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<h3 class="section" id="Rational-Approximations-1"><span>17.7 Rational Approximations<a class="copiable-link" href="#Rational-Approximations-1"> ¶</a></span></h3>
<a class="anchor" id="XREFrat"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rat"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">rat</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-rat"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rat-1"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">rat</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">tol</var>)</code><a class="copiable-link" href="#index-rat-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rat-2"><span><code class="def-type">[<var class="var">n</var>, <var class="var">d</var>] =</code> <strong class="def-name">rat</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-rat-2"> ¶</a></span></dt>
<dd>
<p>Find a rational approximation of <var class="var">x</var> to within the tolerance defined by
<var class="var">tol</var>.
</p>
<p>If unspecified, the default tolerance is <code class="code">1e-6 * norm (<var class="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 class="var">x</var> such
that <code class="code"><var class="var">x</var> = <var class="var">n</var> ./ <var class="var">d</var></code>.
</p>
<p>For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">s = rat (pi)
⇒ s = 3 + 1/(7 + 1/16)
[n, d] = rat (pi)
⇒ n = 355
⇒ d = 113
n / d - pi
⇒ 2.6676e-07
</pre></div></div>
<p>Complex inputs are similar:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">s = rat (0.5 + i * pi)
⇒ s = complex (1 + 1/(-2), 3 + 1/(7 + 1/16))
[n, d] = rat (0.5 + i * pi)
⇒ n = 113 + 710i
⇒ d = 226
n / d - (0.5 + i * pi)
⇒ 0 + 2.6676e-07i
</pre></div></div>
<p>Programming Notes:
</p>
<p>1. With one output <code class="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 class="code">rats</code>.
</p>
<p>2. The string output produced by <code class="code">rat</code> can be passed to <code class="code">eval</code>
to get back the original input up to the tolerance used.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrats">rats</a>, <a class="ref" href="Terminal-Output.html#XREFformat">format</a>.
</p></dd></dl>
<a class="anchor" id="XREFrats"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rats"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">rats</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-rats"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-rats-1"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">rats</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">len</var>)</code><a class="copiable-link" href="#index-rats-1"> ¶</a></span></dt>
<dd><p>Convert <var class="var">x</var> into a rational approximation represented as a string.
</p>
<p>A rational approximation to a floating point number is a simple fraction
with numerator <var class="var">N</var> and denominator <var class="var">D</var> such that
<code class="code"><var class="var">x</var> = <var class="var">N</var>/<var class="var">D</var></code>.
</p>
<p>The optional second argument defines the maximum length of the string
representing the elements of <var class="var">x</var>. By default, <var class="var">len</var> is 13.
</p>
<p>If the length of the smallest possible rational approximation exceeds
<var class="var">len</var>, an asterisk (*) padded with spaces will be returned instead.
</p>
<p>Example conversion from matrix to string, and back again.
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">r = rats (hilb (4));
x = str2num (r)
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrat">rat</a>, <a class="ref" href="Terminal-Output.html#XREFformat">format</a>.
</p></dd></dl>
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