File: maxima_23.html

package info (click to toggle)
maxima 5.10.0-6
links: PTS
area: main
in suites: etch, etch-m68k
size: 44,268 kB
ctags: 17,987
sloc: lisp: 152,894; fortran: 14,667; perl: 14,204; tcl: 10,103; sh: 3,376; makefile: 2,202; ansic: 471; awk: 7
file content (794 lines) | stat: -rw-r--r-- 33,502 bytes
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html401/loose.dtd">
<html>
<!-- Created on September, 20 2006 by texi2html 1.76 -->
<!--
Written by: Lionel Cons <Lionel.Cons@cern.ch> (original author)
            Karl Berry  <karl@freefriends.org>
            Olaf Bachmann <obachman@mathematik.uni-kl.de>
            and many others.
Maintained by: Many creative people <dev@texi2html.cvshome.org>
Send bugs and suggestions to <users@texi2html.cvshome.org>

-->
<head>
<title>Maxima Manual: 23. Numerical</title>

<meta name="description" content="Maxima Manual: 23. Numerical">
<meta name="keywords" content="Maxima Manual: 23. Numerical">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="texi2html 1.76">
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
pre.display {font-family: serif}
pre.format {font-family: serif}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: serif; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: serif; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.sansserif {font-family:sans-serif; font-weight:normal;}
ul.toc {list-style: none}
body
{
    color: black;
    background: white; 
    margin-left: 8%;
    margin-right: 13%;
}

h1
{
    margin-left: +8%;
    font-size: 150%;
    font-family: sans-serif
}

h2
{
    font-size: 125%;
    font-family: sans-serif
}

h3
{
    font-size: 100%;
    font-family: sans-serif
}

h2,h3,h4,h5,h6 { margin-left: +4%; }

div.textbox
{
    border: solid;
    border-width: thin;
    /* width: 100%; */
    padding-top: 1em;
    padding-bottom: 1em;
    padding-left: 2em;
    padding-right: 2em
}

div.titlebox
{
    border: none;
    padding-top: 1em;
    padding-bottom: 1em;
    padding-left: 2em;
    padding-right: 2em;
    background: rgb(200,255,255);
    font-family: sans-serif
}

div.synopsisbox
{
    border: none;
    padding-top: 1em;
    padding-bottom: 1em;
    padding-left: 2em;
    padding-right: 2em;
     background: rgb(255,220,255);
    /*background: rgb(200,255,255); */
    /* font-family: fixed */
}

pre.example
{
    border: none;
    padding-top: 1em;
    padding-bottom: 1em;
    padding-left: 1em;
    padding-right: 1em;
    background: rgb(247,242,180); /* kind of sandy */
    /* background: rgb(200,255,255); */ /* sky blue */
    font-family: "Lucida Console", monospace
}

div.spacerbox
{
    border: none;
    padding-top: 2em;
    padding-bottom: 2em
}

div.image
{
    margin: 0;
    padding: 1em;
    text-align: center;
}
-->
</style>

<link rel="icon" href="http://maxima.sourceforge.net/favicon.ico"/>
</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">

<a name="Numerical"></a>
<a name="SEC76"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="maxima_22.html#SEC75" title="Previous section in reading order"> &lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC77" title="Next section in reading order"> &gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima_22.html#SEC74" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next chapter"> &gt;&gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h1 class="chapter"> 23. Numerical </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC77">23.1 Introduction to Numerical</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">   
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC78">23.2 Fourier packages</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">                     
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC79">23.3 Definitions for Numerical</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">   
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC80">23.4 Definitions for Fourier Series</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
</table>

<hr size="6">
<a name="Introduction-to-Numerical"></a>
<a name="SEC77"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC76" title="Previous section in reading order"> &lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC78" title="Next section in reading order"> &gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="#SEC76" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC76" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next chapter"> &gt;&gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 23.1 Introduction to Numerical </h2>

