell">&#X23A1;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A3;</td><td class="dcell">&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">&#X3C0;</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">4</td></tr>
</table></td><td class="dcell">,</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">3*&#X3C0;</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">4</td></tr>
</table></td><td class="dcell">&#X23A4;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A6;</td></tr>
</table><p>
You can configure <span style="font-family:monospace">Xcas</span> to find all solutions (see section
<a href="#config">2.3</a>, &#X201C;Configuration&#X201D;). If you do that, then
</p><blockquote class="quote"><span style="font-family:monospace">solve(cos(x) + sin(x) = 0, x)
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell">&#X23A1;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A3;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">4*<span style="font-style:italic">n</span><sub>0</sub>*&#X3C0;&#X2212;&#X3C0;</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">4</td></tr>
</table></td><td class="dcell">&#X23A4;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A6;</td></tr>
</table><p>
where <span style="font-style:italic">n</span><sub>0</sub> represents an arbitrary integer.</p><p>The <span style="font-family:monospace">solve</span> function can also handle systems of equations.
For this, use a list of equations for the first argument and a list of
variables for the second. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">solve([x^2 + y - 2, x + y^2 - 2],[x,y])
</span></blockquote><p>
you will get all four solutions as a matrix; each row represents one
solution.
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell">&#X23A1;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A2;<br>
&#X23A3;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1,</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;2,</td><td style="text-align:center;white-space:nowrap" >&#X2212;2</td></tr>
<tr><td style="text-align:center;white-space:nowrap" ><table class="display"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td></tr>
</table></td><td class="dcell">+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">,</td></tr>
</table></td><td style="text-align:center;white-space:nowrap" ><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X2212;</td><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td></tr>
</table></td><td class="dcell">+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">2</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">&nbsp;</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;2</td></tr>
</table></td></tr>
<tr><td style="text-align:center;white-space:nowrap" ><table class="display"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X2212;(</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td></tr>
</table></td><td class="dcell">)+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">,</td></tr>
</table></td><td style="text-align:center;white-space:nowrap" ><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X2212;</td><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X2212;(</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td></tr>
</table></td><td class="dcell">)+1</td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">2</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">&nbsp;</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;2
</td></tr>
</table></td></tr>
</table></td><td class="dcell">&#X23A4;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A5;<br>
&#X23A6;</td></tr>
</table><p>To approximate a solution to an equation or system of equations,
<span style="font-family:monospace">Xcas</span> provides the <span style="font-family:monospace">fsolve</span> command. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">fsolve(x^3 -3*x + 1,x)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[&#X2212;1.87938524157,0.347296355334,1.53208888624]
</td></tr>
</table><p>
Algorithms for approximating solutions of equations 
typically involve starting with a given point and finding a sequence
which converges to a solution. The <span style="font-family:monospace">fsolve</span> command can take
a starting point, if you enter
</p><blockquote class="quote"><span style="font-family:monospace">fsolve(x^3 -3*x + 1,x,1)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
0.347296355334
</td></tr>
</table>
<!--TOC subsection id="sec27" Differential equations-->
<h3 id="sec27" class="subsection">4.5&#XA0;&#XA0;Differential equations</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Commands for differential equations</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">desolve</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >exact solution</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">odesolve</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximate solution</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotode</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >graph of solution</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotfield</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >vector field</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">interactive_plotode</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >clickable interface</td></tr>
</table>
</div><p>
<a id="hevea_default194"></a>
<a id="hevea_default195"></a>
<a id="hevea_default196"></a>
<a id="hevea_default197"></a>
<a id="hevea_default198"></a></p><p>The <span style="font-family:monospace">desolve</span> command is used to try to find exact solutions of
differential equations. The first argument is the differential
equation itself, the second argument is the function. The derivative
of an unknown function <span style="font-style:italic">y</span> is denoted <span style="font-family:monospace">diff(y)</span>, which can be
abbreviated <span style="font-family:monospace">y&#X2019;</span>. The second derivative will be
<span style="font-family:monospace">diff(diff(y))</span> or <span style="font-family:monospace">y&#X2019;&#X2019;</span>, etc.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">desolve(x^2*y&#X2019; = y,y)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">c</span><sub>0</sub>&#XA0;*&#XA0;exp(&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span></td></tr>
</table></td><td class="dcell">)
</td></tr>
</table><p>
where <span style="font-style:italic">c</span><sub>0</sub> is an arbitrary constant. By default the variable is <span style="font-style:italic">x</span>,
if you want to use a different variable, put it in the function in the
second argument;
</p><blockquote class="quote"><span style="font-family:monospace">desolve(t^2*y&#X2019; = y,y(t))
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">c</span><sub>0</sub>&#XA0;*&#XA0;exp(&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">t</span></td></tr>
</table></td><td class="dcell">)
</td></tr>
</table><p>If you want to solve a differential equation with initial conditions,
the first argument should be a list with the differential equation and
the conditions. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">desolve([y&#X2019;&#X2019; + 2*y&#X2019; + y = 0, y(0) = 1, y&#X2019;(0) = 2],y)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
exp(&#X2212;<span style="font-style:italic">x</span>)*(3*<span style="font-style:italic">x</span>+1)
</td></tr>
</table><p>To solve a differential equation numerically, you can use the
<span style="font-family:monospace">odesolve</span> command. This will allow you to solve the equation
<span style="font-style:italic">y</span>&#X2032;=<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>) where the graph passes through a point (<span style="font-style:italic">x</span><sub>0</sub>,<span style="font-style:italic">y</span><sub>0</sub>). The
command
</p><blockquote class="quote"><span style="font-family:monospace">odesolve(f(x,y),[x,y],[x_0,y_0],a)
</span></blockquote><p>
will find <span style="font-style:italic">y</span>(<span style="font-style:italic">a</span>) in this case. For example, to calculate <span style="font-style:italic">y</span>(2) where
<span style="font-style:italic">y</span>(<span style="font-style:italic">x</span>) is the solution of <span style="font-style:italic">y</span>&#X2032;(<span style="font-style:italic">x</span>) =sin(<span style="font-style:italic">xy</span>) with <span style="font-style:italic">y</span>(0)=1, you can
enter
</p><blockquote class="quote"><span style="font-family:monospace">odesolve(sin(x*y),[x,y],[0,1],2)
</span></blockquote><p>
The result will be 
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[1.82241255674]
</td></tr>
</table><p>
The <span style="font-family:monospace">plotode</span> command will plot the graph of the solution; if
you enter
</p><blockquote class="quote"><span style="font-family:monospace">plotode(sin(x*y),[x,y],[0,1])
</span></blockquote><p>
you will get
</p><div class="center">
<img src="tutoriel006.png">
</div><p>
The <span style="font-family:monospace">plotfield</span> command will plot the entire vector field;
</p><blockquote class="quote"><span style="font-family:monospace">plotfield(sin(x*y),[x,y])
</span></blockquote><p>
will result in
</p><div class="center">
<img src="tutoriel007.png">
</div><p>
If you use the <span style="font-family:monospace">interactive_odeplot</span> command, you will get the
vector field and you will be able to click on a point to find the
graph of the solution passing through the point.</p>
<!--TOC section id="sec28" Algebra with <span style="font-family:monospace">Xcas</span>-->
<h2 id="sec28" class="section">5&#XA0;&#XA0;Algebra with <span style="font-family:monospace">Xcas</span></h2><!--SEC END -->
<!--TOC subsection id="sec29" Integer arithmetic-->
<h3 id="sec29" class="subsection">5.1&#XA0;&#XA0;Integer arithmetic</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Integers</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">irem</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >remainder</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">iquo</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >quotient</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">iquorem</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >quotient and remainder</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ifactor</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >prime factorization</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ifactors</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >list of prime factors</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">idivis</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >list of divisors</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">gcd</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >greatest common divisor</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">lcm</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >least common multiple</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">iegcd</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Bezout&#X2019;s identity</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">isprime</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >primality test</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">nextprime</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >next prime number</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">previousprime</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >previous prime number</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">a%p</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-style:italic">a</span> modulo <span style="font-style:italic">p</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">powmod(a,n,p)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-style:italic">a</span><sup><span style="font-style:italic">n</span></sup> modulo <span style="font-style:italic">p</span></td></tr>
</table>
</div><p>
<a id="hevea_default199"></a>
<a id="hevea_default200"></a>
<a id="hevea_default201"></a>
<a id="hevea_default202"></a>
<a id="hevea_default203"></a>
<a id="hevea_default204"></a>
<a id="hevea_default205"></a>
<a id="hevea_default206"></a>
<a id="hevea_default207"></a>
<a id="hevea_default208"></a>
<a id="hevea_default209"></a>
<a id="hevea_default210"></a>
<a id="hevea_default211"></a>
<a id="hevea_default212"></a>
<a id="hevea_default213"></a>
<a id="hevea_default214"></a>
<a id="hevea_default215"></a>
<a id="hevea_default216"></a></p><p><span style="font-family:monospace">Xcas</span> has the usual number theoretic functions. The
<span style="font-family:monospace">iquo</span> command will find the integer quotient of two integers
and <span style="font-family:monospace">irem</span> will find the remainder. The <span style="font-family:monospace">iquorem</span>
command will return a list of both the quotient and remainder; if you
enter
</p><blockquote class="quote"><span style="font-family:monospace">iquorem(30,7)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[4,2]
</td></tr>
</table><p>
since 30 divided by 7 is 4 with a remainder of 2.</p><p>The <span style="font-family:monospace">gcd</span> and <span style="font-family:monospace">lcm</span> commands will find the greatest
common divisor and least common multiple of two integers. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">gcd(72,120)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
24
</td></tr>
</table><p>
The greatest common divisor <span style="font-style:italic">d</span> of two integers <span style="font-style:italic">a</span> and <span style="font-style:italic">b</span> can always be
written in the form <span style="font-style:italic">a</span>*<span style="font-style:italic">u</span> + <span style="font-style:italic">b</span>*<span style="font-style:italic">v</span> = <span style="font-style:italic">d</span> for integers <span style="font-style:italic">u</span> and <span style="font-style:italic">v</span>. (This
is known as B&#XE9;zout&#X2019;s Identity.) The <span style="font-family:monospace">iegcd</span> will
return the coefficients <span style="font-style:italic">u</span> and <span style="font-style:italic">v</span> as well as the greatest commond
divisor. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">iegcd(72,120)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[2,&#X2212;1,24]
</td></tr>
</table><p>
since
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">72&#XB7;&#XA0;2&#XA0;+&#XA0;120&#XB7;(&#X2212;1)&#XA0;=&#XA0;24</td></tr>
</table><p>The <span style="font-family:monospace">ifactor</span> command will give the prime factorization of an
integer; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">ifactor(250)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2*5<sup>3</sup>
</td></tr>
</table><p>
You can use <span style="font-family:monospace">ifactors</span> to get a list of the prime factors of an
integer, where in the list each factor is followed by its multiplicity.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">ifactors(250)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[2,1,5,3]
</td></tr>
</table><p>
since 250 has a prime factor of 2 (it has 1 factor of 2) and a
prime factor of 5 (it has 3 factors of 5).
The <span style="font-family:monospace">idivis</span> command will return a complete list of factors;
</p><blockquote class="quote"><span style="font-family:monospace">idivis(250)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[1,2,5,10,25,50,125,250]
</td></tr>
</table><p>The subject of primes is a difficult one, and you should see the
manual for a discussion of how <span style="font-family:monospace">Xcas</span> checks for primes. But
the command <span style="font-family:monospace">isprime</span> will return <span style="font-family:monospace">true</span> or <span style="font-family:monospace">false</span>
depending on whether or not you enter a prime. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">isprime(37)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">true</span>
</td></tr>
</table><p>
since 37 is a prime number. The commands <span style="font-family:monospace">nextprime</span> and
<span style="font-family:monospace">previousprime</span> will find the first prime after (or before) the
number that you give it; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">nextprime(37)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
41
</td></tr>
</table><p>
since the first prime after 37 is 41.</p><p>Integers modulo <span style="font-style:italic">p</span> are defined by putting <span style="font-family:monospace">% p</span> after them.
Once an integer modulo <span style="font-style:italic">p</span> is defined, then any calculations done with
it are done in &#X2124;/<span style="font-style:italic">p</span>&#X2124;. For example, if you define
</p><blockquote class="quote"><span style="font-family:monospace">a := 3 % 5
</span></blockquote><p>
then
</p><blockquote class="quote"><span style="font-family:monospace">a*2
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
1&#XA0;%&#XA0;5
</td></tr>
</table><p>
(since 6 mod 5 is reduced to 1 mod 5); 
</p><blockquote class="quote"><span style="font-family:monospace">1/a
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
2&#XA0;%&#XA0;5
</td></tr>
</table><p>
etc. The <span style="font-family:monospace">powermod</span> or <span style="font-family:monospace">powmod</span> functions can be used
to efficiently calculate powers modulo a number.</p>
<!--TOC subsection id="sec30" Polynomials and rational functions-->
<h3 id="sec30" class="subsection">5.2&#XA0;&#XA0;Polynomials and rational functions</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Polynomials</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">normal</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >normal form (expanded and reduced)</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">expand</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >expanded form</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ptayl</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Taylor form</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">peval</span> or <span style="font-family:monospace">horner</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >evaluation using Horner&#X2019;s method</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">canonical_form</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >canonical form for a trinomial</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">coeff</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >list of coefficients</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">poly2symb</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >transform an algebraic polynomial to list form</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">symb2poly</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >transform the list form of a polynomial to
algebraic form</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">pcoeff</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >return the polynomial (list form) given a list of zeroes</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">degree</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >degree</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">lcoeff</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the coefficient of the leading term</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">valuation</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the lowest degree of the terms</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tcoeff</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the coefficient of the term with the lowest degree</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">factor</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >prime factorization</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">factors</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >list of prime factors</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">divis</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >list of divisors</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">froot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >roots with multiplicities</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">proot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >approximate values of the roots</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sturmab</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the number of roots in an interval</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">getNum</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the numerator of a rational function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">getDenom</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >the denominator of a rational function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">propfrac</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >writes a rational expression as a whole part and a
proper rational part</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">partfrac</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >partial fraction decomposition</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">quo</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >quotient</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">rem</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >remainder</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">gcd</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >greatest common divisor</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">lcm</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >least common multiple</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">egcd</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Bezout&#X2019;s identity
</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">divpc</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Taylor polynomial for a rational expression</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">randpoly</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >random polynomial</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cyclotomic</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >cyclotomic polynomial</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">lagrange</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Lagrange polynomials</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">hermite</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Hermite polynomials</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">laguerre</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Laguerre polynomials</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tchebyshev1</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Tchebyshev polynomials</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tchebyshev2</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Tchebyshev polynomials</td></tr>
</table>
</div><p>
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<a id="hevea_default262"></a></p><p>Various polynomial operations are available in the <span style="font-family:monospace">Polynomials</span>
submenu of the <span style="font-family:monospace">Cmds</span> menu.</p><p>The <span style="font-family:monospace">expand</span> and <span style="font-family:monospace">normal</span> operators will distribute
multiplication across addition, and so expand a polynomial completely
out. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">expand((x+1)*(x+2)^2)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<span style="font-style:italic">x</span><sup>3</sup>+5*<span style="font-style:italic">x</span><sup>2</sup>+8*<span style="font-style:italic">x</span>+4
</td></tr>
</table><p>
Additionally, <span style="font-family:monospace">normal</span> will reduce a rational expression to
lowest terms; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">normal((x-1)^2/(x^2-1))
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>&#X2212;1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>+1</td></tr>
</table></td></tr>
</table><p>The <span style="font-family:monospace">factor</span> operator will factor a polynomial. 
