Numerical integration involving log(exp(f(x))) shows NaN

Hi everyone,

As title mentioned, when I integrated the function log(exp(f(x))) within the range [-100,100], the result showed "-inf"

,but if I changed log(exp(f(x))) into the equivalent form which is f(x), the integral result was a numerical value.

f(x)=-(x^2)/2, btw.

The following code may explain clearly

f1 = @(x) log(exp(-(x.^2)/2));
r1 = integral(f1,-100,100)

Warning: Inf or NaN value encountered.

r1 = -Inf

f2 = @(x) -(x.^2)/2;
r2 = integral(f2,-100,100)

r2 = -3.3333e+05

My question is why does r1 get "-Inf"?

Is there any way to let r1 equal to r2, if f1 has to be the form log(exp(f(x)))?

Thank you very much in advance.

Answer 1

[Animesh]

Hey @YI-WEN PAN,

The possible reason for getting r1=-Inf in the expression log(exp(f(x))), is due to the limited precision of floating point numbers. This can cause numeric underflow.

In this case the value of the term exp(-(x.^2)/2) will become very small as the magnitude of x grows large. Eventually the value will become so small that it will get rounded down to zero. This will cause the entire expression to evaluate to -Inf as log of 0 is negative infinity.

A workaround that you could try is using symbolic integration instead of numerical integration. This would allow you to simplify the expression log(exp(f(x))) and evaluate the integral analytically.

syms x;
f = log(exp(-(x.^2)/2));
r = simplify(int(f, x, -100, 100))

[YI-WEN PAN]

@Animesh Thanks, it works well~

[wenji]

Instead using int, is there any way to avoid this limited precision floating point number? i am working with a triple integration so I am afriad i could not simply use int....my function would be like f=@(a,b,c) exp(a)^-1*log(1+exp(a)*b^-2)

[Torsten]

If you state your problem properly (what is c ? what are the limits of integration ?) - preferably in a new question - maybe someone will be able to help.