<hr size="6">
<a name="Fourier-packages"></a>
<a name="SEC78"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC77" title="Previous section in reading order"> &lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC79" title="Next section in reading order"> &gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="#SEC76" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC76" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next chapter"> &gt;&gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 23.2 Fourier packages </h2>
<p>The <code>fft</code> package comprises functions for the numerical (not symbolic) computation
of the fast Fourier transform.
<code>load (&quot;fft&quot;)</code> loads this package.
See <code>fft</code>.
</p>
<p>The <code>fourie</code> package comprises functions for the symbolic computation
of Fourier series.
<code>load (&quot;fourie&quot;)</code> loads this package.
There are functions in the <code>fourie</code> package to calculate Fourier integral
coefficients and some functions for manipulation of expressions.
See <code>Definitions for Fourier Series</code>.
</p>

<hr size="6">
<a name="Definitions-for-Numerical"></a>
<a name="SEC79"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC78" title="Previous section in reading order"> &lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC80" title="Next section in reading order"> &gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="#SEC76" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC76" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next chapter"> &gt;&gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 23.3 Definitions for Numerical </h2>

<dl>
<dt><u>Function:</u> <b>polartorect</b><i> (<var>magnitude_array</var>, <var>phase_array</var>)</i>
<a name="IDX698"></a>
</dt>
<dd><p>Translates complex values of the form <code>r %e^(%i t)</code> to the form <code>a + b %i</code>.
<code>load (&quot;fft&quot;)</code> loads this function into Maxima. See also <code>fft</code>.
</p>
<p>The magnitude and phase, <code>r</code> and <code>t</code>, are taken from <var>magnitude_array</var> and
<var>phase_array</var>, respectively. The original values of the input arrays are
replaced by the real and imaginary parts, <code>a</code> and <code>b</code>, on return. The outputs are
calculated as
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">a: r cos (t)
b: r sin (t)
</pre></td></tr></table>
<p>The input arrays must be the same size and 1-dimensional.
The array size need not be a power of 2.
</p>
<p><code>polartorect</code> is the inverse function of <code>recttopolar</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>recttopolar</b><i> (<var>real_array</var>, <var>imaginary_array</var>)</i>
<a name="IDX699"></a>
</dt>
<dd><p>Translates complex values of the form <code>a + b %i</code> to the form <code>r %e^(%i t)</code>.
<code>load (&quot;fft&quot;)</code> loads this function into Maxima. See also <code>fft</code>.
</p>
<p>The real and imaginary parts, <code>a</code> and <code>b</code>, are taken from <var>real_array</var> and
<var>imaginary_array</var>, respectively. The original values of the input arrays
are replaced by the magnitude and angle, <code>r</code> and <code>t</code>, on return. The outputs are
calculated as
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">r: sqrt (a^2 + b^2)
t: atan2 (b, a)
</pre></td></tr></table>
<p>The computed angle is in the range <code>-%pi</code> to <code>%pi</code>. 
</p>
<p>The input arrays must be the same size and 1-dimensional.
The array size need not be a power of 2.
</p>
<p><code>recttopolar</code> is the inverse function of <code>polartorect</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ift</b><i> (<var>real_array</var>, <var>imaginary_array</var>)</i>
<a name="IDX700"></a>
</dt>
<dd><p>Fast inverse discrete Fourier transform. <code>load (&quot;fft&quot;)</code> loads this function
into Maxima.
</p>
<p><code>ift</code> carries out the inverse complex fast Fourier transform on
1-dimensional floating point arrays. The inverse transform is defined as
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">x[j]: sum (y[j] exp (+2 %i %pi j k / n), k, 0, n-1)
</pre></td></tr></table>
<p>See <code>fft</code> for more details.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fft</b><i> (<var>real_array</var>, <var>imaginary_array</var>)</i>
<a name="IDX701"></a>
</dt>
<dt><u>Function:</u> <b>ift</b><i> (<var>real_array</var>, <var>imaginary_array</var>)</i>
<a name="IDX702"></a>
</dt>
<dt><u>Function:</u> <b>recttopolar</b><i> (<var>real_array</var>, <var>imaginary_array</var>)</i>
<a name="IDX703"></a>
</dt>
<dt><u>Function:</u> <b>polartorect</b><i> (<var>magnitude_array</var>, <var>phase_array</var>)</i>
<a name="IDX704"></a>
</dt>
<dd><p>Fast Fourier transform and related functions. <code>load (&quot;fft&quot;)</code>
loads these functions into Maxima.
</p>
<p><code>fft</code> and <code>ift</code> carry out the complex fast Fourier transform and
inverse transform, respectively, on 1-dimensional floating
point arrays. The size of <var>imaginary_array</var> must equal the size of <var>real_array</var>.
</p>
<p><code>fft</code> and <code>ift</code> operate in-place. That is, on return from <code>fft</code> or <code>ift</code>,
the original content of the input arrays is replaced by the output.
The <code>fillarray</code> function can make a copy of an array, should it
be necessary.
</p>
<p>The discrete Fourier transform and inverse transform are defined
as follows. Let <code>x</code> be the original data, with
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">x[i]: real_array[i] + %i imaginary_array[i]
</pre></td></tr></table>  
<p>Let <code>y</code> be the transformed data. The forward and inverse transforms are
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">y[k]: (1/n) sum (x[j] exp (-2 %i %pi j k / n), j, 0, n-1)

