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Next: <a rel="next" accesskey="n" href="Finding-Roots.html#Finding-Roots">Finding Roots</a>,
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<h3 class="section">27.1 Evaluating Polynomials</h3>
<p>The value of a polynomial represented by the vector <var>c</var> can be evaluated
at the point <var>x</var> very easily, as the following example shows:
<pre class="example"> N = length(c)-1;
val = dot( x.^(N:-1:0), c );
</pre>
<p class="noindent">While the above example shows how easy it is to compute the value of a
polynomial, it isn't the most stable algorithm. With larger polynomials
you should use more elegant algorithms, such as Horner's Method, which
is exactly what the Octave function <code>polyval</code> does.
<p>In the case where <var>x</var> is a square matrix, the polynomial given by
<var>c</var> is still well-defined. As when <var>x</var> is a scalar the obvious
implementation is easily expressed in Octave, but also in this case
more elegant algorithms perform better. The <code>polyvalm</code> function
provides such an algorithm.
<!-- ./polynomial/polyval.m -->
<p><a name="doc_002dpolyval"></a>
<div class="defun">
— Function File: <var>y</var> = <b>polyval</b> (<var>p, x</var>)<var><a name="index-polyval-2021"></a></var><br>
— Function File: <var>y</var> = <b>polyval</b> (<var>p, x, </var>[]<var>, mu</var>)<var><a name="index-polyval-2022"></a></var><br>
<blockquote><p>Evaluate the polynomial at of the specified values for <var>x</var>. When <var>mu</var>
is present evaluate the polynomial for (<var>x</var>-<var>mu</var>(1))/<var>mu</var>(2).
If <var>x</var> is a vector or matrix, the polynomial is evaluated for each of
the elements of <var>x</var>.
— Function File: [<var>y</var>, <var>dy</var>] = <b>polyval</b> (<var>p, x, s</var>)<var><a name="index-polyval-2023"></a></var><br>
— Function File: [<var>y</var>, <var>dy</var>] = <b>polyval</b> (<var>p, x, s, mu</var>)<var><a name="index-polyval-2024"></a></var><br>
<blockquote><p>In addition to evaluating the polynomial, the second output
represents the prediction interval, <var>y</var> +/- <var>dy</var>, which
contains at least 50% of the future predictions. To calculate the
prediction interval, the structured variable <var>s</var>, originating
form `polyfit', must be present.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpolyfit.html#doc_002dpolyfit">polyfit</a>, <a href="doc_002dpolyvalm.html#doc_002dpolyvalm">polyvalm</a>, <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>, <a href="doc_002dfilter.html#doc_002dfilter">filter</a>, <a href="doc_002dpolyderiv.html#doc_002dpolyderiv">polyderiv</a>, <a href="doc_002dpolyinteg.html#doc_002dpolyinteg">polyinteg</a>.
</p></blockquote></div>
<!-- ./polynomial/polyvalm.m -->
<p><a name="doc_002dpolyvalm"></a>
<div class="defun">
— Function File: <b>polyvalm</b> (<var>c, x</var>)<var><a name="index-polyvalm-2025"></a></var><br>
<blockquote><p>Evaluate a polynomial in the matrix sense.
<p><code>polyvalm (</code><var>c</var><code>, </code><var>x</var><code>)</code> will evaluate the polynomial in the
matrix sense, i.e., matrix multiplication is used instead of element by
element multiplication as is used in polyval.
<p>The argument <var>x</var> must be a square matrix.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpolyval.html#doc_002dpolyval">polyval</a>, <a href="doc_002dpoly.html#doc_002dpoly">poly</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dconv.html#doc_002dconv">conv</a>, <a href="doc_002ddeconv.html#doc_002ddeconv">deconv</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>, <a href="doc_002dfilter.html#doc_002dfilter">filter</a>, <a href="doc_002dpolyderiv.html#doc_002dpolyderiv">polyderiv</a>, <a href="doc_002dpolyinteg.html#doc_002dpolyinteg">polyinteg</a>.
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