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<a name="Orthogonal-Collocation"></a>
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Next: <a href="Functions-of-Multiple-Variables.html#Functions-of-Multiple-Variables" accesskey="n" rel="next">Functions of Multiple Variables</a>, Previous: <a href="Functions-of-One-Variable.html#Functions-of-One-Variable" accesskey="p" rel="prev">Functions of One Variable</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Orthogonal-Collocation-1"></a>
<h3 class="section">23.2 Orthogonal Collocation</h3>
<a name="XREFcolloc"></a><dl>
<dt><a name="index-colloc"></a><em>[<var>r</var>, <var>amat</var>, <var>bmat</var>, <var>q</var>] =</em> <strong>colloc</strong> <em>(<var>n</var>, "left", "right")</em></dt>
<dd><p>Compute derivative and integral weight matrices for orthogonal collocation.
</p>
<p>Reference: J. Villadsen, M. L. Michelsen,
<cite>Solution of Differential Equation Models by Polynomial Approximation</cite>.
</p></dd></dl>
<p>Here is an example of using <code>colloc</code> to generate weight matrices
for solving the second order differential equation
<var>u</var>’ - <var>alpha</var> * <var>u</var>” = 0 with the boundary conditions
<var>u</var>(0) = 0 and <var>u</var>(1) = 1.
</p>
<p>First, we can generate the weight matrices for <var>n</var> points (including
the endpoints of the interval), and incorporate the boundary conditions
in the right hand side (for a specific value of
<var>alpha</var>).
</p>
<div class="example">
<pre class="example">n = 7;
alpha = 0.1;
[r, a, b] = colloc (n-2, "left", "right");
at = a(2:n-1,2:n-1);
bt = b(2:n-1,2:n-1);
rhs = alpha * b(2:n-1,n) - a(2:n-1,n);
</pre></div>
<p>Then the solution at the roots <var>r</var> is
</p>
<div class="example">
<pre class="example">u = [ 0; (at - alpha * bt) \ rhs; 1]
⇒ [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]
</pre></div>
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