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">factor(x^3+6*x^2+3*x-10
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(<span style="font-style:italic">x</span>&#X2212;1)*(<span style="font-style:italic">x</span>+2)*(<span style="font-style:italic">x</span>+5)
</td></tr>
</table><p>
The result often depends on the number field being used. For example, over the
rational numbers the polynomial <span style="font-style:italic">x</span><sup>4</sup> &#X2212; 1 factors as 
(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)(<span style="font-style:italic">x</span><sup>2</sup> + 1), while over the complex numbers it factors as 
(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)(<span style="font-style:italic">x</span>&#X2212;<span style="font-style:italic">i</span>)(<span style="font-style:italic">x</span>+<span style="font-style:italic">i</span>). If the coefficients of a polynomial are exact
fractions, then the factoring will be over the rationals. To factor
over the complex numbers, you can configure <span style="font-family:monospace">Xcas</span> to do
complex factorization (see section <a href="#config">2.3</a>, &#X201C;Configuration&#X201D;)
or use the <span style="font-family:monospace">cfactor</span> command.
If the coefficients are in &#X2124;/<span style="font-style:italic">p</span>&#X2124; then the polynomial will be
factored over &#X2124;/<span style="font-style:italic">p</span>&#X2124;.</p>
<!--TOC subsection id="sec31" Trigonometry-->
<h3 id="sec31" class="subsection">5.3&#XA0;&#XA0;Trigonometry</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Trigonom&#XE9;trie</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tlin</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >linearize</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tcollect</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >linearize and regroup</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">texpand</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >expand</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">trig2exp</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >trigonometric to exponential</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">exp2trig</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >exponential to trigonometric</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">hyp2exp</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >hyperbolic to exponential</td></tr>
</table>
</div><p>
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<a id="hevea_default268"></a></p><p><span style="font-family:monospace">Xcas</span> has the usual trigonometic functions, both circular and
hyperbolic, as well as their inverses. It also has commands for
manipulating trigonometric expressions; these are in the
<span style="font-family:monospace">Trigo</span> submenus of the <span style="font-family:monospace">Expression</span> menu.</p><p>One example is the <span style="font-family:monospace">tlin</span> command will write products and powers of sines and
cosines as linear combinations of sin(<span style="font-style:italic">n</span> <span style="font-style:italic">x</span>)s and cos(<span style="font-style:italic">n</span> <span style="font-style:italic">x</span>)s. If
you enter
</p><blockquote class="quote"><span style="font-family:monospace">tlin(2*sin(x)^2*cos(3*x))
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">cos(<span style="font-style:italic">x</span>)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">&#XA0;+&#XA0;cos(3*<span style="font-style:italic">x</span>)&#XA0;&#X2212;&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">cos(5*<span style="font-style:italic">x</span>)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td></tr>
</table><p>The <span style="font-family:monospace">texpand</span> command will take expressions involving 
sin(<span style="font-style:italic">n</span> <span style="font-style:italic">x</span>) and cos(<span style="font-style:italic">n</span> <span style="font-style:italic">x</span>) and write them in terms of powers if
sin(<span style="font-style:italic">x</span>) and cos(<span style="font-style:italic">x</span>). If you enter
</p><blockquote class="quote"><span style="font-family:monospace">texpand(sin(2*x)^2*cos(3*x))
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
16*cos(<span style="font-style:italic">x</span>)<sup>5</sup>*sin(<span style="font-style:italic">x</span>)<sup>2</sup>&#X2212;12*cos(<span style="font-style:italic">x</span>)<sup>3</sup>*sin(<span style="font-style:italic">x</span>)<sup>2</sup>
</td></tr>
</table>
<!--TOC subsection id="sec32" Vectors and matrices-->
<h3 id="sec32" class="subsection">5.4&#XA0;&#XA0;Vectors and matrices</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Vectors and matrices</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">v*w</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >scalar product</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cross(v,w)</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >cross product</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">A*B</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >matrix product</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">A.*B</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >term by term product</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">1/A</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >inverse</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tran</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >transpose</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">rank</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >rank</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">det</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >determinant</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ker</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >basis for the kernel</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">image</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >base for the image</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">idn</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >identity matrix</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">ranm</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >matrix with random coefficients</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">makematrix</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >make a matrix from a function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">matrix</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >make a matrix from a function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">blockmatrix</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >combine matrices</td></tr>
</table>
</div><p>
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<a id="hevea_default287"></a></p><p>A vector is a list of numbers, such as <span style="font-family:monospace">[2,3,5]</span>, and a matrix
is a list of vectors all of the same length, such as
<span style="font-family:monospace">[[1,2,3],[4,5,6]]</span>.</p><p>The usual matrix operations (addition, scalar multiplication, matrix
multiplication) are done with the usual operators <span style="font-family:monospace">+</span> and
<span style="font-family:monospace">*</span>. If you define
</p><blockquote class="quote"><span style="font-family:monospace">A := [[1,2,3],[4,5,6],[7,8,9]]
</span></blockquote><blockquote class="quote"><span style="font-family:monospace">B := [[1,1,1],[2,2,2]]
</span></blockquote><p>
then
</p><blockquote class="quote"><span style="font-family:monospace">3*A
</span></blockquote><p>
will give you
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >3</td><td style="text-align:center;white-space:nowrap" >6</td><td style="text-align:center;white-space:nowrap" >9</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >12</td><td style="text-align:center;white-space:nowrap" >15</td><td style="text-align:center;white-space:nowrap" >18</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >21</td><td style="text-align:center;white-space:nowrap" >24</td><td style="text-align:center;white-space:nowrap" >27
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">B*A
</span></blockquote><p>
will give you
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >12</td><td style="text-align:center;white-space:nowrap" >15</td><td style="text-align:center;white-space:nowrap" >18</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >24</td><td style="text-align:center;white-space:nowrap" >30</td><td style="text-align:center;white-space:nowrap" >36
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>
A vector can be regarded as a matrix with one row, except that if a matrix
is multiplied on the right by a vector, the vector will be regarded as
a column. In particular, if <span style="font-family:monospace">v</span> and <span style="font-family:monospace">w</span> are vectors of
the same length, then <span style="font-family:monospace">v*w</span> will return the scalar product.</p><p>The <span style="font-family:monospace">idn</span> command will create an identity matrix;
</p><blockquote class="quote"><span style="font-family:monospace">idn(2)
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >0</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >1
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>
You can also use <span style="font-family:monospace">makemat</span> or <span style="font-family:monospace">matrix</span> commands to build
a matrix. They both require a real-valued function of two variables,
the number of rows and the number of columns. The indices start at 0,
and with the <span style="font-family:monospace">makemat</span> the function comes first, with
<span style="font-family:monospace">matrix</span> the function comes last. Both
</p><blockquote class="quote"><span style="font-family:monospace">makemat((j,k)-&gt;j+k,3,2)
</span></blockquote><p>
and
</p><blockquote class="quote"><span style="font-family:monospace">matrix(3,2,(j,k)-&gt;j+k)
</span></blockquote><p>
produce
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >2</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >3
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>Several matrices can be combined into a larger matrix with the
<span style="font-family:monospace">blockmatrix</span> command. To arrange <span style="font-style:italic">m</span> * <span style="font-style:italic">n</span> matrices
into <span style="font-style:italic">m</span> rows and <span style="font-style:italic">n</span> columns, you give <span style="font-family:monospace">blockmatrix</span> the
values <span style="font-style:italic">m</span>, <span style="font-style:italic">n</span> and a list of the matrices. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">A := [[1,2,3],[4,5,6]]
</span></blockquote><blockquote class="quote"><span style="font-family:monospace">B := [[1,2],[2,3]]
</span></blockquote><p>
then
</p><blockquote class="quote"><span style="font-family:monospace">blockmatrix(2,2,[A,B,B,A])
</span></blockquote><p>
will give you
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >3</td><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >2</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >4</td><td style="text-align:center;white-space:nowrap" >5</td><td style="text-align:center;white-space:nowrap" >6</td><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >3</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >3</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >3</td><td style="text-align:center;white-space:nowrap" >4</td><td style="text-align:center;white-space:nowrap" >5</td><td style="text-align:center;white-space:nowrap" >5
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>You can get the elements from a matrix by following the matrix with
the indices in brackets, separated by commas. For <span style="font-family:monospace">A</span> as above,
</p><blockquote class="quote"><span style="font-family:monospace">A[1,2]
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
6
</td></tr>
</table><p>
You can extract a submatrix<a id="hevea_default288"></a> by using intervals of indices (the
beginning and end index separated by two periods);
</p><blockquote class="quote"><span style="font-family:monospace">A[0..1,1..2]
</span></blockquote><p>
returns
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >2</td><td style="text-align:center;white-space:nowrap" >3</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >5</td><td style="text-align:center;white-space:nowrap" >6
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>Note that if you change one value of a matrix in <span style="font-family:monospace">Xcas</span>, the
entire matrix will be copied. If a program modifies parts of a large
matrix one element at a time, this time can add up.</p>
<!--TOC subsection id="sec33" Linear systems-->
<h3 id="sec33" class="subsection">5.5&#XA0;&#XA0;Linear systems</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Linear systems</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">linsolve</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >solution of a linear system</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">simult</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >solutions of many linear systems</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">rref</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >Gauss-Jordan reduction</td></tr>
</table>
</div><p>
<a id="hevea_default289"></a>
<a id="hevea_default290"></a>
<a id="hevea_default291"></a>
<a id="hevea_default292"></a></p><p>The <span style="font-family:monospace">linsolve</span> command will solve a system of linear equations;
its syntax is the same as that of <span style="font-family:monospace">solve</span> (see section
<a href="#solve">4.4</a>, &#X201C;Solving equations&#X201D;).
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">linsolve([2*x + 3*y = 4, 5*x + 4*y = 3],[x,y])
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[&#X2212;1,2]
</td></tr>
</table><p>The <span style="font-family:monospace">simult</span> command can also solve a system of linear
equations; more generally, it can solve several systems with the same
coefficient matrix. To solve the systems
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">A</span><span style="font-weight:bold"><span style="font-style:italic">x</span></span>&#XA0;=&#XA0;<span style="font-weight:bold"><span style="font-style:italic">b</span></span><sub>1</sub>,&#X2026;,<span style="font-style:italic">A</span><span style="font-weight:bold"><span style="font-style:italic">x</span></span>=<span style="font-weight:bold"><span style="font-style:italic">b</span></span><sub><span style="font-style:italic">k</span></sub></td></tr>
</table><p>
you can enter
</p><blockquote class="quote"><span style="font-family:monospace">simult(A,B)
</span></blockquote><p>
where
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">B</span>&#XA0;=&#XA0;</td><td class="dcell">&#X239B;<br>
&#X239D;</td><td class="dcell"><span style="font-weight:bold"><span style="font-style:italic">b</span></span><sub>1</sub>&#XA0;&#X22EF;&#XA0;<span style="font-weight:bold"><span style="font-style:italic">b</span></span><sub><span style="font-style:italic">k</span></sub></td><td class="dcell">&#X239E;<br>
&#X23A0;</td></tr>
</table><p>
The result will be a matrix whose <span style="font-style:italic">j</span>th column is the solution of
<span style="font-style:italic">A</span><span style="font-weight:bold"><span style="font-style:italic">x</span></span>=<span style="font-weight:bold"><span style="font-style:italic">b</span></span><sub><span style="font-style:italic">j</span></sub>.
For example, if you want to solve the systems
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23AA;<br>
&#X23A8;<br>
&#X23AA;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" >&#XA0;<span style="font-style:italic">x</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span></td><td style="text-align:left;white-space:nowrap" >&#X2212;</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span></td><td style="text-align:left;white-space:nowrap" >=</td><td style="text-align:right;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:left;white-space:nowrap" >&#XA0;<span style="font-style:italic">x</span></td><td style="text-align:left;white-space:nowrap" >&#X2212;</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span></td><td style="text-align:left;white-space:nowrap" >=</td><td style="text-align:right;white-space:nowrap" >1&#XA0;</td></tr>
<tr><td style="text-align:left;white-space:nowrap" >&#XA0;&#X2212;<span style="font-style:italic">x</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span></td><td style="text-align:left;white-space:nowrap" >=</td><td style="text-align:right;white-space:nowrap" >&#X2212;2&#XA0;
</td></tr>
</table></td></tr>
</table><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23AA;<br>
&#X23A8;<br>
&#X23AA;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" >&#XA0;<span style="font-style:italic">x</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span></td><td style="text-align:left;white-space:nowrap" >&#X2212;</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span></td><td style="text-align:left;white-space:nowrap" >=</td><td style="text-align:right;white-space:nowrap" >&#X2212;2</td></tr>
<tr><td style="text-align:left;white-space:nowrap" >&#XA0;<span style="font-style:italic">x</span></td><td style="text-align:left;white-space:nowrap" >&#X2212;</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span></td><td style="text-align:left;white-space:nowrap" >=</td><td style="text-align:right;white-space:nowrap" >1&#XA0;</td></tr>
<tr><td style="text-align:left;white-space:nowrap" >&#XA0;&#X2212;<span style="font-style:italic">x</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span></td><td style="text-align:left;white-space:nowrap" >+</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span></td><td style="text-align:left;white-space:nowrap" >=</td><td style="text-align:right;white-space:nowrap" >1&#XA0;
</td></tr>
</table></td></tr>
</table><p>
which both have the same matrix of coefficients
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >&#XA0;&#XA0;1</td><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >&#X2212;1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#XA0;&#XA0;1</td><td style="text-align:center;white-space:nowrap" >&#X2212;1</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#XA0;&#XA0;&#X2212;1</td><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >1
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>
you can create the matrix which has one column for each system
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >&#XA0;&#XA0;1</td><td style="text-align:center;white-space:nowrap" >&#X2212;2</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#XA0;&#XA0;1</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#XA0;&#XA0;&#X2212;2</td><td style="text-align:center;white-space:nowrap" >1
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">simult([[1,1,-1],[1,-1,1],[-1,1,1]],[[1,-2],[1,1],[-2,1]])
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >&#X2212;1/2</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;1/2</td><td style="text-align:center;white-space:nowrap" >&#X2212;1/2</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;1/2</td><td style="text-align:center;white-space:nowrap" >1
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>
The solution to the first system is the first column,
<span style="font-style:italic">x</span>=1,<span style="font-style:italic">y</span>=&#X2212;1/2,<span style="font-style:italic">z</span>=&#X2212;1/2, and the solution to the second system is the
second column, <span style="font-style:italic">x</span>=&#X2212;1/2,<span style="font-style:italic">y</span>=&#X2212;1/2,<span style="font-style:italic">z</span>=1.</p><p>When there are no solutions, <span style="font-family:monospace">linsolve</span> will return the empty
list while <span style="font-family:monospace">simult</span> will return an error. When there are
infinitely many solutions, <span style="font-family:monospace">linsolve</span> will return formulas for
all solutions while <span style="font-family:monospace">simult</span> will return one solution.</p>
<!--TOC subsection id="sec34" Matrix reduction-->
<h3 id="sec34" class="subsection">5.6&#XA0;&#XA0;Matrix reduction</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Matrix reduction</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">jordan</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >diagonalization or Jordan reduction</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">pcar</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >characteristic polynomial (list form)</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">pmin</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >minimal polynomial (list form)</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">eigenvals</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >eigenvalues</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">eigenvects</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >eigenvectors</td></tr>
</table>
</div><p>
<a id="hevea_default293"></a>
<a id="hevea_default294"></a>
<a id="hevea_default295"></a>
<a id="hevea_default296"></a>
<a id="hevea_default297"></a>
<a id="hevea_default298"></a>
<a id="hevea_default299"></a>
<a id="hevea_default300"></a>
<a id="hevea_default301"></a>
<a id="hevea_default302"></a>
<a id="hevea_default303"></a></p><p>The <span style="font-family:monospace">jordan</span> command will take a matrix <span style="font-style:italic">A</span> and returns a
transition matrix <span style="font-style:italic">P</span> and a matrix <span style="font-style:italic">J</span> in Jordan canonical form, so
that <span style="font-style:italic">P</span><sup>&#X2212;1</sup> <span style="font-style:italic">A</span> <span style="font-style:italic">P</span> = <span style="font-style:italic">J</span>. In particular, if <span style="font-style:italic">A</span> is diagonalizable, then
<span style="font-style:italic">J</span> will be diagonal with the eigenvalues of <span style="font-style:italic">A</span> on the diagonal and
the columns of <span style="font-style:italic">P</span> will be the corresponding eigenvectors.
