collocation

collocation, an Octave code which exemplifies the collocation method, a general technique which begins with an equation satisfied by a function f(x) defined over a continuous domain, and uses collocation to produce a function g(x) from some specified collocation function space, which solves the equation exactly, but only at a discrete set of points.

While some functional equations can be solved exactly, a computational technique must generally be designed to deal with a discrete set of data. The hope is that, if the correct family of collocation spaces is chosen, and enough points are used, and are wisely placed, that the collocation function g(x) will converge to the unknown function f(x), and will do so rapidly.

The most common example of collocation is interpolation, which comes in two versions:

  1. at a prescribed set of N points X(I), values Y(I) are given; construct a g(x) such that g(x(i))=y(i). The only choice the user has, then, is the method for transforming N pairs of data into a function defined for all X.
  2. at any point X, a value Y(X) may be requested. Determine the number of points N, and suitable locations for the points X, and then construct a g(x) such that g(x(i))=y(i). Since N and X are selectable by the user, the idea is to make these choices efficiently (not too many) and effectively (so that g(x) is a good approximation to f(x), even at points where the value Y(X) was not requested in advance.)

A more involved case involves two point boundary value problems. Here, a function y(x) is supposed to exist over an interval [a,b], with known values at the endpoints, satisfying a second order differential equation in between, which might have the form: 

        y(a) = ya
        y''(x) = f(x) for a < x < b
        y(b) = yb
A collocation approach would select N points between A and B, represent the collocation function G(X) in some way that involves N unknown coefficients, and then write out the collocation equations as a linear system that can be solved for the coefficients.

For instance, if we take the interval to be [0,4], F(X) to be EXP(X), and use N = 5 points, equally spaced, and assume G(X) has the form C0+C1*X+C2*X^2+C3*X^3+C4*X^4, our collocation equations become: 

        C0 + C1 * 0 +     C2 * 0^2 +     C3 * 0^3 +      C4 * 0^4 = YA
                      2 * C2       + 6 * C3 * 1^1 + 12 * C4 * 1^2 = EXP(1)
                      2 * C2       + 6 * C3 * 2^1 + 12 * C4 * 2^2 = EXP(2)
                      2 * C2       + 6 * C3 * 3^1 + 12 * C4 * 3^2 = EXP(3)
        C0 + C1 * 4 +     C2 * 4^2 +     C3 * 4^3 +      C4 * 4^4 = Yb
which can be rewritten as: 
        | 1   0   0   0   0 |   C0      YA
        | 0   0   2   6  12 |   C1      EXP(1)
        | 0   0   2  12  48 | * C2  =   EXP(2)
        | 0   0   2  18 108 |   C3      EXP(3)
        | 1   4  16  64 256 |   C4      YB
and this linear system can be solved for C, so that we know the form of our collocation function G(X).

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

collocation is available in a MATLAB version and an Octave version.

Related Data and Programs:

collocation_test

lagrange_interp_1d, an Octave code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

Reference:

Source Code:

Last revised on 13 June 2023.