x[j]:       sum (y[j] exp (+2 %i %pi j k / n), k, 0, n-1)
</pre></td></tr></table>
<p>Suitable arrays can be allocated by the <code>array</code> function. For example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">array (my_array, float, n-1)$
</pre></td></tr></table>
<p>declares a 1-dimensional array with n elements, indexed from 0 through
n-1 inclusive. The number of elements n must be equal to 2^m for some m.
</p>
<p><code>fft</code> can be applied to real data (imaginary array all zeros) to obtain
sine and cosine coefficients. After calling <code>fft</code>, the sine and cosine
coefficients, say <code>a</code> and <code>b</code>, can be calculated as
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">a[0]: real_array[0]
b[0]: 0
</pre></td></tr></table>
<p>and
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">a[j]: real_array[j] + real_array[n-j]
b[j]: imaginary_array[j] - imaginary_array[n-j]
</pre></td></tr></table>
<p>for j equal to 1 through n/2-1, and
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">a[n/2]: real_array[n/2]
b[n/2]: 0
</pre></td></tr></table>
<p><code>recttopolar</code> translates complex values of the form <code>a + b %i</code> to
the form <code>r %e^(%i t)</code>. See <code>recttopolar</code>.
</p>
<p><code>polartorect</code> translates complex values of the form <code>r %e^(%i t)</code>
to the form <code>a + b %i</code>. See <code>polartorect</code>.
</p>
<p><code>demo (&quot;fft&quot;)</code> displays a demonstration of the <code>fft</code> package.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>fortindent</b>
<a name="IDX705"></a>
</dt>
<dd><p>Default value: 0
</p>
<p><code>fortindent</code> controls the left margin indentation of
expressions printed out by the <code>fortran</code> command.  0 gives normal
printout (i.e., 6 spaces), and positive values will causes the
expressions to be printed farther to the right.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fortran</b><i> (<var>expr</var>)</i>
<a name="IDX706"></a>
</dt>
<dd><p>Prints <var>expr</var> as a Fortran statement.
The output line is indented with spaces.
If the line is too long, <code>fortran</code> prints continuation lines.
<code>fortran</code> prints the exponentiation operator <code>^</code> as <code>**</code>,
and prints a complex number <code>a + b %i</code> in the form <code>(a,b)</code>.
</p>
<p><var>expr</var> may be an equation. If so, <code>fortran</code> prints an assignment
statement, assigning the right-hand side of the equation to the left-hand side.
In particular, if the right-hand side of <var>expr</var> is the name of a matrix,
then <code>fortran</code> prints an assignment statement for each element of the matrix.
</p>
<p>If <var>expr</var> is not something recognized by <code>fortran</code>,
the expression is printed in <code>grind</code> format without complaint.
<code>fortran</code> does not know about lists, arrays, or functions.
</p>
<p><code>fortindent</code> controls the left margin of the printed lines.
0 is the normal margin (i.e., indented 6 spaces). Increasing <code>fortindent</code>
causes expressions to be printed further to the right.
</p>
<p>When <code>fortspaces</code> is <code>true</code>, <code>fortran</code> fills out
each printed line with spaces to 80 columns.
</p>
<p><code>fortran</code> evaluates its arguments;
quoting an argument defeats evaluation.
<code>fortran</code> always returns <code>done</code>.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: (a + b)^12$
(%i2) fortran (expr);
      (b+a)**12                                                                 
(%o2)                         done
(%i3) fortran ('x=expr);
      x = (b+a)**12                                                             
(%o3)                         done
(%i4) fortran ('x=expand (expr));
      x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792        
     1   *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b        
     2   **3+66*a**10*b**2+12*a**11*b+a**12                                     
(%o4)                         done
(%i5) fortran ('x=7+5*%i);
      x = (7,5)                                                                 
(%o5)                         done
(%i6) fortran ('x=[1,2,3,4]);
      x = [1,2,3,4]                                                             
(%o6)                         done
(%i7) f(x) := x^2$
(%i8) fortran (f);
      f                                                                         
(%o8)                         done
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>fortspaces</b>
<a name="IDX707"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>fortspaces</code> is <code>true</code>, <code>fortran</code> fills out
each printed line with spaces to 80 columns.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>horner</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX708"></a>
</dt>
<dt><u>Function:</u> <b>horner</b><i> (<var>expr</var>)</i>
<a name="IDX709"></a>
</dt>
<dd><p>Returns a rearranged representation of <var>expr</var> as
in Horner's rule, using <var>x</var> as the main variable if it is specified.
<code>x</code> may be omitted in which case the main variable of the canonical rational expression
form of <var>expr</var> is used.
</p>
<p><code>horner</code> sometimes improves stability if <code>expr</code> is
to be numerically evaluated.  It is also useful if Maxima is used to
generate programs to be run in Fortran. See also <code>stringout</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
                           2
(%o1)            1.0E-155 x  - 5.5 x + 5.2E+155
(%i2) expr2: horner (%, x), keepfloat: true;
(%o2)            (1.0E-155 x - 5.5) x + 5.2E+155
(%i3) ev (expr, x=1e155);
Maxima encountered a Lisp error:

 floating point overflow

Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
(%i4) ev (expr2, x=1e155);
(%o4)                       7.0E+154
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>find_root</b><i> (<var>f</var>(<var>x</var>), <var>x</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX710"></a>
</dt>
<dt><u>Function:</u> <b>find_root</b><i> (<var>f</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX711"></a>
</dt>
<dd><p>Finds the zero of function <var>f</var> as variable <var>x</var> varies over the range <code>[<var>a</var>, <var>b</var>]</code>.
The function must have a
different sign at each endpoint.  If this condition is not met, the
action of the function is governed by <code>find_root_error</code>.  If
<code>find_root_error</code> is <code>true</code> then an error occurs, otherwise the value of
<code>find_root_error</code> is returned (thus for plotting <code>find_root_error</code> might be set to
0.0).  Otherwise (given that Maxima can evaluate the first argument
in the specified range, and that it is continuous) <code>find_root</code> is
guaranteed to come up with the zero (or one of them if there is more
than one zero).  The accuracy of <code>find_root</code> is governed by
<code>find_root_abs</code> and <code>find_root_rel</code> which must be non-negative floating
point numbers.  <code>find_root</code> will stop when the first arg evaluates to
something less than or equal to <code>find_root_abs</code> or if successive
approximants to the root differ by no more than <code>find_root_rel * &lt;one of the approximants&gt;</code>.
The default values of <code>find_root_abs</code> and <code>find_root_rel</code> are
0.0 so <code>find_root</code> gets as good an answer as is possible with the
single precision arithmetic we have.  The first arg may be an
equation.  The order of the last two args is irrelevant.  Thus
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">find_root (sin(x) = x/2, x, %pi, 0.1);
</pre></td></tr></table>
<p>is equivalent to
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">find_root (sin(x) = x/2, x, 0.1, %pi);
</pre></td></tr></table>
<p>The method used is a binary search in the range specified by the last
two args.  When it thinks the function is close enough to being
linear, it starts using linear interpolation.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) f(x) := sin(x) - x/2;
                                        x
(%o1)                  f(x) := sin(x) - -
                                        2
(%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
(%o2)                   1.895494267033981
(%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
(%o3)                   1.895494267033981
(%i4) find_root (f(x), x, 0.1, %pi);
(%o4)                   1.895494267033981
(%i5) find_root (f, 0.1, %pi);
(%o5)                   1.895494267033981
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>find_root_abs</b>
<a name="IDX712"></a>
</dt>
<dd><p>Default value: 0.0
</p>
<p><code>find_root_abs</code> is the accuracy of the <code>find_root</code> command is
governed by <code>find_root_abs</code> and <code>find_root_rel</code> which must be
non-negative floating point numbers.  <code>find_root</code> will stop when the
first arg evaluates to something less than or equal to <code>find_root_abs</code> or if
successive approximants to the root differ by no more than <code>find_root_rel * &lt;one of the approximants&gt;</code>.
The default values of <code>find_root_abs</code> and
<code>find_root_rel</code> are 0.0 so <code>find_root</code> gets as good an answer as is possible
with the single precision arithmetic we have.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>find_root_error</b>
<a name="IDX713"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>find_root_error</code> governs the behavior of <code>find_root</code>.
When <code>find_root</code> is called, it determines whether or not the function
to be solved satisfies the condition that the values of the
function at the endpoints of the interpolation interval are opposite
in sign.  If they are of opposite sign, the interpolation proceeds.
If they are of like sign, and <code>find_root_error</code> is <code>true</code>, then an error is
signaled.  If they are of like sign and <code>find_root_error</code> is not <code>true</code>, the
value of <code>find_root_error</code> is returned.  Thus for plotting, <code>find_root_error</code>
might be set to 0.0.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>find_root_rel</b>
<a name="IDX714"></a>
</dt>
<dd><p>Default value: 0.0
</p>
<p><code>find_root_rel</code> is the accuracy of the <code>find_root</code> command is
governed by <code>find_root_abs</code> and <code>find_root_rel</code> which must be
non-negative floating point numbers.  <code>find_root</code> will stop when the
first arg evaluates to something less than or equal to <code>find_root_abs</code> or if
successive approximants to the root differ by no more than <code>find_root_rel * &lt;one of the approximants&gt;</code>.
The default values of <code>find_root_abs</code> and
<code>find_root_rel</code> are 0.0 so <code>find_root</code> gets as good an answer as is possible
with the single precision arithmetic we have.
</p>
</dd></dl>