If you enter
</p><blockquote class="quote"><span style="font-family:monospace">jordan([[4,1],[-8,-5]])
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;1</td><td style="text-align:center;white-space:nowrap" >&#X2212;8&#XA0;</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td><td class="dcell">,
</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >3</td><td style="text-align:center;white-space:nowrap" >0</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >&#X2212;4&#XA0;</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table><p>
This means that 3 and &#X2212;4 (the diagonal elements of the second
matrix) are the eigenvalues of 
(</p><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >4</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;8</td><td style="text-align:center;white-space:nowrap" >&#X2212;5</td></tr>
</table><p>) and the corresponding
eigenvectors are (</p><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;1</td></tr>
</table><p>) and
(</p><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;8</td></tr>
</table><p>) (the columns of the first matrix).
For diagonalizable matrices you can also get this information with the
<span style="font-family:monospace">eigenvals</span> and <span style="font-family:monospace">eigenvects</span> commands;
</p><blockquote class="quote"><span style="font-family:monospace">eigenvals([[4,1],[-8,-5]])
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
(3,&#X2212;4)
</td></tr>
</table><p>
and 
</p><blockquote class="quote"><span style="font-family:monospace">eigenvects([[4,1],[-8,-5]])
</span></blockquote><p>
will return
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;1</td><td style="text-align:center;white-space:nowrap" >&#X2212;8&#XA0;</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table><p>For matrices with exact and symbolic values, the only eigenvalues used
are those computable with <span style="font-family:monospace">solve</span>; for matrices with floating
point numbers, a numerical algorithm is used to find the eigenvalues.
This algorithm may fail in some cases where there are very close
eigenvalues or eigenvalues with multiplicity greater than one.</p><p>If a function is defined by a polynomial, you can evaluate it with an
argument of a square matrix. If a function is given by a series, the
Jordan form of the matrix can be used to define the value of the
function at a matrix. For example, you can find the exponential of a
square matrix;
</p><blockquote class="quote"><span style="font-family:monospace">exp([[0,-1],[1,2]])
</span></blockquote><p>
will result in
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">

</td><td class="dcell"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >&#X2212;exp(1)</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >exp(1)</td><td style="text-align:center;white-space:nowrap" >2*exp(1)
</td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X23A0;</td></tr>
</table></td></tr>
</table>
<!--TOC section id="sec35" Graphs-->
<h2 id="sec35" class="section">6&#XA0;&#XA0;Graphs</h2><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Plotting graphs</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >graph an expression of one variable</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotfunc</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >graph an expression of one or two variables</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">tangent</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >tangent to a curve</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotparam</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >parametric curve</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotpolar</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >polar plotting</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotimplicit</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >implicit curve</td></tr>
</table>
</div><p>
<a id="hevea_default304"></a>
<a id="hevea_default305"></a>
<a id="hevea_default306"></a>
<a id="hevea_default307"></a>
<a id="hevea_default308"></a>
<a id="hevea_default309"></a>
<a id="hevea_default310"></a>
<a id="hevea_default311"></a>
<a id="hevea_default312"></a></p><p>The <span style="font-family:monospace">Graphic</span> menu has entries for several graphing commands;
if you choose one you will be given a template you can fill out to
produce a graphic. The appropriate command will be placed on the
command line. If you want several graphs in the same window, you can
put the commands on the same command line separated by semicolons.</p><p>Once you have created a graphic window, there will be a panel to the
right with buttons allowing you to control the image. The default
parameters for graphs, such as the size of the graphs, are
configurable (see section <a href="#config">2.3</a>, &#X201C;Configuration&#X201D;).</p><p>As well as being displayed in the <span style="font-family:monospace">Xcas</span> window, the two
dimensional graphics also appear in the DispG (Display Graphics)
window. You can bring that window up with the
<span style="font-family:monospace">Cfg</span>&#X25B8;<span style="font-family:monospace">Show</span>&#X25B8;<span style="font-family:monospace">DispG</span> menu item.
This window will contain all two dimensional graphs; they can be
cleared with the <span style="font-family:monospace">ClrGraph</span> command. </p>
<!--TOC subsection id="sec36" Curves-->
<h3 id="sec36" class="subsection">6.1&#XA0;&#XA0;Curves</h3><!--SEC END --><p>The simplest way to draw graphs is with the templates from the
<span style="font-family:monospace">Graphic</span> menu, but there are command line equivalents. </p><p>The command line instruction for graphing a function is the
<span style="font-family:monospace">plot</span> command. It takes an expression (or list of
expressions) followed by the variable. To use a domain different than
the default, you can indicate the range of the variable by setting it
equal to an interval. If you are plotting several curves, you can
distinguish them by giving them different colors; you can do this with
a third argument <span style="font-family:monospace">color=</span> followed by a list of colors.
If you plot
</p><blockquote class="quote"><span style="font-family:monospace">plot([x^2,x^3],x=-1..1,color=[red,blue])
</span></blockquote><p>
you will get
</p><div class="center">
<img src="tutoriel008.png">
</div><p>You can draw parameterized curves with the <span style="font-family:monospace">plotparam</span> command.
The coordinates of the curve must be given as a single complex
expression; the <span style="font-style:italic">x</span> coordinate will be the real part of the expression
and the <span style="font-style:italic">y</span> coordinate will be the imaginary part. For example, if
you enter
</p><blockquote class="quote"><span style="font-family:monospace">plotparam(sin(t) + i*cos(t),t)
</span></blockquote><p>
you will get the circle
</p><div class="center">
<img src="tutoriel009.png">
</div><p>
(Since the <span style="font-style:italic">x</span>- and <span style="font-style:italic">y</span>-axes are scaled differently, the circle
looks elliptical.)</p><p>You can draw a curve using polar coordinates with the
<span style="font-family:monospace">plotpolar</span> command. This takes the form
<span style="font-family:monospace">plotpolar(f(theta),theta,theta-min,theta-max)</span>.</p><p>You can draw an implicitly defined curve with the
<span style="font-family:monospace">plotimplicit</span> command; the command
<span style="font-family:monospace">plotimplicit(</span><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>)<span style="font-family:monospace">,</span><span style="font-style:italic">x</span><span style="font-family:monospace">,</span><span style="font-style:italic">y</span><span style="font-family:monospace">)</span> will draw the curve
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>)=0. For example, the command
</p><blockquote class="quote"><span style="font-family:monospace">plotimplicit(x^2 + y^2 = 1,x,y)
</span></blockquote><p>
will draw a circle.</p><p>The <span style="font-family:monospace">tangent</span> command will draw the tangent line to a curve, if
you tell it the curve and the point. To draw the tangent line to the
graph <span style="font-style:italic">y</span>=<span style="font-style:italic">x</span><sup>2</sup> at <span style="font-style:italic">x</span>=1, for example, you can enter
</p><blockquote class="quote"><span style="font-family:monospace">tangent(plotfunc(x^2,x),1)
</span></blockquote><p>
You will get
</p><div class="center">
<img src="tutoriel010.png">
</div>
<!--TOC subsection id="sec37" Plane geometry-->
<h3 id="sec37" class="subsection">6.2&#XA0;&#XA0;Plane geometry</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">2D graphical objects</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">legend</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >place text starting from a given point</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">point</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >determine a point given a complex number or two
coordinates</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">segment</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >segment determined by 2 points</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">circle</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >circle determined by a point and radius</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">inter</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >find the intersection of curves</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">equation</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >return the cartesian equation of a curve</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">parameq</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >return the parametric equation of a curve</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">polygonplot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >draw a polygonal line</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">scatterplot</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >draw a cloud of dots</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">polygon</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >draw a closed polygon</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">open_polygon</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >draw an open polygon</td></tr>
</table>
</div><p>
<a id="hevea_default313"></a>
<a id="hevea_default314"></a>
<a id="hevea_default315"></a>
<a id="hevea_default316"></a>
<a id="hevea_default317"></a>
<a id="hevea_default318"></a>
<a id="hevea_default319"></a>
<a id="hevea_default320"></a>
<a id="hevea_default321"></a>
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<a id="hevea_default328"></a>
<a id="hevea_default329"></a></p><p>Among its other capabilities, <span style="font-family:monospace">Xcas</span> works with plane geometry.
The <span style="font-family:monospace">Geo</span>&#X25B8;<span style="font-family:monospace">New figure 2D</span> menu item or the
<span style="font-family:monospace">Alt+g</span> key will bring up the screen for plane geometry.</p><p>A point on the geometry screen can be specified with the
<span style="font-family:monospace">point</span> command, which can take either an ordered pair or real
numbers or a complex number as argument. The <span style="font-family:monospace">Geo</span> menu
contains many commands for drawing geometric objects, such as
<span style="font-family:monospace">circle</span> (which takes a point and a radius as arguments) and
<span style="font-family:monospace">polygon</span> (which takes a sequence of points as arguments).</p><p>Some functions, such as <span style="font-family:monospace">polygonplot</span> and <span style="font-family:monospace">scatterplot</span>,
take lists of <span style="font-style:italic">x</span>-coordinates and <span style="font-style:italic">y</span>-coordinates as arguments. For
example,
</p><blockquote class="quote"><span style="font-family:monospace">polygonplot([0,2,0],[0,0,2])
</span></blockquote><p>
and 
</p><blockquote class="quote"><span style="font-family:monospace">open_polygon(point(0,0),point(2,0),point(0,2))
</span></blockquote><p>
will both draw the same segments.</p><p>The <span style="font-family:monospace">legend</span> can be used to place text on the screen; one
simple way of using it is to give it a point and text as arguments.</p>
<!--TOC subsection id="sec38" 3D graphical objects-->
<h3 id="sec38" class="subsection">6.3&#XA0;&#XA0;3D graphical objects</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">3D graphical objects</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotfunc</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >graph of a function</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plotparam</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >parametric surface</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">point</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >point</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">plane</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >plane</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">sphere</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >sphere with a center and radius</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">cone</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >cone with a center, axis and opening angle</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">inter</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >intersection </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">polygon</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >polygon </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">open_polygon</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >open polygon</td></tr>
</table>
</div><p>
<a id="hevea_default330"></a>
<a id="hevea_default331"></a>
<a id="hevea_default332"></a>
<a id="hevea_default333"></a>
<a id="hevea_default334"></a>
<a id="hevea_default335"></a>
<a id="hevea_default336"></a>
<a id="hevea_default337"></a>
<a id="hevea_default338"></a>
<a id="hevea_default339"></a>
<a id="hevea_default340"></a>
<a id="hevea_default341"></a></p><p><span style="font-family:monospace">Xcas</span> can also handle three-dimensions graphically, either by
drawing curves and graphs in three-dimensions or by drawing
three-dimensional geometric objects. A three-dimensional screen can
be brought up with the <span style="font-family:monospace">Geo</span>&#X25B8;<span style="font-family:monospace">New figure 3D</span>
menu item or the <span style="font-family:monospace">Alt+h</span> key.
There will controls for the view window to the right of the screen.
You can rotate the visualization cube by using the mouse outside of
the cube or by clicking in the cube and using the <span style="font-family:monospace">x</span>,
<span style="font-family:monospace">y</span> and <span style="font-family:monospace">z</span> keys to rotate and the <span style="font-family:monospace">+</span> and
<span style="font-family:monospace">-</span> keys for zooming.</p><p>You can draw a graph <span style="font-style:italic">z</span> = <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>) with the <span style="font-family:monospace">plotfunc</span> command,
which takes an expression and a list of two variables. Like graphs of
functions of one variable, you can use a domain different than the
default by giving the variables their own intervals. The command
</p><blockquote class="quote"><span style="font-family:monospace">plotfunc(y^2 - x^2,[x=-1..1,y=-1..1])
</span></blockquote><p>
will give you the graph
</p><div class="center">
<img src="tutoriel011.png">
</div><p>The <span style="font-family:monospace">plotparam</span> command can be used to draw a parameterized
curves and surfaces in three-dimensions. If the first argument is a
list of three expressions involving two variables and the next two
arguments are the variables (with optional intervals), then
<span style="font-family:monospace">plotparam</span> will plot the surface. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">plotparam([u,u+v,v],u,v)
</span></blockquote><p>
you will get
</p><div class="center">
<img src="tutoriel012.png">
</div><p>
If the first argument is a list of three expressions involving one
variable and the next argument is the variable, then
<span style="font-family:monospace">plotparam</span> will plot the curve. If you enter
</p><blockquote class="quote"><span style="font-family:monospace">plotparam([cos(t),sin(t),t],t)
</span></blockquote><p>
you will get
</p><div class="center">
<img src="tutoriel013.png">
</div><p>A point in three-dimensions is given with the <span style="font-family:monospace">point</span> command
with three arguments. Commands like <span style="font-family:monospace">polygon</span> and
<span style="font-family:monospace">open_polygon</span> work in three-dimensions as well as two, as
well as additional commands such as <span style="font-family:monospace">sphere</span>. A plane can be
drawn with the <span style="font-family:monospace">plane</span> command, which can be given either three
points, a line and two points, or an equation of the form <span style="font-family:monospace">a*x
+ b*y + c*z = d</span>.</p>
<!--TOC section id="sec39" Programming-->
<h2 id="sec39" class="section">7&#XA0;&#XA0;Programming</h2><!--SEC END -->
<!--TOC subsection id="sec40" The language-->
<h3 id="sec40" class="subsection">7.1&#XA0;&#XA0;The language</h3><!--SEC END --><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Instructions for </span><span style="font-weight:bold"><span style="font-family:monospace">Xcas</span></span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">a:=2;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >assignment </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">input("a=",a);</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >input expression </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">textinput("a=",a);</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >string input </td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">print("a=",a);</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >output</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">return(a);</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >return value</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">break;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >break out of loop</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">continue;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >go to the next iteration</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">if (&lt;condition&gt;) </span><span style="font-family:monospace">&lt;inst&gt;</span><span style="font-family:monospace">;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >if&#X2026;then</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">if (&lt;condition&gt;) </span><span style="font-family:monospace">&lt;inst1&gt;</span><span style="font-family:monospace"> else </span><span style="font-family:monospace">&lt;inst2&gt;</span><span style="font-family:monospace">;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >	 if&#X2026;then &#X2026;else</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">for (j:= a;j&lt;=b;j++) </span><span style="font-family:monospace">&lt;inst&gt;</span><span style="font-family:monospace">;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >for loop</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">for (j:= a;j&lt;=b;j:=j+p) </span><span style="font-family:monospace">&lt;inst&gt;</span><span style="font-family:monospace">;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >for loop</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">repeat &lt;inst&gt; until &lt;condition&gt;;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >repeat loop</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">while (&lt;condition&gt;) </span><span style="font-family:monospace">&lt;inst&gt;</span><span style="font-family:monospace">;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >while loop</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">do &lt;inst1&gt; if (&lt;condition&gt;) break;&lt;inst2&gt; od;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >do loop</td></tr>
</table>
</div><p>
<a id="hevea_default342"></a>
<a id="hevea_default343"></a>
<a id="hevea_default344"></a>
<a id="hevea_default345"></a></p><div class="center">
<table border=1  style="border-spacing:0;" class="cellpadding1"><tr><td style="text-align:center;border:solid 1px;white-space:nowrap"  colspan=2><span style="font-weight:bold">Boolean operators</span></td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">==</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >test for equality</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">!=</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >test for inequality</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">&lt;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >test for strictly less than</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">&gt;</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >test for strictly greater than</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">&lt;=</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >tes for less than or equal</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">&gt;=</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >test for greater than or equal to</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">&amp;&amp;</span>, <span style="font-family:monospace">and</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >infixed &#X201C;and&#X201D;</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">||</span>, <span style="font-family:monospace">or</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >infixed &#X201C;or&#X201D;</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">true</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >boolean true (same as 1)</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">false</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >boolean false (same as 0)</td></tr>
<tr><td style="vertical-align:top;text-align:left;border:solid 1px;" ><span style="font-family:monospace">not</span>, <span style="font-family:monospace">!</span></td><td style="vertical-align:top;text-align:left;border:solid 1px;" >&#X201C;not&#X201D;</td></tr>
</table>
</div><p>
<a id="hevea_default346"></a>
<a id="hevea_default347"></a>
<a id="hevea_default348"></a>
<a id="hevea_default349"></a>
<a id="hevea_default350"></a>
<a id="hevea_default351"></a>
<a id="hevea_default352"></a>
<a id="hevea_default353"></a></p><p>You can extend <span style="font-family:monospace">Xcas</span> by adding desired functions with its
built-in programming language. The main features of the language are:</p><ul class="itemize"><li class="li-itemize">
It is a functional language. The argument of a function can
be another function; in which case you can either give the name of
a function or the definition of the function. If 
<span style="font-family:monospace">f(x) := x^2</span>, then
<span style="font-family:monospace">function_diff(f)</span> is the same as
<span style="font-family:monospace">function_diff(x-&gt;x^2)</span>.</li><li class="li-itemize">There is no distinction between a program and a function. A
function returns the value of the last evaluated statement or what
follows the reserved word <span style="font-family:monospace">return</span>.</li><li class="li-itemize">The language is untyped. Any variable can take on any value; the
only different types of variables are global variables, which are
not declared, and local variables, which are declared at the
beginning of a function.
