(octave.info)Orthogonal Collocation

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23.2 Orthogonal Collocation
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 -- : [R, AMAT, BMAT, Q] = colloc (N, "left", "right")
     Compute derivative and integral weight matrices for orthogonal
     collocation.

     Reference: J. Villadsen, M. L. Michelsen, ‘Solution of Differential
     Equation Models by Polynomial Approximation’.

   Here is an example of using ‘colloc’ to generate weight matrices for
solving the second order differential equation U’ - ALPHA * U” = 0 with
the boundary conditions U(0) = 0 and U(1) = 1.

   First, we can generate the weight matrices for N points (including
the endpoints of the interval), and incorporate the boundary conditions
in the right hand side (for a specific value of ALPHA).

     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);

   Then the solution at the roots R is

     u = [ 0; (at - alpha * bt) \ rhs; 1]
          ⇒ [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]