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<h3 class="section">28.5 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 just is what the <code>polyfit</code> function does.
<!-- polyfit scripts/polynomial/polyfit.m -->
<p><a name="doc_002dpolyfit"></a>
<div class="defun">
— Function File: <var>p</var> = <b>polyfit</b> (<var>x, y, n</var>)<var><a name="index-polyfit-2729"></a></var><br>
— Function File: [<var>p</var>, <var>s</var>] = <b>polyfit</b> (<var>x, y, n</var>)<var><a name="index-polyfit-2730"></a></var><br>
— Function File: [<var>p</var>, <var>s</var>, <var>mu</var>] = <b>polyfit</b> (<var>x, y, n</var>)<var><a name="index-polyfit-2731"></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 to the points
<code>[</code><var>x</var><code>, </code><var>y</var><code>]</code>.
<p>The polynomial coefficients are returned in a row vector.
<p>The optional output <var>s</var> 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_002dpolyaffine.html#doc_002dpolyaffine">polyaffine</a>, <a href="doc_002droots.html#doc_002droots">roots</a>, <a href="doc_002dvander.html#doc_002dvander">vander</a>, <a href="doc_002dzscore.html#doc_002dzscore">zscore</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 piecewise 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>
<!-- mkpp scripts/polynomial/mkpp.m -->
<p><a name="doc_002dmkpp"></a>
<div class="defun">
— Function File: <var>pp</var> = <b>mkpp</b> (<var>breaks, coefs</var>)<var><a name="index-mkpp-2732"></a></var><br>
— Function File: <var>pp</var> = <b>mkpp</b> (<var>breaks, coefs, d</var>)<var><a name="index-mkpp-2733"></a></var><br>
<blockquote>
<p>Construct a piecewise polynomial (pp) structure from sample points
<var>breaks</var> and coefficients <var>coefs</var>. <var>breaks</var> must be a vector of
strictly increasing values. The number of intervals is given by
<var>ni</var><code> = length (</code><var>breaks</var><code>) - 1</code>.
When <var>m</var> is the polynomial order <var>coefs</var> must be of
size: <var>ni</var> x <var>m</var> + 1.
<p>The i-th row of <var>coefs</var>,
<var>coefs</var><code> (</code><var>i</var><code>,:)</code>, contains the coefficients for the polynomial
over the <var>i</var>-th interval, ordered from highest (<var>m</var>) to
lowest (<var>0</var>).
<p><var>coefs</var> may also be a multi-dimensional array, specifying a vector-valued
or array-valued polynomial. In that case the polynomial order is defined
by the length of the last dimension of <var>coefs</var>.
The size of first dimension(s) are given by the scalar or
vector <var>d</var>. If <var>d</var> is not given it is set to <code>1</code>.
In any case <var>coefs</var> is reshaped to a 2-D matrix of
size <code>[</code><var>ni</var><code>*prod(</code><var>d</var> <var>m</var><code>)] </code>
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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>, <a href="doc_002dpchip.html#doc_002dpchip">pchip</a>, <a href="doc_002dppder.html#doc_002dppder">ppder</a>, <a href="doc_002dppint.html#doc_002dppint">ppint</a>, <a href="doc_002dppjumps.html#doc_002dppjumps">ppjumps</a>.
</p></blockquote></div>
<!-- unmkpp scripts/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-2734"></a></var><br>
<blockquote>
<p>Extract the components of a piecewise polynomial structure <var>pp</var>.
The components are:
<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>.
<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>, <a href="doc_002dpchip.html#doc_002dpchip">pchip</a>.
</p></blockquote></div>
<!-- ppval scripts/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-2735"></a></var><br>
<blockquote><p>Evaluate the piecewise polynomial structure <var>pp</var> at the points <var>xi</var>.
If <var>pp</var> describes a scalar polynomial function, the result is an
array of the same shape as <var>xi</var>.
Otherwise, the size of the result is <code>[pp.dim, length(</code><var>xi</var><code>)]</code> if
<var>xi</var> is a vector, or <code>[pp.dim, size(</code><var>xi</var><code>)]</code> if it is a
multi-dimensional array.
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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>, <a href="doc_002dpchip.html#doc_002dpchip">pchip</a>.
</p></blockquote></div>
<!-- ppder scripts/polynomial/ppder.m -->
<p><a name="doc_002dppder"></a>
<div class="defun">
— Function File: ppd = <b>ppder</b> (<var>pp</var>)<var><a name="index-ppder-2736"></a></var><br>
— Function File: ppd = <b>ppder</b> (<var>pp, m</var>)<var><a name="index-ppder-2737"></a></var><br>
<blockquote><p>Compute the piecewise <var>m</var>-th derivative of a piecewise polynomial
struct <var>pp</var>. If <var>m</var> is omitted the first derivative is calculated.
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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_002dppint.html#doc_002dppint">ppint</a>.
</p></blockquote></div>
<!-- ppint scripts/polynomial/ppint.m -->
<p><a name="doc_002dppint"></a>
<div class="defun">
— Function File: <var>ppi</var> = <b>ppint</b> (<var>pp</var>)<var><a name="index-ppint-2738"></a></var><br>
— Function File: <var>ppi</var> = <b>ppint</b> (<var>pp, c</var>)<var><a name="index-ppint-2739"></a></var><br>
<blockquote><p>Compute the integral of the piecewise polynomial struct <var>pp</var>.
<var>c</var>, if given, is the constant of integration.
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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_002dppder.html#doc_002dppder">ppder</a>.
</p></blockquote></div>
<!-- ppjumps scripts/polynomial/ppjumps.m -->
<p><a name="doc_002dppjumps"></a>
<div class="defun">
— Function File: <var>jumps</var> = <b>ppjumps</b> (<var>pp</var>)<var><a name="index-ppjumps-2740"></a></var><br>
<blockquote><p>Evaluate the boundary jumps of a piecewise polynomial.
If there are n intervals, and the dimensionality of <var>pp</var> is
d, the resulting array has dimensions <code>[d, n-1]</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmkpp.html#doc_002dmkpp">mkpp</a>.
</p></blockquote></div>
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