</li></ul><p>A function declaration looks like
</p><pre class="verbatim">   function_name (var1, var2, ...) := {
   local var_loc1, var_loc2, ... ;
     statement1;
     statement2;
     ...
   }
</pre><p>
The syntax is similar to <span style="font-family:monospace">C++</span>, although many variants are
recognized, particularly in compatibility mode.
Recall that <span style="font-family:monospace">i</span> is &#X221A;<span style="text-decoration:overline">&#X2212;1</span> and cannot be used for a loop
variable. 
The conditional tests are Booleans, which are the results of the usual
Boolean operators.</p><p>A program can capture runtime errors with a
<span style="font-family:monospace">try</span>&#X2013;<span style="font-family:monospace">catch</span> construction, which takes the form
</p><pre class="verbatim">   try 
     {
      block to catch errors
     }
   catch (variable)
     {
      block to execute when an error is caught
     }
</pre><p>
For example, the following will catch an error caused by incorrect
matrix multiplication:
</p><pre class="verbatim">   try 
     { A := idn(2) * idn(3) }
   catch (error)
     { print("The error is " + error) }
</pre>
<!--TOC subsection id="sec41" Some examples-->
<h3 id="sec41" class="subsection">7.2&#XA0;&#XA0;Some examples</h3><!--SEC END --><p>To write a program, it is a good idea to use the program editor that
comes with <span style="font-family:monospace">Xcas</span>, which provides a template and commands
helpful for writing programs. You can open this editor with the
<span style="font-family:monospace">New Program</span> item in the <span style="font-family:monospace">Prg</span> menu or the
<span style="font-family:monospace">Alt+p</span> key.</p><p>Consider the following program, which takes two integers and returns the
quotient and remainder of the Euclidean division algorithm (like the
<span style="font-family:monospace">iquorem</span> function).
</p><pre class="verbatim">   idiv2(a,b) := {
     local q,r;
     if (b != 0) {
       q := iquo(a,b);
       r := irem(a,b);
       }
     else {
       q := 0;
       r := a;
       }
     return [q,r];
   }
</pre><p>
If you enter this into the editor, you can test it with the
<span style="font-family:monospace">OK</span> button. You can then use the function in the command
line; if you enter
</p><blockquote class="quote"><span style="font-family:monospace">idiv2(25,15)
</span></blockquote><p>
you will get
</p><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
[1,10]
</td></tr>
</table><p>
You can save it in a file; the name <span style="font-family:monospace">idiv2.cxx</span> would be a good
name. You can then use it in later session with the command
</p><blockquote class="quote"><span style="font-family:monospace">read("idiv2.cxx")
</span></blockquote><p>
or by opening it in the program editor and validating it with the
<span style="font-family:monospace">OK</span> button.</p><p>Here are some more programs that you can play with. This first one
computes the GCD of two integers iteratively.
</p><pre class="verbatim">   pgcdi(a,b) := {
     local r;
     while (b != 0) {
       r := irem(a,b);
       a := b;
       b := r;
       }
     return a;
   }:;
</pre><p>
The second one computes the GCD recursively.
</p><pre class="verbatim">   pgcdr(a,b) := {
     if (b == 0) return a;
     return pgcdr(b, irem(a,b));
   }:;
</pre><p>If a program doesn&#X2019;t work the way you expect, you can run it in
step-by-step mode with the debug command. For more details, consult
the <span style="font-family:monospace">Interface</span> item of the <span style="font-family:monospace">Help</span> menu. For example,
you can start the debugging by typing
</p><blockquote class="quote"><span style="font-family:monospace">debug(idiv2(25,15))
</span></blockquote><p>
The debugger will automatically display the values of the parameters
<span style="font-family:monospace">a</span> and <span style="font-family:monospace">b</span> and local variables <span style="font-family:monospace">q</span> and
<span style="font-family:monospace">r</span> when executing the program line by line with the
<span style="font-family:monospace">sst</span> button.</p>
<!--TOC subsection id="sec42" Programming style-->
<h3 id="sec42" class="subsection">7.3&#XA0;&#XA0;Programming style</h3><!--SEC END --><p>The <span style="font-family:monospace">Xcas</span> programming language is interpreted, not compiled.
The run time of an <span style="font-family:monospace">Xcas</span> program is affected by the number of
instructions rather than the number of lines. </p><p>The speed of a program does not always match up with the clarity of
the program; compromises are often necessary. For the most part, the
calculation time isn&#X2019;t an issue; interpreted languages are often used
to test algorithms and create models. Full scale applications are
written in a compiled language like <span style="font-family:monospace">C++</span>. A <span style="font-family:monospace">C++</span> can
use <span style="font-family:monospace">giac</span> for the formal calculations.</p><p>When you are trying to write a fast program, you may want to take into
account the number of instructions and the speed of the instructions.
For example, it is in general faster to create lists and sequences
than it is to program loops. Recall than in <span style="font-family:monospace">Xcas</span> you can
find out how long it takes to run a command by entering
</p><blockquote class="quote"><span style="font-family:monospace">time(</span><span style="font-family:monospace"><span style="font-style:italic">command</span></span><span style="font-family:monospace">)
</span></blockquote>
<!--TOC section id="sec43" Exercises-->
<h2 id="sec43" class="section">8&#XA0;&#XA0;Exercises</h2><!--SEC END --><p>There are usually several ways to get the same result in
<span style="font-family:monospace">Xcas</span>. We will try to use the simplest approaches.</p><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;1</span>&#XA0;&#XA0;Verify the following identities.
<ol class="enumerate" type=1><li class="li-enumerate">
(2<sup>1/3</sup>+4<sup>1/3</sup>)<sup>3</sup>&#X2212;6(2<sup>1/3</sup>+4<sup>1/3</sup>)=6
</li><li class="li-enumerate">&#X3C0; /4 = 4arctan(1/5)&#X2212;arctan(1/239)
</li><li class="li-enumerate">sin(5<span style="font-style:italic">x</span>) = 5sin(<span style="font-style:italic">x</span>)&#X2212;20sin<sup>3</sup>(<span style="font-style:italic">x</span>)+16sin<sup>5</sup>(<span style="font-style:italic">x</span>)
</li><li class="li-enumerate">(tan(<span style="font-style:italic">x</span>)+tan(<span style="font-style:italic">y</span>))cos(<span style="font-style:italic">x</span>)cos(<span style="font-style:italic">y</span>) = sin(<span style="font-style:italic">x</span>+<span style="font-style:italic">y</span>)
</li><li class="li-enumerate">cos<sup>6</sup>(<span style="font-style:italic">x</span>)+sin<sup>6</sup>(<span style="font-style:italic">x</span>) = 1&#X2212;3sin<sup>2</sup>(<span style="font-style:italic">x</span>)cos<sup>2</sup>(<span style="font-style:italic">x</span>)
</li><li class="li-enumerate">ln(tan(<span style="font-style:italic">x</span>/2+&#X3C0;/4)) = argsinh(tan(<span style="font-style:italic">x</span>))
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;2</span>&#XA0;&#XA0;Transform the rational expression
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>4</sup>+<span style="font-style:italic">x</span><sup>3</sup>&#X2212;4<span style="font-style:italic">x</span><sup>2</sup>&#X2212;4<span style="font-style:italic">x</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>4</sup>+<span style="font-style:italic">x</span><sup>3</sup>&#X2212;<span style="font-style:italic">x</span><sup>2</sup>&#X2212;<span style="font-style:italic">x</span></td></tr>
</table></td></tr>
</table>
into the following:
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>+2)(<span style="font-style:italic">x</span>+1)(<span style="font-style:italic">x</span>&#X2212;2)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>3</sup>+<span style="font-style:italic">x</span><sup>2</sup>&#X2212;<span style="font-style:italic">x</span>&#X2212;1</td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;&#XA0;
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>4</sup>+<span style="font-style:italic">x</span><sup>3</sup>&#X2212;4<span style="font-style:italic">x</span><sup>2</sup>&#X2212;4<span style="font-style:italic">x</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)<sup>2</sup></td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;&#XA0;
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>+2)(<span style="font-style:italic">x</span>&#X2212;2)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)</td></tr>
</table></td><td class="dcell">&#XA0;,
</td></tr>
</table>
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)</td></tr>
</table></td><td class="dcell">&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">4</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)</td></tr>
</table></td><td class="dcell">&#XA0;.
</td></tr>
</table>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;3</span>&#XA0;&#XA0;Transform the rational expression
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">2</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>3</sup>&#X2212;<span style="font-style:italic">yx</span><sup>2</sup>&#X2212;<span style="font-style:italic">yx</span>+<span style="font-style:italic">y</span><sup>2</sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>3</sup>&#X2212;<span style="font-style:italic">yx</span><sup>2</sup>&#X2212;<span style="font-style:italic">x</span>+<span style="font-style:italic">y</span></td></tr>
</table></td></tr>
</table>
into the following
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">2</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup>&#X2212;<span style="font-style:italic">y</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup>&#X2212;1</td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;&#XA0;
2</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup>&#X2212;<span style="font-style:italic">y</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(<span style="font-style:italic">x</span>&#X2212;1)(<span style="font-style:italic">x</span>+1)</td></tr>
</table></td><td class="dcell">
&#XA0;,
</td></tr>
</table>
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">2&#X2212;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">y</span>&#X2212;1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>&#X2212;1</td></tr>
</table></td><td class="dcell">+</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">y</span>&#X2212;1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>+1</td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;&#XA0;
2&#X2212;2</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">y</span>&#X2212;1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span><sup>2</sup>&#X2212;1</td></tr>
</table></td><td class="dcell">&#XA0;.
</td></tr>
</table>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;4</span>&#XA0;&#XA0;For each of the following definitions of a function <span style="font-style:italic">f</span>
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)&#XA0;=&#XA0;</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" ><span style="font-style:italic">e</span><sup><span style="font-style:italic">x</span></sup>&#X2212;1</td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;&#XA0;
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)&#XA0;=&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">x</span></td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >1+<span style="font-style:italic">x</span><sup>2</sup></td></tr>
</table></td></tr>
</table></td></tr>
</table></td><td class="dcell">
&#XA0;,
</td></tr>
</table>
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)&#XA0;=&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">1+sin(<span style="font-style:italic">x</span>)+cos(<span style="font-style:italic">x</span>)</td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;&#XA0;
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)&#XA0;=&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">ln(<span style="font-style:italic">x</span>)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>(<span style="font-style:italic">x</span><sup>2</sup>+1)<sup>2</sup></td></tr>
</table></td><td class="dcell">
&#XA0;.
</td></tr>
</table>
<ol class="enumerate" type=1><li class="li-enumerate">
Find an antiderivative <span style="font-style:italic">F</span>.
</li><li class="li-enumerate">Find <span style="font-style:italic">F</span>&#X2032;(<span style="font-style:italic">x</span>) and show that <span style="font-style:italic">F</span>&#X2032;(<span style="font-style:italic">x</span>) can be simplified to <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>).
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;5</span>&#XA0;&#XA0;For each of the following integrals
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:xx-large">&#X222B;</span></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">&#X2212;1</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">&#X2212;2</td></tr>
</table></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span></td></tr>
</table></td><td class="dcell">&#XA0;<span style="font-style:italic">dx</span>&#XA0;,&#XA0;
</td><td class="dcell"><span style="font-size:xx-large">&#X222B;</span></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">1</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">0</td></tr>
</table></td><td class="dcell">&#XA0;<span style="font-style:italic">x</span>arctan(<span style="font-style:italic">x</span>)&#XA0;<span style="font-style:italic">dx</span>&#XA0;,
</td></tr>
</table>
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-size:xx-large">&#X222B;</span></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">&#X3C0;/2</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">0</td></tr>
</table></td><td class="dcell">&#XA0;</td><td class="dcell"><span style="font-size:x-large">&#X221A;</span></td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td style="text-align:center;white-space:nowrap" >cos(<span style="font-style:italic">x</span>)</td></tr>
</table></td><td class="dcell">&#XA0;<span style="font-style:italic">dx</span>&#XA0;,&#XA0;
</td><td class="dcell"><span style="font-size:xx-large">&#X222B;</span></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">&#X3C0;/2</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">0</td></tr>
</table></td><td class="dcell">&#XA0;<span style="font-style:italic">x</span><sup>4</sup>sin(<span style="font-style:italic">x</span>)cos(<span style="font-style:italic">x</span>)&#XA0;<span style="font-style:italic">dx</span>&#XA0;.
</td></tr>
</table> 
<ol class="enumerate" type=1><li class="li-enumerate">
Find the exact value, and find an approximation.
</li><li class="li-enumerate">For <span style="font-style:italic">n</span>=100 and <span style="font-style:italic">n</span>=1000, do the following.
For each <span style="font-style:italic">j</span>=0,&#X2026;,<span style="font-style:italic">n</span>, let <span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span></sub>=<span style="font-style:italic">a</span>+<span style="font-style:italic">j</span>(<span style="font-style:italic">b</span>&#X2212;<span style="font-style:italic">a</span>)/<span style="font-style:italic">n</span> and <span style="font-style:italic">y</span><sub><span style="font-style:italic">j</span></sub>=<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span></sub>).
Find an approximate value for the integral by using the left endpoint
rule:
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">n</span>&#X2212;1</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-size:xx-large">&#X2211;</span></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">j</span>=0</td></tr>
</table></td><td class="dcell">&#XA0;<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span></sub>)(<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span>+1</sub>&#X2212;<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span></sub>)&#XA0;.
</td></tr>
</table>
</li><li class="li-enumerate">Do the same as the previous part, except use the trapezoid
method:
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">n</span>&#X2212;1</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-size:xx-large">&#X2211;</span></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">j</span>=0</td></tr>
</table></td><td class="dcell">&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">2</td></tr>
</table></td><td class="dcell">(<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span></sub>)+<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span>+1</sub>))(<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span>+1</sub>&#X2212;<span style="font-style:italic">x</span><sub><span style="font-style:italic">j</span></sub>)&#XA0;.
