MATLAB complicated integration

Question

I have an integration function which does not have indefinite integral expression.

Specifically, the function is f(y)=h(y)+integral(@(x) exp(-x-1/x),0,y) where h(y) is a simple function.

Matlab numerically computes f(y) well, but I want to compute the following function.

g(w)=w*integral(1-f(y).^(1/w),0,inf) where w is a real number in [0,1].

The problem for computing g(w) is handling f(y).^(1/w) numerically.

How can I calculate g(w) with MATLAB? Is it impossible?

Answer 1

Expressions containing e^(-1/x) are generally difficult to compute near x = 0. Actually, I am surprised that Matlab computes f(y) well in the first place. I'd suggest trying to compute g(w)=w*integral(1-f(y).^(1/w),epsilon,inf) for epsilon greater than zero, then gradually decreasing epsilon toward 0 to check if you can get numerical convergence at all. Convergence is certainly not guaranteed!

Answer 2

You can calculate g(w) using the functions you have, but you need to add the ( ArrayValued , true ) name-value pair. The option allows you to specify a vector-valued w and allows the nested integral call to receive a vector of y values, which is how integral naturally works.

f = @(y) h(y)+integral(@(x) exp(-x-1/x),0,y,'ArrayValued',true);
g = @(w) w .* integral(1-f(y).^(1./w),0,Inf,'ArrayValued',true);

At least, that works on my R2014b installation.

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Note: While h(y) may be simple, if it's integral over the positive real line does not converge, g(w) will more than likely not converge (I don't think I need to qualify that, but I'll hedge my bets).