improper integral

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kostas
kostas 2011-4-5
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?
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Asatur Khurshudyan
Asatur Khurshudyan 2012-12-30
I guess you have to calculate the principal value (by Cauchy) of that integral, otherwise you`ll have Nan`s or Infinity`s. I also have such a problem with 3D plotting improper integrals with infinite upper limit. Have anyone did something like that?
Roger Stafford
Roger Stafford 2012-12-30
For a = 0 this becomes just the integral of a normal distribution density function over its full range, and of course it must then be equal to 1. It is when a is greater than zero that the integral becomes more interesting. For negative a the integral is divergent.

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采纳的回答

Andrew Newell
Andrew Newell 2011-4-5
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.

更多回答（1 个）

Matt Fig
Matt Fig 2011-4-5
quadgk(@(x) fun(x,0,1),-inf,inf) % 0 is a, 1 is b.
But for a = 0, b = 1, we encounter an infinite number or nan. Are these numbers correct? If so, you could try this:
quadgk(@(x) fun(x,0,1),-10,10) % Exponentially decreasing values.
Or look at QUADL.
quadl(@(x) fun(x,0,1),-200,200,1e-15)

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