</td></tr>
</table>
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;6</span>&#XA0;&#XA0;Define the function <span style="font-style:italic">f</span> by 
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>)=cos(<span style="font-style:italic">xy</span>).
<ol class="enumerate" type=1><li class="li-enumerate">
Let <span style="font-style:italic">x</span><sub>0</sub>=<span style="font-style:italic">y</span><sub>0</sub>=&#X3C0;/4. Define the function that maps (<span style="font-style:italic">u</span>,<span style="font-style:italic">v</span>,<span style="font-style:italic">t</span>) to
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>0</sub>+<span style="font-style:italic">ut</span>,<span style="font-style:italic">y</span><sub>0</sub>+<span style="font-style:italic">vt</span>)&#XA0;.</td></tr>
</table>
</li><li class="li-enumerate">Define the function <span style="font-style:italic">g</span> which is the partial derivative of the
preceding function with respect to <span style="font-style:italic">t</span> (so this will be a directional
derivative of <span style="font-style:italic">f</span>).
</li><li class="li-enumerate">Find the gradient of <span style="font-style:italic">f</span> at (<span style="font-style:italic">x</span><sub>0</sub>,<span style="font-style:italic">y</span><sub>0</sub>), then find the scalar product
of this gradient with the vector (<span style="font-style:italic">u</span>,<span style="font-style:italic">v</span>). Write this result in terms
of <span style="font-style:italic">g</span>.
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;7</span>&#XA0;&#XA0;Consider <span style="font-style:italic">x</span><sup>3</sup>&#X2212;(<span style="font-style:italic">a</span>&#X2212;1)<span style="font-style:italic">x</span><sup>2</sup>+<span style="font-style:italic">a</span><sup>2</sup><span style="font-style:italic">x</span>&#X2212;<span style="font-style:italic">a</span><sup>3</sup>=0 as an equation in <span style="font-style:italic">x</span>.
<ol class="enumerate" type=1><li class="li-enumerate">
Graph the solution <span style="font-style:italic">x</span> as a function of <span style="font-style:italic">a</span> using <span style="font-family:monospace">plotimplicit</span>.
</li><li class="li-enumerate">Find the three solutions of the equation. You can use <span style="font-family:monospace">rootof</span>
to find the first solution, then use <span style="font-family:monospace">quo</span> to factor out the
first solution. You can then find the last two solutions by solving
the resulting second degree equation. (You can use <span style="font-family:monospace">coeff</span> to
find the discriminant of the equation.)
</li><li class="li-enumerate">For the values of <span style="font-style:italic">a</span> which give three real roots, graph each of the
roots in different colors on the same graph.
(You can use <span style="font-family:monospace">resultant</span> to find the values of <span style="font-style:italic">a</span> for which the
equation has a multiple root; these values are the possible bounds of
intervals for <span style="font-style:italic">a</span> where each of the roots is real.)
</li><li class="li-enumerate">Find the solutions for <span style="font-style:italic">a</span>=0,1,2.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;8</span>&#XA0;&#XA0;For each of the limits
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">&nbsp;</td></tr>
<tr><td class="dcell" style="text-align:center">lim</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>&#X2192;&#XA0;0</td></tr>
</table></td><td class="dcell">&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">sin(<span style="font-style:italic">x</span>)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span></td></tr>
</table></td><td class="dcell">
&#XA0;,&#XA0;
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">&nbsp;</td></tr>
<tr><td class="dcell" style="text-align:center">lim</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>&#X2192;&#XA0;0<sup>+</sup></td></tr>
</table></td><td class="dcell">&#XA0;(sin(<span style="font-style:italic">x</span>))<sup>1/<span style="font-style:italic">x</span></sup>
&#XA0;,&#XA0;
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">&nbsp;</td></tr>
<tr><td class="dcell" style="text-align:center">lim</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>&#X2192;&#XA0;+&#X221E;</td></tr>
</table></td><td class="dcell">&#XA0;(1+1/<span style="font-style:italic">x</span>)<sup><span style="font-style:italic">x</span></sup>
&#XA0;,&#XA0;
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">&nbsp;</td></tr>
<tr><td class="dcell" style="text-align:center">lim</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span>&#X2192;&#XA0;+&#X221E;</td></tr>
</table></td><td class="dcell">&#XA0;(2<sup><span style="font-style:italic">x</span></sup>+3<sup><span style="font-style:italic">x</span></sup>)<sup>1/<span style="font-style:italic">x</span></sup>
</td></tr>
</table>
<ol class="enumerate" type=1><li class="li-enumerate">
Find the exact value.
</li><li class="li-enumerate">Find a value of <span style="font-style:italic">x</span> such that the distance from <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>) to the limit is
less than 10<sup>&#X2212;3</sup>.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;9</span>&#XA0;&#XA0;For each function <span style="font-style:italic">f</span>, find ranges for the <span style="font-style:italic">x</span> coordinates and the <span style="font-style:italic">y</span>
coordinates that give the most informative graph.
<ol class="enumerate" type=1><li class="li-enumerate">
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)=1/<span style="font-style:italic">x</span>.
</li><li class="li-enumerate"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)=<span style="font-style:italic">e</span><sup><span style="font-style:italic">x</span></sup>.
</li><li class="li-enumerate"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)=1/sin(<span style="font-style:italic">x</span>).
</li><li class="li-enumerate"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)=<span style="font-style:italic">x</span>/sin(<span style="font-style:italic">x</span>).
</li><li class="li-enumerate"><span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)=sin(<span style="font-style:italic">x</span>)/<span style="font-style:italic">x</span>.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;10</span>&#XA0;&#XA0;Let
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>)=3<span style="font-style:italic">x</span><sup>2</sup>+1+1/&#X3C0;<sup>4</sup>ln((&#X3C0;&#X2212;<span style="font-style:italic">x</span>)<sup>2</sup>).
<ol class="enumerate" type=1><li class="li-enumerate">
Verify that this function takes negative values on &#X211D;<sup>+</sup>.
Graph the function over the interval [0,5].
</li><li class="li-enumerate">Find &#X454; &gt;0 such that <span style="font-family:monospace">Xcas</span> gives the correct graph of the
function over the interval [&#X3C0;&#X2212;&#X454;,&#X3C0;+&#X454;]. 
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;11</span>&#XA0;&#XA0;&#XA0;
<ol class="enumerate" type=1><li class="li-enumerate">
Graph the function exp(<span style="font-style:italic">x</span>) over the interval [&#X2212;1,1]. 
On the same graph, plot the Taylor polynomials (of orders 1,2,3 and 4)
for this function centered at <span style="font-style:italic">x</span>=0.
</li><li class="li-enumerate">Same question for the interval [1,2]. 
</li><li class="li-enumerate">Graph the function sin(<span style="font-style:italic">x</span>) on the interval [&#X2212;&#X3C0;,&#X3C0;].
On the same graph, plot the Taylor polynomials (of orders 1,3 and 5)
for this function centered at <span style="font-style:italic">x</span>=0.
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;12</span>&#XA0;&#XA0;Plot the following graphs on the same window, with <span style="font-style:italic">x</span> and <span style="font-style:italic">y</span>
coordinates from 0 to 1.
<ol class="enumerate" type=1><li class="li-enumerate">
The line <span style="font-style:italic">y</span>=<span style="font-style:italic">x</span>.
</li><li class="li-enumerate">The graph of the function <span style="font-style:italic">f</span>&#XA0;: &#XA0;<span style="font-style:italic">x</span>&#X21A6; 1/6+<span style="font-style:italic">x</span>/3+<span style="font-style:italic">x</span><sup>2</sup>/2.
</li><li class="li-enumerate">The tangent line to the graph of <span style="font-style:italic">f</span> at <span style="font-style:italic">x</span>=1.
</li><li class="li-enumerate">The vertical line segment from the <span style="font-style:italic">x</span>-axis to the point where the
graph of <span style="font-style:italic">f</span> intersects the line <span style="font-style:italic">y</span>=<span style="font-style:italic">x</span>, and a horizontal line segment
from the <span style="font-style:italic">y</span>-axis to that point of intersection.
</li><li class="li-enumerate">The labels &#X201C;fixed point&#X201D; and &#X201C;tangent&#X201D;, at the appropriate positions.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;13</span>&#XA0;&#XA0;The goal of this exercise is to graph a family of functions on the
same screen. You will need to choose the number of curves, the
interval to graph over to obtain the most informative graphic.
<ol class="enumerate" type=1><li class="li-enumerate">
Functions <span style="font-style:italic">f</span><sub><span style="font-style:italic">a</span></sub>(<span style="font-style:italic">x</span>) = <span style="font-style:italic">x</span><sup><span style="font-style:italic">a</span></sup><span style="font-style:italic">e</span><sup>&#X2212;<span style="font-style:italic">x</span></sup> for <span style="font-style:italic">a</span> from &#X2212;1 to 1. 
</li><li class="li-enumerate">Functions <span style="font-style:italic">f</span><sub><span style="font-style:italic">a</span></sub>(<span style="font-style:italic">x</span>)=1/(<span style="font-style:italic">x</span>&#X2212;<span style="font-style:italic">a</span>)<sup>2</sup> for <span style="font-style:italic">a</span> from &#X2212;1 to 1.
</li><li class="li-enumerate">Functions <span style="font-style:italic">f</span><sub><span style="font-style:italic">a</span></sub>(<span style="font-style:italic">x</span>)=sin(<span style="font-style:italic">ax</span>), for <span style="font-style:italic">a</span> from 0 to 2.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;14</span>&#XA0;&#XA0;Graph each of the following curves. You will need to choose a range
of values for the parameter to make sure you have the complete graph.
<ol class="enumerate" type=1><li class="li-enumerate">
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23A8;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">x</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >sin(<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >cos<sup>3</sup>(<span style="font-style:italic">t</span>)
</td></tr>
</table></td><td class="dcell">
</td></tr>
</table>
</li><li class="li-enumerate"><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23A8;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">x</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >sin(4&#XA0;<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >cos<sup>3</sup>(6&#XA0;<span style="font-style:italic">t</span>)
</td></tr>
</table></td><td class="dcell">
</td></tr>
</table>
</li><li class="li-enumerate"><table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23A8;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">x</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >sin(132&#XA0;<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >cos<sup>3</sup>(126&#XA0;<span style="font-style:italic">t</span>)
</td></tr>
</table></td><td class="dcell">
</td></tr>
</table>
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;15</span>&#XA0;&#XA0;The goal of this exercise is to visualize in different ways the
surface of the graph <span style="font-style:italic">z</span>=<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>)=<span style="font-style:italic">x</span>&#XA0;<span style="font-style:italic">y</span><sup>2</sup>. You will need to have a
3-d geometry window open.
<ol class="enumerate" type=1><li class="li-enumerate">
Use <span style="font-family:monospace">plotfunc</span> to draw an informative graph, choosing an
appropriate domain and number of steps.
</li><li class="li-enumerate">Create an editable parameter <span style="font-style:italic">a</span> with <span style="font-family:monospace">assume</span>. Draw the curve
<span style="font-style:italic">z</span>=<span style="font-style:italic">f</span>(<span style="font-style:italic">a</span>,<span style="font-style:italic">y</span>) and vary the parameter with the mouse.
</li><li class="li-enumerate">Create an editable parameter <span style="font-style:italic">b</span> with <span style="font-family:monospace">assume</span>. Draw the curve
<span style="font-style:italic">z</span>=<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">b</span>) and vary the parameter with the mouse.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;16</span>&#XA0;&#XA0;The goal of this exercise is to visualize a cone in different ways.
<ol class="enumerate" type=1><li class="li-enumerate">
Draw the surface given by <span style="font-style:italic">z</span>=1&#X2212;&#X221A;<span style="text-decoration:overline"><span style="font-style:italic">x</span></span><sup><span style="text-decoration:overline">2</span></sup><span style="text-decoration:overline">+</span><span style="text-decoration:overline"><span style="font-style:italic">y</span></span><sup><span style="text-decoration:overline">2</span></sup>.
</li><li class="li-enumerate">Sketch the parameterized surface defined by
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23AA;<br>
&#X23A8;<br>
&#X23AA;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">x</span>(<span style="font-style:italic">u</span>,<span style="font-style:italic">v</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">u</span>&#XA0;cos(<span style="font-style:italic">v</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span>(<span style="font-style:italic">u</span>,<span style="font-style:italic">v</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">u</span>&#XA0;sin(<span style="font-style:italic">v</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span>(<span style="font-style:italic">u</span>,<span style="font-style:italic">v</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >1&#X2212;<span style="font-style:italic">u</span>&#XA0;.
</td></tr>
</table></td><td class="dcell">
</td></tr>
</table>
</li><li class="li-enumerate">For a sufficiently large value of <span style="font-style:italic">a</span>, draw the curve parameterized by
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23AA;<br>
&#X23A8;<br>
&#X23AA;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">x</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">t</span>&#XA0;cos(<span style="font-style:italic">a</span>&#XA0;<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">t</span>&#XA0;sin(<span style="font-style:italic">a</span>&#XA0;<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >1&#X2212;<span style="font-style:italic">t</span>&#XA0;.
</td></tr>
</table></td><td class="dcell">
</td></tr>
</table></li><li class="li-enumerate">Draw the family of curves parameterized by
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X23A7;<br>
&#X23AA;<br>
&#X23A8;<br>
&#X23AA;<br>
&#X23A9;</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">x</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">a</span>&#XA0;cos(<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">y</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">a</span>&#XA0;sin(<span style="font-style:italic">t</span>)</td></tr>
<tr><td style="text-align:left;white-space:nowrap" ><span style="font-style:italic">z</span>(<span style="font-style:italic">t</span>)</td><td style="text-align:center;white-space:nowrap" >=</td><td style="text-align:left;white-space:nowrap" >1&#X2212;<span style="font-style:italic">a</span>&#XA0;.
</td></tr>
</table></td><td class="dcell">
</td></tr>
</table>
</li><li class="li-enumerate">Draw the cone using the <span style="font-family:monospace">cone</span> function.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;17</span>&#XA0;&#XA0;&#XA0;
<ol class="enumerate" type=1><li class="li-enumerate">
Generate a list &#X2113; of 100 integers randomly generated between 1 and 9.
</li><li class="li-enumerate">Verify that all values in &#X2113; are in {1,&#X2026;,9}.
</li><li class="li-enumerate">Extract from the list &#X2113; all values greater than or equal to 5.
</li><li class="li-enumerate">For each <span style="font-style:italic">k</span>=1,&#X2026;,9, find the number of values in &#X2113; which are
equal to <span style="font-style:italic">k</span>.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;18</span>&#XA0;&#XA0;For a real number <span style="font-style:italic">x</span>, the continued fraction for <span style="font-style:italic">x</span> of order <span style="font-style:italic">n</span> is
a list of integers [<span style="font-style:italic">a</span><sub>0</sub>,&#X2026;,<span style="font-style:italic">a</span><sub><span style="font-style:italic">n</span></sub>] created in the following way:
<ul class="itemize"><li class="li-itemize">
let <span style="font-style:italic">x</span><sub>0</sub> = <span style="font-style:italic">x</span>.
</li><li class="li-itemize">let <span style="font-style:italic">a</span><sub>0</sub> be the integer part of <span style="font-style:italic">x</span><sub>0</sub>.
</li><li class="li-itemize">let <span style="font-style:italic">x</span><sub>1</sub> = 1/(<span style="font-style:italic">x</span><sub>0</sub>&#X2212;<span style="font-style:italic">a</span><sub>0</sub>).