<hr size="6">
<a name="Definitions-for-Fourier-Series"></a>
<a name="SEC80"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC79" title="Previous section in reading order"> &lt; </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next section in reading order"> &gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="#SEC76" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
<td valign="middle" align="left">[<a href="#SEC76" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next chapter"> &gt;&gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 23.4 Definitions for Fourier Series </h2>

<dl>
<dt><u>Function:</u> <b>equalp</b><i> (<var>x</var>, <var>y</var>)</i>
<a name="IDX715"></a>
</dt>
<dd><p>Returns <code>true</code> if <code>equal (<var>x</var>, <var>y</var>)</code> otherwise <code>false</code> (doesn't give an
error message like <code>equal (x, y)</code> would do in this case).
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>remfun</b><i> (<var>f</var>, <var>expr</var>)</i>
<a name="IDX716"></a>
</dt>
<dt><u>Function:</u> <b>remfun</b><i> (<var>f</var>, <var>expr</var>, <var>x</var>)</i>
<a name="IDX717"></a>
</dt>
<dd><p><code>remfun (<var>f</var>, <var>expr</var>)</code>
replaces all occurrences of <code><var>f</var> (<var>arg</var>)</code> by <var>arg</var> in <var>expr</var>.
</p>
<p><code>remfun (<var>f</var>, <var>expr</var>, <var>x</var>)</code>
replaces all occurrences of <code><var>f</var> (<var>arg</var>)</code> by <var>arg</var> in <var>expr</var>
only if <var>arg</var> contains the variable <var>x</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>funp</b><i> (<var>f</var>, <var>expr</var>)</i>
<a name="IDX718"></a>
</dt>
<dt><u>Function:</u> <b>funp</b><i> (<var>f</var>, <var>expr</var>, <var>x</var>)</i>
<a name="IDX719"></a>
</dt>
<dd><p><code>funp (<var>f</var>, <var>expr</var>)</code>
returns <code>true</code> if <var>expr</var> contains the function <var>f</var>.
</p>
<p><code>funp (<var>f</var>, <var>expr</var>, <var>x</var>)</code>
returns <code>true</code> if <var>expr</var> contains the function <var>f</var> and the variable
<var>x</var> is somewhere in the argument of one of the instances of <var>f</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>absint</b><i> (<var>f</var>, <var>x</var>, <var>halfplane</var>)</i>
<a name="IDX720"></a>
</dt>
<dt><u>Function:</u> <b>absint</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX721"></a>
</dt>
<dt><u>Function:</u> <b>absint</b><i> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX722"></a>
</dt>
<dd><p><code>absint (<var>f</var>, <var>x</var>, <var>halfplane</var>)</code>
returns the indefinite integral of <var>f</var> with respect to
<var>x</var> in the given halfplane (<code>pos</code>, <code>neg</code>, or <code>both</code>).
<var>f</var> may contain expressions of the form
<code>abs (x)</code>, <code>abs (sin (x))</code>, <code>abs (a) * exp (-abs (b) * abs (x))</code>.
</p>
<p><code>absint (<var>f</var>, <var>x</var>)</code> is equivalent to <code>absint (<var>f</var>, <var>x</var>, pos)</code>.
</p>
<p><code>absint (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>)</code>
returns the definite integral of <var>f</var> with respect to <var>x</var> from <var>a</var> to <var>b</var>.
<var>f</var> may include absolute values.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fourier</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX723"></a>
