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<span id="Rational-Approximations-1"></span><h3 class="section">17.7 Rational Approximations</h3>
<span id="XREFrat"></span><dl class="def">
<dt id="index-rat"><span class="category">: </span><span><em><var>s</var> =</em> <strong>rat</strong> <em>(<var>x</var>)</em><a href='#index-rat' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-rat-1"><span class="category">: </span><span><em><var>s</var> =</em> <strong>rat</strong> <em>(<var>x</var>, <var>tol</var>)</em><a href='#index-rat-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-rat-2"><span class="category">: </span><span><em>[<var>n</var>, <var>d</var>] =</em> <strong>rat</strong> <em>(…)</em><a href='#index-rat-2' class='copiable-anchor'> ¶</a></span></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>
<span id="XREFrats"></span><dl class="def">
<dt id="index-rats"><span class="category">: </span><span><em><var>s</var> =</em> <strong>rats</strong> <em>(<var>x</var>)</em><a href='#index-rats' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-rats-1"><span class="category">: </span><span><em><var>s</var> =</em> <strong>rats</strong> <em>(<var>x</var>, <var>len</var>)</em><a href='#index-rats-1' class='copiable-anchor'> ¶</a></span></dt>
<dd><p>Convert <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>N</var> and denominator <var>D</var> such that
<code><var>x</var> = <var>N</var>/<var>D</var></code>.
</p>
<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>Example conversion from matrix to string, and back again.
</p>
<div class="example">
<pre class="example">r = rats (hilb (4));
x = str2num (r)
</pre></div>
<p><strong>See also:</strong> <a href="#XREFrat">rat</a>, <a href="Terminal-Output.html#XREFformat">format</a>.
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