Question

I’m trying to compute a rather ugly integral using MATLAB. What I’m having problem with though is a part where I multiply a very big number (>10^300) with a very small number (<10^-300). MATLAB returns ‘inf’ for this even though it should be in the range of 0-0.0005. This is what I have

    besselFunction = @(u)besseli(qb,2*sqrt(lambda*(theta + mu)).*u);
    exponentFuncion = @(u)exp(-u.*(lambda + theta + mu));

where qb = 5, lambda = 12, theta = 10, mu = 3. And what I want to find is

    besselFunction(u)*exponentFunction(u)

for all real values of u. The problem is that whenever u>28 it will be evaluated as ‘inf’. I’ve heared, and tried, to use MATLAB function ‘vpa’ but it doesn’t seem to work well when I want to use functions…

Any tips will be appreciated at this point!

Answer 1

I’d use logarithms.

Let x = Bessel function of u and y = x*exp(-u) (simpler than your equation, but similar).

Since log(v*w) = log(v) + log(w), then log(y) = log(x) + log(exp(-u))

This simplifies to

log(y) = log(x) - u

This will be better behaved numerically.

The other key will be to not evaluate that Bessel function that turns into a large number and passing it to a math function to get the log. Better to write your own that returns the logarithm of the Bessel function directly. Look at a reference like Abramowitz and Stegun to try and find one.