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<h3 class="section">27.5 Polynomial Interpolation</h3>
<p>Octave comes with good support for various kinds of interpolation,
most of which are described in <a href="Interpolation.html#Interpolation">Interpolation</a>. One simple alternative
to the functions described in the aforementioned chapter, is to fit
a single polynomial to some given data points. To avoid a highly
fluctuating polynomial, one most often wants to fit a low-order polynomial
to data. This usually means that it is necessary to fit the polynomial
in a least-squares sense, which is what the <code>polyfit</code> function does.
<!-- ./polynomial/polyfit.m -->
<p><a name="doc_002dpolyfit"></a>
<div class="defun">
— Function File: [<var>p</var>, <var>s</var>, <var>mu</var>] = <b>polyfit</b> (<var>x, y, n</var>)<var><a name="index-polyfit-2048"></a></var><br>
<blockquote><p>Return the coefficients of a polynomial <var>p</var>(<var>x</var>) of degree
<var>n</var> that minimizes the least-squares-error of the fit.
<p>The polynomial coefficients are returned in a row vector.
<p>The second output is a structure containing the following fields:
<dl>
<dt>‘<samp><span class="samp">R</span></samp>’<dd>Triangular factor R from the QR decomposition.
<br><dt>‘<samp><span class="samp">X</span></samp>’<dd>The Vandermonde matrix used to compute the polynomial coefficients.
<br><dt>‘<samp><span class="samp">df</span></samp>’<dd>The degrees of freedom.
<br><dt>‘<samp><span class="samp">normr</span></samp>’<dd>The norm of the residuals.
<br><dt>‘<samp><span class="samp">yf</span></samp>’<dd>The values of the polynomial for each value of <var>x</var>.
</dl>
<p>The second output may be used by <code>polyval</code> to calculate the
statistical error limits of the predicted values.
<p>When the third output, <var>mu</var>, is present the
coefficients, <var>p</var>, are associated with a polynomial in
<var>xhat</var> = (<var>x</var>-<var>mu</var>(1))/<var>mu</var>(2).
Where <var>mu</var>(1) = mean (<var>x</var>), and <var>mu</var>(2) = std (<var>x</var>).
This linear transformation of <var>x</var> improves the numerical
stability of the fit.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpolyval.html#doc_002dpolyval">polyval</a>, <a href="doc_002dresidue.html#doc_002dresidue">residue</a>.
</p></blockquote></div>
<p>In situations where a single polynomial isn't good enough, a solution
is to use several polynomials pieced together. The function <code>mkpp</code>
creates a piece-wise polynomial, <code>ppval</code> evaluates the function
created by <code>mkpp</code>, and <code>unmkpp</code> returns detailed information
about the function.
<p>The following example shows how to combine two linear functions and a
quadratic into one function. Each of these functions is expressed
on adjoined intervals.
<pre class="example"> x = [-2, -1, 1, 2];
p = [ 0, 1, 0;
1, -2, 1;
0, -1, 1 ];
pp = mkpp(x, p);
xi = linspace(-2, 2, 50);
yi = ppval(pp, xi);
plot(xi, yi);
</pre>
<!-- ./polynomial/ppval.m -->
<p><a name="doc_002dppval"></a>
<div class="defun">
— Function File: <var>yi</var> = <b>ppval</b> (<var>pp, xi</var>)<var><a name="index-ppval-2049"></a></var><br>
<blockquote><p>Evaluate piece-wise polynomial <var>pp</var> at the points <var>xi</var>.
If <var>pp</var><code>.d</code> is a scalar greater than 1, or an array,
then the returned value <var>yi</var> will be an array that is
<code>d1, d1, ..., dk, length (</code><var>xi</var><code>)]</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmkpp.html#doc_002dmkpp">mkpp</a>, <a href="doc_002dunmkpp.html#doc_002dunmkpp">unmkpp</a>, <a href="doc_002dspline.html#doc_002dspline">spline</a>.
</p></blockquote></div>
<!-- ./polynomial/mkpp.m -->
<p><a name="doc_002dmkpp"></a>
<div class="defun">
— Function File: <var>pp</var> = <b>mkpp</b> (<var>x, p</var>)<var><a name="index-mkpp-2050"></a></var><br>
— Function File: <var>pp</var> = <b>mkpp</b> (<var>x, p, d</var>)<var><a name="index-mkpp-2051"></a></var><br>
<blockquote>
<p>Construct a piece-wise polynomial structure from sample points
<var>x</var> and coefficients <var>p</var>. The i-th row of <var>p</var>,
<var>p</var><code> (</code><var>i</var><code>,:)</code>, contains the coefficients for the polynomial
over the <var>i</var>-th interval, ordered from highest to
lowest. There must be one row for each interval in <var>x</var>, so
<code>rows (</code><var>p</var><code>) == length (</code><var>x</var><code>) - 1</code>.
<p>You can concatenate multiple polynomials of the same order over the
same set of intervals using <var>p</var><code> = [ </code><var>p1</var><code>; </code><var>p2</var><code>;
...; </code><var>pd</var><code> ]</code>. In this case, <code>rows (</code><var>p</var><code>) == </code><var>d</var><code>
* (length (</code><var>x</var><code>) - 1)</code>.
<p><var>d</var> specifies the shape of the matrix <var>p</var> for all except the
last dimension. If <var>d</var> is not specified it will be computed as
<code>round (rows (</code><var>p</var><code>) / (length (</code><var>x</var><code>) - 1))</code> instead.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dunmkpp.html#doc_002dunmkpp">unmkpp</a>, <a href="doc_002dppval.html#doc_002dppval">ppval</a>, <a href="doc_002dspline.html#doc_002dspline">spline</a>.
</p></blockquote></div>
<!-- ./polynomial/unmkpp.m -->
<p><a name="doc_002dunmkpp"></a>
<div class="defun">
— Function File: [<var>x</var>, <var>p</var>, <var>n</var>, <var>k</var>, <var>d</var>] = <b>unmkpp</b> (<var>pp</var>)<var><a name="index-unmkpp-2052"></a></var><br>
<blockquote>
<p>Extract the components of a piece-wise polynomial structure <var>pp</var>.
These are as follows:
<dl>
<dt><var>x</var><dd>Sample points.
<br><dt><var>p</var><dd>Polynomial coefficients for points in sample interval. <var>p</var><code>
(</code><var>i</var><code>, :)</code> contains the coefficients for the polynomial over
interval <var>i</var> ordered from highest to lowest. If <var>d</var><code> >
1</code>, <var>p</var><code> (</code><var>r</var><code>, </code><var>i</var><code>, :)</code> contains the coefficients for
the r-th polynomial defined on interval <var>i</var>. However, this is
stored as a 2-D array such that <var>c</var><code> = reshape (</code><var>p</var><code> (:,
</code><var>j</var><code>), </code><var>n</var><code>, </code><var>d</var><code>)</code> gives <var>c</var><code> (</code><var>i</var><code>, </code><var>r</var><code>)</code>
is the j-th coefficient of the r-th polynomial over the i-th interval.
<br><dt><var>n</var><dd>Number of polynomial pieces.
<br><dt><var>k</var><dd>Order of the polynomial plus 1.
<br><dt><var>d</var><dd>Number of polynomials defined for each interval.
</dl>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmkpp.html#doc_002dmkpp">mkpp</a>, <a href="doc_002dppval.html#doc_002dppval">ppval</a>, <a href="doc_002dspline.html#doc_002dspline">spline</a>.
</p></blockquote></div>
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