MATLAB: Improper integral

Question

Hi!!I need help in the calculation of the integral of the function below

    function out = fun(u,a,b)
          out = exp(-a.* exp(2.*u)).*exp(-(u + b.^2).^2./2.*b.^2)/...
              (sqrt(2.*pi.*b.^2));
     end

a=0; b=1; *quad('fun',-Inf,Inf,[],[],a,b);

Is there any idea? where is the mistake?

Answer 1

I don't think quad ever allowed infinite limits. quadgk does. The answer you need depends on how old your MATLAB version is. I am going to express an answer in more recent syntax. If you try

a=0; b=1; 
fun = @(u) exp(-a.* exp(2.*u)).*exp(-(u + b.^2).^2./2.*b.^2)/...
  (sqrt(2.*pi.*b.^2));
quad(fun,-Inf,Inf)

then you get this message:

Warning: Infinite or Not-a-Number function value encountered.

and the answer is NaN. The likely reason is that there is underflow in these exponential expressions:

>> fun(1000)
ans =
   NaN

However, if you plot this function, you can see that this function approaches zero very rapidly. For example,

>> fun(0)
ans = 
    0

Therefore, you can do this calculation:

quadl(fun,-100,100)
ans = 
    1.0000

This is 1 to an accuracy of about 10^-8.

Note: the fun = ... line is an anonymous function.