Functions of Multiple Variables - Untitled
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Next: Orthogonal Collocation, Previous: Functions of One Variable, Up: Numerical Integration

### 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 the function

         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 y. Since `quad` is written in Fortran it cannot be called recursively. This means that `quad` cannot integrate a function that calls `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 points by the sum over `i=1:n` and `j=1:n` of `q(i)*q(j)*f(r(i),r(j))`, where q and r is as returned by `colloc(n)`. The 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.