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[Help-gsl] Re: Looking for good codes for 2D numerical integrals...

From: Michael
Subject: [Help-gsl] Re: Looking for good codes for 2D numerical integrals...
Date: Thu, 28 Jun 2007 04:44:52 -0700

Hi Tommy,

Here is my 2D integral. As you can see, only one (the inner integral)
is a Fourier type. Can I still use your change-of-coordinate trick?
Also I have to consider the efficiency...


My 2D integration is:

Integrate( F(v) * FourierTransform[g(t)] (v), v from -infinity to
+infinity ).


g(t)'s evaluation is costly. So I plan to fix the parameter for g(t), and
compute the Fourier Transform of g(t) only once, and store in memroy. And
then in calibration loop, I only vary the parameters for F(v). And each time
for each different set of parameters of F(v), I compute the dot-product of
F(v) sample points and the FT[g(t)] sample points to obtain approximation to
the integral.

My question is: how to improve the accuracy of FFT-based integration? I know
it's inefficient, but is there any remedy at least?

Moreover, is there a better adaptive quadature based "smart" integration
method that can help me deal with the above situation efficiently? I am
thinking of doing a cache for the Fourier Transform of g(t), which is
FT[g(t)](v), since adaptive quadature based integration may sample different
point of FT[g(t)](v) each time... but perhaps the overhead introduced in the
cache may outweigh the smart adaptive integration itself...

Any suggestions? Thanks a lot!

On 6/28/07, Tommy Nordgren <address@hidden> wrote:

        For the fourier-type integral, change to polar coordinates, then only 1
of the integration variables (the radius) will be of infinite range.
        Then you can integrate over the angle with the GSL numerical quadrature
functions, where the function to be integrated is computed as a
Fourier integral,
in the radius, the angle held constant
"Home is not where you are born, but where your heart finds peace" -
Tommy Nordgren, "The dying old crone"

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