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<a name="Evaluating-Polynomials"></a>
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<a name="Evaluating-Polynomials-1"></a>
<h3 class="section">28.1 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:
</p>
<div class="example">
<pre class="example">N = length (c) - 1;
val = dot (x.^(N:-1:0), c);
</pre></div>
<p>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>
<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.
</p>
<a name="XREFpolyval"></a><dl>
<dt><a name="index-polyval"></a><em><var>y</var> =</em> <strong>polyval</strong> <em>(<var>p</var>, <var>x</var>)</em></dt>
<dt><a name="index-polyval-1"></a><em><var>y</var> =</em> <strong>polyval</strong> <em>(<var>p</var>, <var>x</var>, [], <var>mu</var>)</em></dt>
<dt><a name="index-polyval-2"></a><em>[<var>y</var>, <var>dy</var>] =</em> <strong>polyval</strong> <em>(<var>p</var>, <var>x</var>, <var>s</var>)</em></dt>
<dt><a name="index-polyval-3"></a><em>[<var>y</var>, <var>dy</var>] =</em> <strong>polyval</strong> <em>(<var>p</var>, <var>x</var>, <var>s</var>, <var>mu</var>)</em></dt>
<dd>
<p>Evaluate the polynomial <var>p</var> at the specified values of <var>x</var>.
</p>
<p>If <var>x</var> is a vector or matrix, the polynomial is evaluated for each of
the elements of <var>x</var>.
</p>
<p>When <var>mu</var> is present, evaluate the polynomial for
(<var>x</var>-<var>mu</var>(1))/<var>mu</var>(2).
</p>
<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 from <code>polyfit</code>, must be
supplied.
</p>
<p><strong>See also:</strong> <a href="#XREFpolyvalm">polyvalm</a>, <a href="Derivatives-_002f-Integrals-_002f-Transforms.html#XREFpolyaffine">polyaffine</a>, <a href="Polynomial-Interpolation.html#XREFpolyfit">polyfit</a>, <a href="Finding-Roots.html#XREFroots">roots</a>, <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>.
</p></dd></dl>
<a name="XREFpolyvalm"></a><dl>
<dt><a name="index-polyvalm"></a><em></em> <strong>polyvalm</strong> <em>(<var>c</var>, <var>x</var>)</em></dt>
<dd><p>Evaluate a polynomial in the matrix sense.
</p>
<p><code>polyvalm (<var>c</var>, <var>x</var>)</code> will evaluate the polynomial in the
matrix sense, i.e., matrix multiplication is used instead of element by
element multiplication as used in <code>polyval</code>.
</p>
<p>The argument <var>x</var> must be a square matrix.
</p>
<p><strong>See also:</strong> <a href="#XREFpolyval">polyval</a>, <a href="Finding-Roots.html#XREFroots">roots</a>, <a href="Miscellaneous-Functions.html#XREFpoly">poly</a>.
</p></dd></dl>
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