22.3 Functions of Multiple Variables

Octave does not have built-in functions for computing the integral of functions of multiple variables directly. 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 process can be simplified with the dblquad and triplequad functions for integrals over two and three variables. For example

I =  dblquad (@(x, y) sin(pi.*x.*y).*sqrt(x.*y), 0, 1, 0, 1)
      ⇒ 0.30022

Function File: dblquad (f, xa, xb, ya, yb, tol, quadf, …)

Numerically evaluate a double integral. The function over with to integrate is defined by f, and the interval for the integration is defined by [xa, xb, ya, yb]. The function f must accept a vector x and a scalar y, and return a vector of the same length as x.

If defined, tol defines the absolute tolerance to which to which to integrate each sub-integral.

Additional arguments, are passed directly to f. To use the default value for tol one may pass an empty matrix.

See also: triplequad, quad, quadv, quadl, quadgk, trapz.

Function File: triplequad (f, xa, xb, ya, yb, za, zb, tol, quadf, …)

Numerically evaluate a triple integral. The function over which to integrate is defined by f, and the interval for the integration is defined by [xa, xb, ya, yb, za, zb]. The function f must accept a vector x and a scalar y, and return a vector of the same length as x.

If defined, tol defines the absolute tolerance to which to which to integrate each sub-integral.

Additional arguments, are passed directly to f. To use the default value for tol one may pass an empty matrix.

See also: dblquad, quad, quadv, quadl, quadgk, trapz.

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 generalization 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.