Integration

Does matlab have a function of Integration? For example, I have a Gaussian density function, if I use the integration function on the density function over the time of (-infinity, +infinity), then we should obtain one. Thank you very much

Answer 1

[Walter Roberson]

MATLAB has several numeric integration routines, such as quad(), quadv(), quadgk(), triplequad()

The Symbolic Toolbox offers symbolic integration.

Answer 2

[Mike Hosea]

QUADGK can handle infinite limits.

    >> quadgk(@(x)exp(-x.^2./2)./sqrt(2*pi),-inf,inf)
    ans =
       1.000000000000038

The default tolerances used there were, BTW, AbsTol = 1e-10 and RelTol = 1e-6. Since the numerical answer is about 1, and RelTol > AbsTol, only the RelTol was important. We can crank down both tolerances

    >> quadgk(@(x)exp(-x.^2./2)./sqrt(2*pi),-inf,inf,'AbsTol',5e-14,'RelTol',5e-14)
    ans =
         1

Getting 1 exactly in that case was just luck. Evaluating CDFs is one example where pure absolute error control might make sense to you:

    >> quadgk(@(x)exp(-x.^2./2)./sqrt(2*pi),-inf,inf,'AbsTol',1e-10,'RelTol',0)
    ans =
       1.000000000000000

If you have a very small variance, the integrator might not see the non-zero part of the curve.

    >> sigma = 1e-10; f = @(x)exp(-x.^2./(2*sigma))./sqrt(2*pi*sigma)
    f = 
        @(x)exp(-x.^2./(2*sigma))./sqrt(2*pi*sigma)
    >> quadgk(f,-inf,inf,'AbsTol',1e-10,'RelTol',0)
    ans =
         0

This is because the function was zero everywhere the integrator "looked". If you know you have a problem like that, you can cope with it in one of two ways. You can split the interval where the action is:

    >> quadgk(f,-inf,0,'AbsTol',1e-10,'RelTol',0)+quadgk(f,0,inf,'AbsTol',1e-10,'RelTol',0)
    ans =
         1

or you can toss in some waypoints to force the integrator to use a finer mesh where you believe the action is:

    >> quadgk(f,-inf,inf,'AbsTol',1e-10,'RelTol',0,'Waypoints',linspace(-1e-3,1e-3,101))
    ans =
       1.000000000000000

although I have to say that using Waypoints for this purpose may require some trial and error. Obviously for such a function there is hardly any need to integrate from -inf to inf because the integrand evaluates to exactly zero for abs(x) => 4e-4 or so.

    >> quadgk(f,-4e-4,4e-4,'AbsTol',1e-10,'RelTol',0)
    ans =
         1