<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html> <!-- Created by GNU Texinfo 6.5, http://www.gnu.org/software/texinfo/ --> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <title>Rational Approximations (GNU Octave (version 5.1.0))</title> <meta name="description" content="Rational Approximations (GNU Octave (version 5.1.0))"> <meta name="keywords" content="Rational Approximations (GNU Octave (version 5.1.0))"> <meta name="resource-type" content="document"> <meta name="distribution" content="global"> <meta name="Generator" content="makeinfo"> <link href="index.html#Top" rel="start" title="Top"> <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> <link href="Arithmetic.html#Arithmetic" rel="up" title="Arithmetic"> <link href="Coordinate-Transformations.html#Coordinate-Transformations" rel="next" title="Coordinate Transformations"> <link href="Special-Functions.html#Special-Functions" rel="prev" title="Special Functions"> <style type="text/css"> <!-- a.summary-letter {text-decoration: none} blockquote.indentedblock {margin-right: 0em} blockquote.smallindentedblock {margin-right: 0em; font-size: smaller} blockquote.smallquotation {font-size: smaller} div.display {margin-left: 3.2em} div.example {margin-left: 3.2em} div.lisp {margin-left: 3.2em} div.smalldisplay {margin-left: 3.2em} div.smallexample {margin-left: 3.2em} div.smalllisp {margin-left: 3.2em} kbd {font-style: oblique} pre.display {font-family: inherit} pre.format {font-family: inherit} pre.menu-comment {font-family: serif} pre.menu-preformatted {font-family: serif} pre.smalldisplay {font-family: inherit; font-size: smaller} pre.smallexample {font-size: smaller} pre.smallformat {font-family: inherit; font-size: smaller} pre.smalllisp {font-size: smaller} span.nolinebreak {white-space: nowrap} span.roman {font-family: initial; font-weight: normal} span.sansserif {font-family: sans-serif; font-weight: normal} ul.no-bullet {list-style: none} --> </style> <link rel="stylesheet" type="text/css" href="octave.css"> </head> <body lang="en"> <a name="Rational-Approximations"></a> <div class="header"> <p> Next: <a href="Coordinate-Transformations.html#Coordinate-Transformations" accesskey="n" rel="next">Coordinate Transformations</a>, Previous: <a href="Special-Functions.html#Special-Functions" accesskey="p" rel="prev">Special Functions</a>, Up: <a href="Arithmetic.html#Arithmetic" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p> </div> <hr> <a name="Rational-Approximations-1"></a> <h3 class="section">17.7 Rational Approximations</h3> <a name="XREFrat"></a><dl> <dt><a name="index-rat"></a><em><var>s</var> =</em> <strong>rat</strong> <em>(<var>x</var>)</em></dt> <dt><a name="index-rat-1"></a><em><var>s</var> =</em> <strong>rat</strong> <em>(<var>x</var>, <var>tol</var>)</em></dt> <dt><a name="index-rat-2"></a><em>[<var>n</var>, <var>d</var>] =</em> <strong>rat</strong> <em>(…)</em></dt> <dd> <p>Find a rational approximation of <var>x</var> to within the tolerance defined by <var>tol</var>. </p> <p>If unspecified, the default tolerance is <code>1e-6 * norm (<var>x</var>(:), 1)</code>. </p> <p>When called with one output argument, return a string containing a continued fraction expansion (multiple terms). </p> <p>When called with two output arguments, return numeric matrices for the numerator and denominator of a fractional representation of <var>x</var> such that <code><var>x</var> = <var>n</var> ./ <var>d</var></code>. </p> <p>For example: </p> <div class="example"> <pre class="example">s = rat (pi) ⇒ s = 3 + 1/(7 + 1/16) [n, d] = rat (pi) ⇒ n = 355 ⇒ d = 113 n / d - pi ⇒ 0.00000026676 </pre></div> <p>Programming Note: With one output <code>rat</code> produces a string which is a continued fraction expansion. To produce a string which is a simple fraction (one numerator, one denominator) use <code>rats</code>. </p> <p><strong>See also:</strong> <a href="#XREFrats">rats</a>, <a href="Terminal-Output.html#XREFformat">format</a>. </p></dd></dl> <a name="XREFrats"></a><dl> <dt><a name="index-rats"></a><em></em> <strong>rats</strong> <em>(<var>x</var>, <var>len</var>)</em></dt> <dd><p>Convert <var>x</var> into a rational approximation represented as a string. </p> <p>The string can be converted back into a matrix as follows: </p> <div class="example"> <pre class="example">r = rats (hilb (4)); x = str2num (r) </pre></div> <p>The optional second argument defines the maximum length of the string representing the elements of <var>x</var>. By default <var>len</var> is 9. </p> <p>If the length of the smallest possible rational approximation exceeds <var>len</var>, an asterisk (*) padded with spaces will be returned instead. </p> <p><strong>See also:</strong> <a href="Terminal-Output.html#XREFformat">format</a>, <a href="#XREFrat">rat</a>. </p></dd></dl> </body> </html>