</li><li class="li-itemize">for <span style="font-style:italic">k</span>=1,&#X2026;,<span style="font-style:italic">n</span>, let <span style="font-style:italic">a</span><sub><span style="font-style:italic">k</span></sub> be the integer part of <span style="font-style:italic">x</span><sub><span style="font-style:italic">k</span></sub> and let
<span style="font-style:italic">x</span><sub><span style="font-style:italic">k</span>+1</sub> = 1/(<span style="font-style:italic">x</span><sub><span style="font-style:italic">k</span></sub> &#X2212; <span style="font-style:italic">a</span><sub><span style="font-style:italic">k</span></sub>).
</li></ul>
The list [<span style="font-style:italic">a</span><sub>0</sub>,&#X2026;,<span style="font-style:italic">a</span><sub><span style="font-style:italic">n</span></sub>] is associated with the fraction
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">u</span><sub><span style="font-style:italic">n</span></sub>&#XA0;=&#XA0;<span style="font-style:italic">a</span><sub>0</sub>+</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">a</span><sub>1</sub>+
</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">a</span><sub>2</sub>+</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell">&#X22F1;+</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">a</span><sub><span style="font-style:italic">n</span></sub></td></tr>
</table></td></tr>
</table></td></tr>
</table></td></tr>
</table></td></tr>
</table></td></tr>
</table></td></tr>
</table></td></tr>
</table>
For <span style="font-style:italic">x</span>&#X2208;{&#X3C0;,&#X221A;<span style="text-decoration:overline">2</span>, <span style="font-style:italic">e</span>} and <span style="font-style:italic">n</span>&#X2208; {5,10}&#XA0;:
<ol class="enumerate" type=1><li class="li-enumerate">
Find [<span style="font-style:italic">a</span><sub>0</sub>,&#X2026;,<span style="font-style:italic">a</span><sub><span style="font-style:italic">n</span></sub>].
</li><li class="li-enumerate">Compare your result with the value given by <span style="font-family:monospace">Xcas</span>&#X2019;s <span style="font-family:monospace">dfc</span>
function.
</li><li class="li-enumerate">Find <span style="font-style:italic">u</span><sub><span style="font-style:italic">n</span></sub> and the numeric value of <span style="font-style:italic">x</span>&#X2212;<span style="font-style:italic">u</span><sub><span style="font-style:italic">n</span></sub>.
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;19</span>&#XA0;&#XA0;Write (without using a loop) the following sequences:
<ol class="enumerate" type=1><li class="li-enumerate">
The numbers from 1 to 3 in steps of 0.1.
</li><li class="li-enumerate">The numbers from 3 to 1 in steps of &#X2212;0.1.
</li><li class="li-enumerate">The squares of the first 10 integers.
</li><li class="li-enumerate">Numbers of the form (&#X2212;1)<sup><span style="font-style:italic">n</span></sup> <span style="font-style:italic">n</span><sup>2</sup> for <span style="font-style:italic">n</span>=1,&#X2026;,10.
</li><li class="li-enumerate">10 &#X201C;0&#X201D;s followed by 10 &#X201C;1&#X201D;s.
</li><li class="li-enumerate">3 &#X201C;0&#X201D;s followed by 3 &#X201C;1&#X201D;s, followed by 3 &#X201C;2&#X201D;,&#X2026;, 
followed by 3 &#X201C;9&#X201D;s. 
</li><li class="li-enumerate">&#X201C;1&#X201D; followed by 1 &#X201C;0&#X201D;, followed by a &#X201C;2&#X201D; followed by 2 &#X201C;0&#X201D;s,
&#X2026;, followed by &#X201C;8&#X201D; followed by 8 &#X201C;0&#X201D;s, followed by &#X201C;9&#X201D;.
</li><li class="li-enumerate">1 &#X201C;1&#X201D; followed by 2 &#X201C;2&#X201D;s, followed by 3 &#X201C;3&#X201D;s,&#X2026;,
followed by 9 &#X201C;9&#X201D;s.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;20</span>&#XA0;&#XA0;&#XA0;
<ol class="enumerate" type=1><li class="li-enumerate">
Define the following polynomials of degree 6.
<ol class="enumerate" type=a><li class="li-enumerate">
A polynomial whose roots are the integers from 1 to 6.
</li><li class="li-enumerate">A polynomial whose roots are 0 (triple root), 1
(double root) and 2 (simple root).
</li><li class="li-enumerate">The polynomial (<span style="font-style:italic">x</span><sup>2</sup>&#X2212;1)<sup>3</sup>.
</li><li class="li-enumerate">The polynomial <span style="font-style:italic">x</span><sup>6</sup>&#X2212;1.
</li></ol>
</li><li class="li-enumerate">Write (without using the <span style="font-family:monospace">companion</span> function)
the companion matrix <span style="font-style:italic">A</span> for each of the polynomials in (1).
Recall the the companion matrix for the polynomial
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">P</span>=<span style="font-style:italic">x</span><sup><span style="font-style:italic">d</span></sup>+<span style="font-style:italic">a</span><sub><span style="font-style:italic">d</span>&#X2212;1</sub><span style="font-style:italic">x</span><sup><span style="font-style:italic">d</span>&#X2212;1</sup>+&#X22EF;+<span style="font-style:italic">a</span><sub>1</sub><span style="font-style:italic">x</span>+<span style="font-style:italic">a</span><sub>0</sub>&#XA0;,
</td></tr>
</table>
is
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">
<a id="compagnon"></a>
<span style="font-style:italic">A</span>&#XA0;=&#XA0;
</td><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239C;<br>
&#X239C;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell">
</td><td class="dcell"><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >1</td><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >&#X2026;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >0</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X22EE;</td><td style="text-align:center;white-space:nowrap" >&#X22F1;</td><td style="text-align:center;white-space:nowrap" >&#X22F1;</td><td style="text-align:center;white-space:nowrap" >&#X22F1;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&#X22EE;</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X22EE;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&#X22F1;</td><td style="text-align:center;white-space:nowrap" >&#X22F1;</td><td style="text-align:center;white-space:nowrap" >0</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >&#X2026;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&#X2026;</td><td style="text-align:center;white-space:nowrap" >0</td><td style="text-align:center;white-space:nowrap" >1</td></tr>
<tr><td style="text-align:center;white-space:nowrap" >&#X2212;<span style="font-style:italic">a</span><sub>0</sub></td><td style="text-align:center;white-space:nowrap" >&#X2212;<span style="font-style:italic">a</span><sub>1</sub></td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&#X2026;</td><td style="text-align:center;white-space:nowrap" >&nbsp;</td><td style="text-align:center;white-space:nowrap" >&#X2212;<span style="font-style:italic">a</span><sub><span style="font-style:italic">d</span>&#X2212;1</sub>
</td></tr>
</table></td><td class="dcell">
</td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X239F;<br>
&#X239F;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td><td class="dcell">&#XA0;.
&#XA0;&#XA0;&#XA0;&#XA0;(1)</td></tr>
</table> 
</li><li class="li-enumerate">Find the eigenvalues of the matrix <span style="font-style:italic">A</span>.
</li><li class="li-enumerate">Find the characteristic polynomial of <span style="font-style:italic">A</span>.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;21</span>&#XA0;&#XA0;&#XA0;
<ol class="enumerate" type=1><li class="li-enumerate">
For variables <span style="font-style:italic">a</span> and <span style="font-style:italic">b</span>, write the square matrix <span style="font-style:italic">A</span> of order 4 with
<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>=<span style="font-style:italic">a</span> if <span style="font-style:italic">j</span>=<span style="font-style:italic">k</span> and <span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>=<span style="font-style:italic">b</span> if <span style="font-style:italic">j</span> &#X2260; <span style="font-style:italic">k</span>.
</li><li class="li-enumerate">Find and factor the characteristic polynomial of <span style="font-style:italic">A</span>.
</li><li class="li-enumerate">Find an orthogonal matrix <span style="font-style:italic">P</span> such that <span style="font-style:italic">P</span><sup><span style="font-style:italic">T</span></sup> <span style="font-style:italic">A</span> <span style="font-style:italic">P</span> is a diagonal
matrix. 
</li><li class="li-enumerate">Use your answer to the previous question to define the function that
maps an integer <span style="font-style:italic">n</span> to the matrix <span style="font-style:italic">A</span><sup><span style="font-style:italic">n</span></sup>.
</li><li class="li-enumerate">Find <span style="font-style:italic">A</span><sup><span style="font-style:italic">k</span></sup> for <span style="font-style:italic">k</span>=1,&#X2026;,6 by finding the matrix products. Check
that the function given in the previous part gives the same results.
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;22</span>&#XA0;&#XA0;&#XA0;
<ol class="enumerate" type=1><li class="li-enumerate">
Find the square matrix <span style="font-style:italic">N</span> of order 6 given by <span style="font-style:italic">n</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>=1 if
<span style="font-style:italic">k</span>=<span style="font-style:italic">j</span>+1 and <span style="font-style:italic">n</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>=0 if <span style="font-style:italic">k</span> &#X2260; <span style="font-style:italic">j</span>+1. 
</li><li class="li-enumerate">Find <span style="font-style:italic">N</span><sup><span style="font-style:italic">p</span></sup> for <span style="font-style:italic">p</span>=1,&#X2026;,6. 
</li><li class="li-enumerate">Write the matrix <span style="font-style:italic">A</span> = <span style="font-style:italic">xI</span>+<span style="font-style:italic">N</span>, where <span style="font-style:italic">x</span> is a variable.
</li><li class="li-enumerate">Find <span style="font-style:italic">A</span><sup><span style="font-style:italic">p</span></sup> for <span style="font-style:italic">p</span>=1,&#X2026;,6. 
</li><li class="li-enumerate">Find exp(<span style="font-style:italic">At</span>) as a function of <span style="font-style:italic">x</span> and <span style="font-style:italic">t</span>&#XA0;:
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">exp(<span style="font-style:italic">At</span>)&#XA0;=&#XA0;<span style="font-style:italic">I</span>+</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">&#X221E;</td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-size:xx-large">&#X2211;</span></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">p</span>=1</td></tr>
</table></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">t</span><sup><span style="font-style:italic">p</span></sup></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center"><span style="font-style:italic">p</span>!</td></tr>
</table></td><td class="dcell">&#XA0;<span style="font-style:italic">A</span><sup><span style="font-style:italic">p</span></sup>&#XA0;.
</td></tr>
</table>
</li></ol> 
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;23</span>&#XA0;&#XA0;Define the following functions without using a loop.
<ol class="enumerate" type=1><li class="li-enumerate">
The function <span style="font-style:italic">f</span> which takes as input an integer <span style="font-style:italic">n</span> and two real
numbers <span style="font-style:italic">a</span> and <span style="font-style:italic">b</span> and returns the <span style="font-style:italic">n</span>&#XD7; <span style="font-style:italic">n</span> matrix <span style="font-style:italic">A</span> whose
diagonal terms are all equal to <span style="font-style:italic">a</span> and whose non-diagonal entries are
equal to <span style="font-style:italic">b</span>.
</li><li class="li-enumerate">The function <span style="font-style:italic">g</span> which takes as input an integer <span style="font-style:italic">n</span> and three real
numbers <span style="font-style:italic">a</span>,<span style="font-style:italic">b</span> and <span style="font-style:italic">c</span>m and returns the matrix 
<span style="font-style:italic">A</span>=(<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>)<sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span>=1,&#X2026;,<span style="font-style:italic">n</span></sub> whose diagonal elements are equal to
<span style="font-style:italic">a</span>, whose terms <span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">j</span>+1</sub> are equal to <span style="font-style:italic">b</span> and whose terms <span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>+1,<span style="font-style:italic">j</span></sub> 
are equal to <span style="font-style:italic">c</span>, and whose remaining terms are 0.
</li><li class="li-enumerate">The function <span style="font-style:italic">H</span> which takes as input an integer <span style="font-style:italic">n</span> and returns
Hilbert&#X2019;s Matrix; the matrix <span style="font-style:italic">A</span>=(<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>)<sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span>=1,&#X2026;,<span style="font-style:italic">n</span></sub> where
<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub> = 1/(<span style="font-style:italic">j</span>+<span style="font-style:italic">k</span>+1).
Compare the execution time of your function with that of the
<span style="font-family:monospace">hilbert</span> function.
</li><li class="li-enumerate">The function <span style="font-style:italic">V</span> which takes as input a vector <span style="font-style:italic">x</span>=[<span style="font-style:italic">x</span><sub>1</sub>,&#X2026;,<span style="font-style:italic">x</span><sub><span style="font-style:italic">n</span></sub>] and
returns Vandermonde&#X2019;s Matrix; the matrix
<span style="font-style:italic">A</span>=(<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>)<sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span>=1,&#X2026;,<span style="font-style:italic">n</span></sub> where <span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub> = <span style="font-style:italic">x</span><sub><span style="font-style:italic">k</span></sub><sup><span style="font-style:italic">j</span>&#X2212;1</sup>.
Compare the execution time of your function with that of the
<span style="font-family:monospace">vandermonde</span> function.
</li><li class="li-enumerate">The function <span style="font-style:italic">T</span> which takes as input a vector <span style="font-style:italic">x</span>=[<span style="font-style:italic">x</span><sub>1</sub>,&#X2026;,<span style="font-style:italic">x</span><sub><span style="font-style:italic">n</span></sub>] and
returns the Toeplitz Matrix; the matrix
<span style="font-style:italic">A</span>=(<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub>)<sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span>=1,&#X2026;,<span style="font-style:italic">n</span></sub> where
<span style="font-style:italic">a</span><sub><span style="font-style:italic">j</span>,<span style="font-style:italic">k</span></sub> = <span style="font-style:italic">x</span><sub>|<span style="font-style:italic">j</span>&#X2212;<span style="font-style:italic">k</span>|+1</sub> .
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;24</span>&#XA0;&#XA0;Write the following functions, which take as input a function <span style="font-style:italic">f</span>:&#X211D;
&#X2192; &#X211D; and three real numbers <span style="font-style:italic">x</span><sub><span style="font-style:italic">min</span></sub>, <span style="font-style:italic">x</span><sub>0</sub> and <span style="font-style:italic">x</span><sub><span style="font-style:italic">max</span></sub> with
<span style="font-style:italic">x</span><sub><span style="font-style:italic">min</span></sub>&#X2264; <span style="font-style:italic">x</span><sub>0</sub> &#X2264; <span style="font-style:italic">x</span><sub><span style="font-style:italic">max</span></sub>. 
<ol class="enumerate" type=1><li class="li-enumerate">
<span style="font-family:monospace">derive</span>&#XA0;: 
This function calculates and graphs the derivative of <span style="font-style:italic">f</span> over the
interval [<span style="font-style:italic">x</span><sub><span style="font-style:italic">min</span></sub>,<span style="font-style:italic">x</span><sub><span style="font-style:italic">max</span></sub>] and returns <span style="font-style:italic">f</span>&#X2032;(<span style="font-style:italic">x</span><sub>0</sub>).
</li><li class="li-enumerate"><span style="font-family:monospace">tangent</span>&#XA0;: 
This function graphs the function <span style="font-style:italic">f</span> on the interval
[<span style="font-style:italic">x</span><sub><span style="font-style:italic">min</span></sub>,<span style="font-style:italic">x</span><sub><span style="font-style:italic">max</span></sub>] and in the same window draws the tangent to the
graph at <span style="font-style:italic">x</span><sub>0</sub>. It returns the equation for the tangent line as a
first degree polynomial.
