|GNU Octave Manual Version 3|
by John W. Eaton, David Bateman, Søren Hauberg
Paperback (6"x9"), 568 pages
RRP £24.95 ($39.95)
21.3 Functions of Multiple Variables
Octave does not have built-in functions for computing the integral of functions of multiple variables. It is however possible to compute the integral of a function of multiple variables using the functions for one-dimensional integrals.
To illustrate how the integration can be performed, we will integrate
f(x, y) = sin(pi*x*y)*sqrt(x*y)
for x and y between 0 and 1.
The first approach creates a function that integrates f with
respect to x, and then integrates that function with respect to
quad is written in Fortran it cannot be called
recursively. This means that
quad cannot integrate a function
quad, and hence cannot be used to perform the double
integration. It is however possible with
quadl, which is what
the following code does.
function I = g(y) I = ones(1, length(y)); for i = 1:length(y) f = @(x) sin(pi.*x.*y(i)).*sqrt(x.*y(i)); I(i) = quadl(f, 0, 1); endfor endfunction I = quadl("g", 0, 1) => 0.30022
The above mentioned approach works but is fairly slow, and that problem
increases exponentially with the dimensionality the problem. Another
possible solution is to use Orthogonal Collocation as described in the
previous section. The integral of a function f(x,y) for
x and y between 0 and 1 can be approximated using n
the sum over
where q and r is as returned by
generalisation to more than two variables is straight forward. The
following code computes the studied integral using n=7 points.
f = @(x,y) sin(pi*x*y').*sqrt(x*y'); n = 7; [t, A, B, q] = colloc(n); I = q'*f(t,t)*q; => 0.30022
It should be noted that the number of points determines the quality of the approximation. If the integration needs to be performed between a and b instead of 0 and 1, a change of variables is needed.
|ISBN 095461206X||GNU Octave Manual Version 3||See the print edition|