<html lang="en"> <head> <title>Orthogonal Collocation - Untitled</title> <meta http-equiv="Content-Type" content="text/html"> <meta name="description" content="Untitled"> <meta name="generator" content="makeinfo 4.13"> <link title="Top" rel="start" href="index.html#Top"> <link rel="up" href="Numerical-Integration.html#Numerical-Integration" title="Numerical Integration"> <link rel="prev" href="Functions-of-Multiple-Variables.html#Functions-of-Multiple-Variables" title="Functions of Multiple Variables"> <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> <meta http-equiv="Content-Style-Type" content="text/css"> <style type="text/css"><!-- pre.display { font-family:inherit } pre.format { font-family:inherit } pre.smalldisplay { font-family:inherit; font-size:smaller } pre.smallformat { font-family:inherit; font-size:smaller } pre.smallexample { font-size:smaller } pre.smalllisp { font-size:smaller } span.sc { font-variant:small-caps } span.roman { font-family:serif; font-weight:normal; } span.sansserif { font-family:sans-serif; font-weight:normal; } --></style> </head> <body> <div class="node"> <a name="Orthogonal-Collocation"></a> <p> Previous: <a rel="previous" accesskey="p" href="Functions-of-Multiple-Variables.html#Functions-of-Multiple-Variables">Functions of Multiple Variables</a>, Up: <a rel="up" accesskey="u" href="Numerical-Integration.html#Numerical-Integration">Numerical Integration</a> <hr> </div> <h3 class="section">22.2 Orthogonal Collocation</h3> <!-- ./DLD-FUNCTIONS/colloc.cc --> <p><a name="doc_002dcolloc"></a> <div class="defun"> — Loadable Function: [<var>r</var>, <var>amat</var>, <var>bmat</var>, <var>q</var>] = <b>colloc</b> (<var>n, "left", "right"</var>)<var><a name="index-colloc-1780"></a></var><br> <blockquote><p>Compute derivative and integral weight matrices for orthogonal collocation using the subroutines given in J. Villadsen and M. L. Michelsen, <cite>Solution of Differential Equation Models by Polynomial Approximation</cite>. </p></blockquote></div> <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>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>). <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> <p>Then the solution at the roots <var>r</var> is <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> </body></html>