</li><li class="li-enumerate"><span style="font-family:monospace">araignee</span>&#XA0;:
This function graphs the function <span style="font-style:italic">f</span> on the [<span style="font-style:italic">x</span><sub><span style="font-style:italic">min</span></sub>,<span style="font-style:italic">x</span><sub><span style="font-style:italic">max</span></sub>], as
well as the line <span style="font-style:italic">y</span>=<span style="font-style:italic">x</span>. It calculates and returns the first 10
iterates of <span style="font-style:italic">f</span> starting at <span style="font-style:italic">x</span><sub>0</sub> (so <span style="font-style:italic">x</span><sub>1</sub> = <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>0</sub>), <span style="font-style:italic">x</span><sub>2</sub> =
<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>1</sub>),&#X2026;). It also draws the sequence of segments, alternately
vertical and horizontal, allowing you to visualize the iterations:
segments joining (<span style="font-style:italic">x</span><sub>0</sub>,0), (<span style="font-style:italic">x</span><sub>0</sub>,<span style="font-style:italic">x</span><sub>1</sub>), (<span style="font-style:italic">x</span><sub>1</sub>,<span style="font-style:italic">x</span><sub>1</sub>),
(<span style="font-style:italic">x</span><sub>1</sub>,<span style="font-style:italic">x</span><sub>2</sub>), (<span style="font-style:italic">x</span><sub>2</sub>,<span style="font-style:italic">x</span><sub>2</sub>), &#X2026;(compare this to the function <span style="font-family:monospace">plotseq</span>)
</li><li class="li-enumerate"><span style="font-family:monospace">newton_graph</span>&#XA0;:
This function graphs the function <span style="font-style:italic">f</span> over the interval [<span style="font-style:italic">x</span><sub><span style="font-style:italic">min</span></sub>,<span style="font-style:italic">x</span><sub><span style="font-style:italic">max</span></sub>].
It also calculates and returns the first ten iterates of the sequence
starting at <span style="font-style:italic">x</span><sub>0</sub> given by Newton&#X2019;s Method: <span style="font-style:italic">x</span><sub>1</sub>=<span style="font-style:italic">x</span><sub>0</sub> &#X2212;<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>0</sub>)/<span style="font-style:italic">f</span>&#X2032;(<span style="font-style:italic">x</span><sub>0</sub>),
<span style="font-style:italic">x</span><sub>2</sub>=<span style="font-style:italic">x</span><sub>1</sub> &#X2212; <span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>1</sub>)/<span style="font-style:italic">f</span>&#X2032;(<span style="font-style:italic">x</span><sub>1</sub>) &#X2026;&#XA0; (The values of the derivative
should be approximated.) This function also graphs in the same window
the segments displaying the iterations: segments joining
(<span style="font-style:italic">x</span><sub>0</sub>,0), (<span style="font-style:italic">x</span><sub>0</sub>,<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>0</sub>)), (<span style="font-style:italic">x</span><sub>1</sub>,0),
(<span style="font-style:italic">x</span><sub>1</sub>,<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>1</sub>)), (<span style="font-style:italic">x</span><sub>2</sub>,0), (<span style="font-style:italic">x</span><sub>2</sub>,<span style="font-style:italic">f</span>(<span style="font-style:italic">x</span><sub>2</sub>)),&#X2026;(compare with the function <span style="font-family:monospace">newton</span>)
</li></ol>
</div><div class="theorem"><span style="font-weight:bold">Exercise&#XA0;25</span>&#XA0;&#XA0;Let <span style="font-style:italic">D</span> be the unit square <span style="font-style:italic">D</span>=(0,1)<sup>2</sup>. Let &#X3A6;
be the function defined on <span style="font-style:italic">D</span> by
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell">&#X3A6;(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>)&#XA0;=&#XA0;(<span style="font-style:italic">z</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>),<span style="font-style:italic">t</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>))=
</td><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">x</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">1+<span style="font-style:italic">y</span></td></tr>
</table></td><td class="dcell">&#XA0;,&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center"><span style="font-style:italic">y</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">1+<span style="font-style:italic">x</span></td></tr>
</table></td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td><td class="dcell">&#XA0;.
</td></tr>
</table>
<ol class="enumerate" type=1><li class="li-enumerate">
Find the inverse of &#X3A6;.
</li><li class="li-enumerate">Find and graph the image under &#X3A6; of the domain <span style="font-style:italic">D</span>: &#X394;=&#X3A6;(<span style="font-style:italic">D</span>).
</li><li class="li-enumerate">Let <span style="font-style:italic">A</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>) be the Jacobian matrix of &#X3A6; at a point (<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>) in
<span style="font-style:italic">D</span>, and <span style="font-style:italic">B</span>(<span style="font-style:italic">z</span>,<span style="font-style:italic">t</span>) the Jacobian matrix of &#X3A6;<sup>&#X2212;1</sup> at a point
(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>) in &#X394;. Calculate these two matrices, and verify that 
<span style="font-style:italic">B</span>(&#X3A6;(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>)) and <span style="font-style:italic">A</span>(<span style="font-style:italic">x</span>,<span style="font-style:italic">y</span>) are inverses of each other.
</li><li class="li-enumerate">Let <span style="font-style:italic">J</span>(<span style="font-style:italic">z</span>,<span style="font-style:italic">t</span>) be the determinant of the matrix <span style="font-style:italic">B</span>. Calculate and simplify 
<span style="font-style:italic">J</span>(<span style="font-style:italic">z</span>,<span style="font-style:italic">t</span>).
</li><li class="li-enumerate">Evaluate
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">I</span><sub>1</sub>=</td><td class="dcell"><span style="font-size:xx-large">&#X222C;</span></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">&nbsp;</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left"><span style="font-style:italic">D</span></td></tr>
</table></td><td class="dcell">&#XA0;</td><td class="dcell">&#X239B;<br>
&#X239C;<br>
&#X239C;<br>
&#X239D;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1+<span style="font-style:italic">x</span>+<span style="font-style:italic">y</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell" style="text-align:center">(1+<span style="font-style:italic">x</span>)(1+<span style="font-style:italic">y</span>)</td></tr>
</table></td><td class="dcell">&#XA0;</td><td class="dcell">&#X239E;<br>
&#X239F;<br>
&#X239F;<br>
&#X23A0;</td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">3</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">&nbsp;</td></tr>
</table></td><td class="dcell">&#XA0;&#XA0;<span style="font-style:italic">dxdy</span>&#XA0;.
</td></tr>
</table> 
</li><li class="li-enumerate">Evaluate
<table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><span style="font-style:italic">I</span><sub>2</sub>=</td><td class="dcell"><span style="font-size:xx-large">&#X222C;</span></td><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:left">&nbsp;</td></tr>
<tr><td class="dcell" style="text-align:left"><br>
<br>
</td></tr>
<tr><td class="dcell" style="text-align:left">&#X394;</td></tr>
</table></td><td class="dcell">&#XA0;(1+<span style="font-style:italic">z</span>)(1+<span style="font-style:italic">t</span>)&#XA0;<span style="font-style:italic">dzdt</span>&#XA0;,
</td></tr>
</table> 
and verify that <span style="font-style:italic">I</span><sub>1</sub>=<span style="font-style:italic">I</span><sub>2</sub>.
</li></ol>
</div><!--TOC section id="sec44" Index-->
<h2 id="sec44" class="section">Index</h2><!--SEC END --><p></p><table style="border-spacing:6px;border-collapse:separate;" class="cellpading0"><tr><td style="vertical-align:top;text-align:left;" ><ul class="indexenv"><li class="li-indexenv">
<span style="font-family:monospace">()</span>, <a href="#hevea_default135">3.5</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">-&gt;</span>, <a href="#hevea_default130">3.4</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">:=</span>, <a href="#hevea_default7">1.3</a>, <a href="#hevea_default45">3.2</a>, <a href="#hevea_default51">3.2</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">=</span>, <a href="#hevea_default59">3.2</a>, <a href="#hevea_default61">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">==</span>, <a href="#hevea_default52">3.2</a>, <a href="#hevea_default54">3.2</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">[]</span>, <a href="#hevea_default136">3.5</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">%{ %}</span>, <a href="#hevea_default137">3.5</a>
<br>
<br>
</li><li class="li-indexenv">All_trig_sol, <a href="#hevea_default22">2.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">abs</span>, <a href="#hevea_default73">3.4</a>
</li><li class="li-indexenv">absolute value, <a href="#hevea_default74">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">acos</span>, <a href="#hevea_default120">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">acosh</span>, <a href="#hevea_default126">3.4</a>
</li><li class="li-indexenv">addition, <a href="#hevea_default24">3.1</a>
</li><li class="li-indexenv">and, <a href="#hevea_default346">7.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">and</span>, <a href="#hevea_default48">3.2</a>
</li><li class="li-indexenv">antiderivatives, <a href="#hevea_default189">4.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">append</span>, <a href="#hevea_default144">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">apply</span>, <a href="#hevea_default145">3.5</a>
</li><li class="li-indexenv">arc<ul class="indexenv"><li class="li-indexenv">
cosine, <a href="#hevea_default121">3.4</a>
</li><li class="li-indexenv">sine, <a href="#hevea_default119">3.4</a>
</li><li class="li-indexenv">tangent, <a href="#hevea_default123">3.4</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">arg</span>, <a href="#hevea_default92">3.4</a>
</li><li class="li-indexenv">argument, <a href="#hevea_default93">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">asc</span>, <a href="#hevea_default160">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">asin</span>, <a href="#hevea_default118">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">asinh</span>, <a href="#hevea_default124">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">assume</span>, <a href="#hevea_default47">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">atan</span>, <a href="#hevea_default122">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">atanh</span>, <a href="#hevea_default128">3.4</a>
<br>
<br>
</li><li class="li-indexenv">Bezout, <a href="#hevea_default212">5.1</a>
</li><li class="li-indexenv">Bezout&#X2019;s identity, <a href="#hevea_default249">5.2</a>
</li><li class="li-indexenv">Booleans, <a href="#hevea_default53">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">blockmatrix</span>, <a href="#hevea_default281">5.4</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">canonical_form</span>, <a href="#hevea_default224">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ceil</span>, <a href="#hevea_default84">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">char</span>, <a href="#hevea_default161">3.6</a>
</li><li class="li-indexenv">characters, <a href="#hevea_default169">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">circle</span>, <a href="#hevea_default317">6.2</a>, <a href="#hevea_default326">6.2</a>
</li><li class="li-indexenv">cloud of dots, <a href="#hevea_default325">6.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">coeff</span>, <a href="#hevea_default225">5.2</a>
</li><li class="li-indexenv">command line, <a href="#hevea_default2">1.1</a>, <a href="#hevea_default23">2.4</a>
</li><li class="li-indexenv">complex roots, <a href="#hevea_default21">2.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">concat</span>, <a href="#hevea_default163">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">cone</span>, <a href="#hevea_default340">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">conj</span>, <a href="#hevea_default90">3.4</a>
</li><li class="li-indexenv">conjugate, <a href="#hevea_default91">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">convert</span>, <a href="#hevea_default69">3.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">coordinates</span>, <a href="#hevea_default94">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">cos</span>, <a href="#hevea_default106">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">cosh</span>, <a href="#hevea_default114">3.4</a>
</li><li class="li-indexenv">cosine, <a href="#hevea_default107">3.4</a>
<ul class="indexenv"><li class="li-indexenv">
hyperbolic, <a href="#hevea_default115">3.4</a>
</li></ul>
</li><li class="li-indexenv">cotangent, <a href="#hevea_default111">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">cumSum</span>, <a href="#hevea_default142">3.5</a>
</li><li class="li-indexenv">cumulative sum, <a href="#hevea_default143">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">curl</span>, <a href="#hevea_default176">4.1</a>
</li><li class="li-indexenv">curve<ul class="indexenv"><li class="li-indexenv">
implicit, <a href="#hevea_default312">6</a>
</li><li class="li-indexenv">parametric, <a href="#hevea_default310">6</a>
</li><li class="li-indexenv">polar, <a href="#hevea_default311">6</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">cyclotomic</span>, <a href="#hevea_default252">5.2</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">Digits</span>, <a href="#hevea_default18">2.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">degree</span>, <a href="#hevea_default229">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">desolve</span>, <a href="#hevea_default194">4.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">det</span>, <a href="#hevea_default274">5.4</a>
</li><li class="li-indexenv">determinant, <a href="#hevea_default287">5.4</a>
</li><li class="li-indexenv">diagonalization, <a href="#hevea_default295">5.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">diff</span>, <a href="#hevea_default170">4.1</a>
</li><li class="li-indexenv">divergence, <a href="#hevea_default175">4.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">divergence</span>, <a href="#hevea_default174">4.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">divis</span>, <a href="#hevea_default235">5.2</a>
</li><li class="li-indexenv">division, <a href="#hevea_default27">3.1</a>
</li><li class="li-indexenv">divisors, <a href="#hevea_default236">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">divpc</span>, <a href="#hevea_default250">5.2</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">e</span>, <a href="#hevea_default32">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">egcd</span>, <a href="#hevea_default248">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">eigenvals</span>, <a href="#hevea_default298">5.6</a>
</li><li class="li-indexenv">eigenvalues, <a href="#hevea_default302">5.6</a>
</li><li class="li-indexenv">eigenvectors, <a href="#hevea_default303">5.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">eigenvects</span>, <a href="#hevea_default299">5.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">else</span>, <a href="#hevea_default345">7.1</a>, <a href="#hevea_default351">7.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">epsilon</span>, <a href="#hevea_default19">2.3</a>, <a href="#hevea_default31">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">equation</span>, <a href="#hevea_default319">6.2</a>, <a href="#hevea_default327">6.2</a>
</li><li class="li-indexenv">equations, <a href="#hevea_default60">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">evalf</span>, <a href="#hevea_default5">1.2</a>, <a href="#hevea_default13">1.6</a>, <a href="#hevea_default29">3.1</a>, <a href="#hevea_default42">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">exact</span>, <a href="#hevea_default30">3.1</a>, <a href="#hevea_default43">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">exp</span>, <a href="#hevea_default98">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">exp2trig</span>, <a href="#hevea_default267">5.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">expand</span>, <a href="#hevea_default9">1.4</a>, <a href="#hevea_default63">3.3</a>, <a href="#hevea_default218">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">expr</span>, <a href="#hevea_default168">3.6</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">False</span>, <a href="#hevea_default58">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">factor</span>, <a href="#hevea_default10">1.4</a>, <a href="#hevea_default66">3.3</a>, <a href="#hevea_default233">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">factorial</span>, <a href="#hevea_default95">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">factors</span>, <a href="#hevea_default234">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">false</span>, <a href="#hevea_default57">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">floor</span>, <a href="#hevea_default83">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">for</span>, <a href="#hevea_default352">7.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">frac</span>, <a href="#hevea_default81">3.4</a>
</li><li class="li-indexenv">fractional part, <a href="#hevea_default82">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">froot</span>, <a href="#hevea_default237">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">fsolve</span>, <a href="#hevea_default191">4.4</a>
</li><li class="li-indexenv">function<ul class="indexenv"><li class="li-indexenv">
apply to a list, <a href="#hevea_default146">3.5</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">function_diff</span>, <a href="#hevea_default171">4.1</a>
</li><li class="li-indexenv">functions, <a href="#hevea_default8">1.3</a>
<br>
<br>
</li><li class="li-indexenv">Gauss-Jordan, <a href="#hevea_default289">5.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">gcd</span>, <a href="#hevea_default209">5.1</a>, <a href="#hevea_default246">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">getDenom</span>, <a href="#hevea_default241">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">getNum</span>, <a href="#hevea_default240">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">grad</span>, <a href="#hevea_default172">4.1</a>
</li><li class="li-indexenv">gradient, <a href="#hevea_default173">4.1</a>
<br>
<br>