</dt>
<dd><p>Returns a list of the Fourier coefficients of <code><var>f</var>(<var>x</var>)</code> defined
on the interval <code>[-%pi, %pi]</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>foursimp</b><i> (<var>l</var>)</i>
<a name="IDX724"></a>
</dt>
<dd><p>Simplifies <code>sin (n %pi)</code> to 0 if <code>sinnpiflag</code> is <code>true</code> and
<code>cos (n %pi)</code> to <code>(-1)^n</code> if <code>cosnpiflag</code> is <code>true</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>sinnpiflag</b>
<a name="IDX725"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>See <code>foursimp</code>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>cosnpiflag</b>
<a name="IDX726"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>See <code>foursimp</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fourexpand</b><i> (<var>l</var>, <var>x</var>, <var>p</var>, <var>limit</var>)</i>
<a name="IDX727"></a>
</dt>
<dd><p>Constructs and returns the Fourier series from the list of
Fourier coefficients <var>l</var> up through <var>limit</var> terms (<var>limit</var>
may be <code>inf</code>). <var>x</var> and <var>p</var> have same meaning as in
<code>fourier</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fourcos</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX728"></a>
</dt>
<dd><p>Returns the Fourier cosine coefficients for <code><var>f</var>(<var>x</var>)</code> defined on <code>[0, %pi]</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>foursin</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX729"></a>
</dt>
<dd><p>Returns the Fourier sine coefficients for <code><var>f</var>(<var>x</var>)</code> defined on <code>[0, %pi]</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>totalfourier</b><i> (<var>f</var>, <var>x</var>, <var>p</var>)</i>
<a name="IDX730"></a>
</dt>
<dd><p>Returns <code>fourexpand (foursimp (fourier (<var>f</var>, <var>x</var>, <var>p</var>)), <var>x</var>, <var>p</var>, 'inf)</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fourint</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX731"></a>
</dt>
<dd><p>Constructs and returns a list of the Fourier integral coefficients of <code><var>f</var>(<var>x</var>)</code>
defined on <code>[minf, inf]</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fourintcos</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX732"></a>
</dt>
<dd><p>Returns the Fourier cosine integral coefficients for <code><var>f</var>(<var>x</var>)</code> on <code>[0, inf]</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fourintsin</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX733"></a>
</dt>
<dd><p>Returns the Fourier sine integral coefficients for <code><var>f</var>(<var>x</var>)</code> on <code>[0, inf]</code>.
</p>
</dd></dl>

<hr size="6">
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC76" title="Beginning of this chapter or previous chapter"> &lt;&lt; </a>]</td>
<td valign="middle" align="left">[<a href="maxima_24.html#SEC81" title="Next chapter"> &gt;&gt; </a>]</td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left"> &nbsp; </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<p>
 <font size="-1">
  This document was generated by <em>Robert Dodier</em> on <em>September, 20 2006</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>.
 </font>
 <br>

</p>
</body>
</html>