</li><li class="li-indexenv">Help index, <a href="#hevea_default12">1.6</a>, <a href="#hevea_default14">1.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">head</span>, <a href="#hevea_default165">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">hermite</span>, <a href="#hevea_default256">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">hessian</span>, <a href="#hevea_default178">4.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">horner</span>, <a href="#hevea_default223">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">hyp2exp</span>, <a href="#hevea_default268">5.3</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">i</span>, <a href="#hevea_default34">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">idivis</span>, <a href="#hevea_default208">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">idn</span>, <a href="#hevea_default277">5.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">egcd</span>, <a href="#hevea_default211">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">if</span>, <a href="#hevea_default344">7.1</a>, <a href="#hevea_default350">7.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ifactor</span>, <a href="#hevea_default205">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ifactors</span>, <a href="#hevea_default207">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">im</span>, <a href="#hevea_default88">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">image</span>, <a href="#hevea_default276">5.4</a>
</li><li class="li-indexenv">imaginary part, <a href="#hevea_default89">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">inf</span>, <a href="#hevea_default36">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">infinity</span>, <a href="#hevea_default35">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">int</span>, <a href="#hevea_default186">4.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">integer</span>, <a href="#hevea_default62">3.2</a>
</li><li class="li-indexenv">integrals, <a href="#hevea_default188">4.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">inter</span>, <a href="#hevea_default318">6.2</a>, <a href="#hevea_default341">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">interactive_plotode</span>, <a href="#hevea_default198">4.5</a>
</li><li class="li-indexenv">interface, <a href="#hevea_default15">2.1</a>
</li><li class="li-indexenv">intersection, <a href="#hevea_default320">6.2</a>
</li><li class="li-indexenv">inverse<ul class="indexenv"><li class="li-indexenv">
cosine hyperbolic, <a href="#hevea_default127">3.4</a>
</li><li class="li-indexenv">sine hyperbolic, <a href="#hevea_default125">3.4</a>
</li><li class="li-indexenv">tangent hyperbolic, <a href="#hevea_default129">3.4</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">iquo</span>, <a href="#hevea_default202">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">iquorem</span>, <a href="#hevea_default204">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">irem</span>, <a href="#hevea_default200">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">isprime</span>, <a href="#hevea_default213">5.1</a>
<br>
<br>
</li><li class="li-indexenv">Jordan form, <a href="#hevea_default294">5.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">jordan</span>, <a href="#hevea_default293">5.6</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">ker</span>, <a href="#hevea_default275">5.4</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">lagrange</span>, <a href="#hevea_default254">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">laguerre</span>, <a href="#hevea_default258">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">laplacian</span>, <a href="#hevea_default177">4.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">lcm</span>, <a href="#hevea_default210">5.1</a>, <a href="#hevea_default247">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">lcoeff</span>, <a href="#hevea_default230">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">legend</span>, <a href="#hevea_default313">6.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">limit</span>, <a href="#hevea_default181">4.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">linsolve</span>, <a href="#hevea_default192">4.4</a>, <a href="#hevea_default290">5.5</a>
</li><li class="li-indexenv">list, <a href="#hevea_default133">3.5</a>
<ul class="indexenv"><li class="li-indexenv">
of coefficients, <a href="#hevea_default150">3.5</a>
</li></ul>
</li></ul></td><td style="vertical-align:top;text-align:left;" ><ul class="indexenv"><li class="li-indexenv">lists, <a href="#hevea_default152">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ln</span>, <a href="#hevea_default100">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">log</span>, <a href="#hevea_default99">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">log10</span>, <a href="#hevea_default103">3.4</a>
</li><li class="li-indexenv">logarithm<ul class="indexenv"><li class="li-indexenv">
base 10, <a href="#hevea_default102">3.4</a>
</li><li class="li-indexenv">natural, <a href="#hevea_default101">3.4</a>
</li></ul>
</li><li class="li-indexenv">loop, <a href="#hevea_default353">7.1</a>
</li><li class="li-indexenv">loops, <a href="#hevea_default343">7.1</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">makematrix</span>, <a href="#hevea_default280">5.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">map</span>, <a href="#hevea_default147">3.5</a>
</li><li class="li-indexenv">matrix<ul class="indexenv"><li class="li-indexenv">
determinant, <a href="#hevea_default286">5.4</a>
</li><li class="li-indexenv">hessian, <a href="#hevea_default179">4.1</a>
</li><li class="li-indexenv">identity, <a href="#hevea_default282">5.4</a>
</li><li class="li-indexenv">image, <a href="#hevea_default285">5.4</a>
</li><li class="li-indexenv">kernel, <a href="#hevea_default284">5.4</a>
</li><li class="li-indexenv">rank, <a href="#hevea_default283">5.4</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">matrix</span>, <a href="#hevea_default279">5.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">max</span>, <a href="#hevea_default76">3.4</a>
</li><li class="li-indexenv">maximum, <a href="#hevea_default77">3.4</a>
</li><li class="li-indexenv">menu bar, <a href="#hevea_default16">2.1</a>, <a href="#hevea_default17">2.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">mid</span>, <a href="#hevea_default139">3.5</a>, <a href="#hevea_default164">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">min</span>, <a href="#hevea_default78">3.4</a>
</li><li class="li-indexenv">minimum, <a href="#hevea_default79">3.4</a>
</li><li class="li-indexenv">multiplication, <a href="#hevea_default26">3.1</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">NULL</span>, <a href="#hevea_default138">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">nextprime</span>, <a href="#hevea_default215">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">normal</span>, <a href="#hevea_default64">3.3</a>, <a href="#hevea_default217">5.2</a>
</li><li class="li-indexenv">not, <a href="#hevea_default348">7.1</a>
</li><li class="li-indexenv">number<ul class="indexenv"><li class="li-indexenv">
prime, <a href="#hevea_default214">5.1</a>
</li></ul>
</li><li class="li-indexenv">numbers<ul class="indexenv"><li class="li-indexenv">
approximate, <a href="#hevea_default4">1.2</a>, <a href="#hevea_default37">3.1</a>
</li><li class="li-indexenv">exact, <a href="#hevea_default3">1.2</a>, <a href="#hevea_default38">3.1</a>, <a href="#hevea_default39">3.1</a>
</li><li class="li-indexenv">hexadecimal, <a href="#hevea_default41">3.1</a>
</li><li class="li-indexenv">octal, <a href="#hevea_default40">3.1</a>
</li></ul>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">odesolve</span>, <a href="#hevea_default195">4.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">open_polygon</span>, <a href="#hevea_default324">6.2</a>, <a href="#hevea_default337">6.3</a>
</li><li class="li-indexenv">or, <a href="#hevea_default347">7.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">or</span>, <a href="#hevea_default49">3.2</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">parameq</span>, <a href="#hevea_default328">6.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">partfrac</span>, <a href="#hevea_default72">3.3</a>, <a href="#hevea_default243">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">pcar</span>, <a href="#hevea_default296">5.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">pcoeff</span>, <a href="#hevea_default228">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">peval</span>, <a href="#hevea_default222">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">pi</span>, <a href="#hevea_default33">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plane</span>, <a href="#hevea_default338">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plotpolar</span>, <a href="#hevea_default308">6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plot</span>, <a href="#hevea_default11">1.5</a>, <a href="#hevea_default304">6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plotfield</span>, <a href="#hevea_default196">4.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plotfunc</span>, <a href="#hevea_default305">6</a>, <a href="#hevea_default331">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plotimplicit</span>, <a href="#hevea_default309">6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plotode</span>, <a href="#hevea_default197">4.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">plotparam</span>, <a href="#hevea_default307">6</a>, <a href="#hevea_default332">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">pmin</span>, <a href="#hevea_default297">5.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">point</span>, <a href="#hevea_default315">6.2</a>, <a href="#hevea_default333">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">poly2symb</span>, <a href="#hevea_default148">3.5</a>, <a href="#hevea_default226">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">polygon</span>, <a href="#hevea_default323">6.2</a>, <a href="#hevea_default336">6.3</a>
</li><li class="li-indexenv">polygonal line, <a href="#hevea_default321">6.2</a>, <a href="#hevea_default334">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">polygonplot</span>, <a href="#hevea_default322">6.2</a>, <a href="#hevea_default335">6.3</a>
</li><li class="li-indexenv">polynomial<ul class="indexenv"><li class="li-indexenv">
characteristic, <a href="#hevea_default300">5.6</a>
</li><li class="li-indexenv">cyclotomic, <a href="#hevea_default253">5.2</a>
</li><li class="li-indexenv">Hermite, <a href="#hevea_default257">5.2</a>
</li><li class="li-indexenv">Lagrange, <a href="#hevea_default255">5.2</a>
</li><li class="li-indexenv">Laguerre, <a href="#hevea_default259">5.2</a>
</li><li class="li-indexenv">minimal, <a href="#hevea_default301">5.6</a>
</li><li class="li-indexenv">Taylor, <a href="#hevea_default220">5.2</a>
</li><li class="li-indexenv">Tchebyshev, <a href="#hevea_default261">5.2</a>
</li></ul>
</li><li class="li-indexenv">power, <a href="#hevea_default28">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">powmod</span>, <a href="#hevea_default199">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">previousprime</span>, <a href="#hevea_default216">5.1</a>
</li><li class="li-indexenv">prime factors, <a href="#hevea_default206">5.1</a>
</li><li class="li-indexenv">product<ul class="indexenv"><li class="li-indexenv">
cross, <a href="#hevea_default270">5.4</a>
</li><li class="li-indexenv">matrix, <a href="#hevea_default271">5.4</a>
</li><li class="li-indexenv">scalar, <a href="#hevea_default269">5.4</a>
</li><li class="li-indexenv">term by term, <a href="#hevea_default272">5.4</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">product</span>, <a href="#hevea_default141">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">proot</span>, <a href="#hevea_default193">4.4</a>, <a href="#hevea_default238">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">propfrac</span>, <a href="#hevea_default242">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ptayl</span>, <a href="#hevea_default219">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">purge</span>, <a href="#hevea_default50">3.2</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">quo</span>, <a href="#hevea_default244">5.2</a>
</li><li class="li-indexenv">quotient, <a href="#hevea_default203">5.1</a>
<br>
<br>
</li><li class="li-indexenv">radian, <a href="#hevea_default20">2.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">randpoly</span>, <a href="#hevea_default251">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">rank</span>, <a href="#hevea_default273">5.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ranm</span>, <a href="#hevea_default278">5.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">ratnormal</span>, <a href="#hevea_default65">3.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">re</span>, <a href="#hevea_default86">3.4</a>
</li><li class="li-indexenv">real part, <a href="#hevea_default87">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">rem</span>, <a href="#hevea_default245">5.2</a>
</li><li class="li-indexenv">remainder, <a href="#hevea_default201">5.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">romberg</span>, <a href="#hevea_default187">4.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">round</span>, <a href="#hevea_default80">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">rref</span>, <a href="#hevea_default292">5.5</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">scatterplot</span>, <a href="#hevea_default329">6.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">segment</span>, <a href="#hevea_default316">6.2</a>
</li><li class="li-indexenv">sequence, <a href="#hevea_default132">3.5</a>
</li><li class="li-indexenv">sequences, <a href="#hevea_default151">3.5</a>
</li><li class="li-indexenv">series, <a href="#hevea_default184">4.2</a>
<ul class="indexenv"><li class="li-indexenv">
Taylor, <a href="#hevea_default185">4.2</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">series</span>, <a href="#hevea_default183">4.2</a>
</li><li class="li-indexenv">set, <a href="#hevea_default134">3.5</a>
</li><li class="li-indexenv">sets, <a href="#hevea_default153">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sign</span>, <a href="#hevea_default75">3.4</a>
</li><li class="li-indexenv">simplifications, <a href="#hevea_default70">3.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">simplify</span>, <a href="#hevea_default67">3.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">simult</span>, <a href="#hevea_default291">5.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sin</span>, <a href="#hevea_default104">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sincos</span>, <a href="#hevea_default71">3.3</a>
</li><li class="li-indexenv">sine, <a href="#hevea_default105">3.4</a>
<ul class="indexenv"><li class="li-indexenv">
hyperbolic, <a href="#hevea_default113">3.4</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">sinh</span>, <a href="#hevea_default112">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">size</span>, <a href="#hevea_default162">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">solve</span>, <a href="#hevea_default190">4.4</a>
</li><li class="li-indexenv">space, <a href="#hevea_default330">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sphere</span>, <a href="#hevea_default339">6.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sqrt</span>, <a href="#hevea_default96">3.4</a>
</li><li class="li-indexenv">square root, <a href="#hevea_default97">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">string</span>, <a href="#hevea_default167">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sturmab</span>, <a href="#hevea_default239">5.2</a>
</li><li class="li-indexenv">sublist, <a href="#hevea_default158">3.5</a>
</li><li class="li-indexenv">sublists, <a href="#hevea_default156">3.5</a>
</li><li class="li-indexenv">submatrix, <a href="#hevea_default288">5.4</a>
</li><li class="li-indexenv">subsequence, <a href="#hevea_default157">3.5</a>
</li><li class="li-indexenv">subsequences, <a href="#hevea_default155">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">subst</span>, <a href="#hevea_default46">3.2</a>
</li><li class="li-indexenv">subtraction, <a href="#hevea_default25">3.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">sum</span>, <a href="#hevea_default140">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">symb2poly</span>, <a href="#hevea_default149">3.5</a>, <a href="#hevea_default227">5.2</a>
<br>
<br>
</li><li class="li-indexenv">Taylor polynomial, <a href="#hevea_default221">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">True</span>, <a href="#hevea_default56">3.2</a>
</li><li class="li-indexenv">tables, <a href="#hevea_default154">3.5</a>, <a href="#hevea_default159">3.5</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tail</span>, <a href="#hevea_default166">3.6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">cot</span>, <a href="#hevea_default110">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tan</span>, <a href="#hevea_default108">3.4</a>
</li><li class="li-indexenv">tangent, <a href="#hevea_default109">3.4</a>
<ul class="indexenv"><li class="li-indexenv">
hyperbolic, <a href="#hevea_default117">3.4</a>
</li></ul>
</li><li class="li-indexenv"><span style="font-family:monospace">tangent</span>, <a href="#hevea_default306">6</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tanh</span>, <a href="#hevea_default116">3.4</a>
</li><li class="li-indexenv"><span style="font-family:monospace">taylor</span>, <a href="#hevea_default182">4.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tchebyshev1</span>, <a href="#hevea_default260">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tchebyshev2</span>, <a href="#hevea_default262">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tcoeff</span>, <a href="#hevea_default232">5.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tcollect</span>, <a href="#hevea_default264">5.3</a>
</li><li class="li-indexenv">test, <a href="#hevea_default342">7.1</a>, <a href="#hevea_default349">7.1</a>
</li><li class="li-indexenv"><span style="font-family:monospace">texpand</span>, <a href="#hevea_default265">5.3</a>
</li><li class="li-indexenv">text, <a href="#hevea_default314">6.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tlin</span>, <a href="#hevea_default263">5.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">trig2exp</span>, <a href="#hevea_default266">5.3</a>
</li><li class="li-indexenv"><span style="font-family:monospace">true</span>, <a href="#hevea_default55">3.2</a>
</li><li class="li-indexenv"><span style="font-family:monospace">tsimplify</span>, <a href="#hevea_default68">3.3</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">unapply</span>, <a href="#hevea_default131">3.4</a>, <a href="#hevea_default180">4.1</a>
<br>
<br>
</li><li class="li-indexenv"><span style="font-family:monospace">valuation</span>, <a href="#hevea_default231">5.2</a>
</li><li class="li-indexenv">variables, <a href="#hevea_default6">1.3</a>, <a href="#hevea_default44">3.2</a>
<br>
<br>
</li><li class="li-indexenv">whole part, <a href="#hevea_default85">3.4</a>
<br>
<br>
</li><li class="li-indexenv">Xcas<ul class="indexenv"><li class="li-indexenv">
getting, <a href="#hevea_default1">1.1</a>
</li><li class="li-indexenv">starting, <a href="#hevea_default0">1.1</a>
</li></ul>
</li></ul></td></tr>
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