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I would like to evaluate the following expression but takes too much time. I would be greatly appreciate for any comments to make it fast.

Clear[a, b];

FullSimplify[
 FourierTransform[
  Exp[-1/2 (1 - s) (a^2 + b^2)]*(α4 1/2 Sqrt[
       6] (1/Sqrt[2] (a + I b))^4 - α1 α4 1/(2 Sqrt[
          6]) (1/Sqrt[2] (a + I b))^3 (-(a^2 + b^2)/2 + 
        4) + α2 α4 1/(2 Sqrt[3]) (1/
          Sqrt[2] (a + I b))^2 ((a^2 + b^2)^2/8 - 2(a^2 + b^2) + 
        6) + α4^2 1/
       24 ((a^2 + b^2)^6/16 - 2 (a^2 + b^2)^3 + 18 (a^2 + b^2)^2 - 
        48 (a^2 + b^2) + 24) + α2 1/
       Sqrt[2] (1/Sqrt[2] (a + I b))^2 - α1 α2 1/
       Sqrt[2] (1/Sqrt[2] (a + I b)) (-(a^2 + b^2)/2 + 
        2) + α2^2 ((a^2 + b^2)^2/8 - (a^2 + b^2) + 
        1) + α2 α4 1/(2 Sqrt[3]) Conjugate[
        1/Sqrt[2] (a + I b)]^2 ((a^2 + b^2)^2/8 - 2 (a^2 + b^2) + 
        6) + α1 (1/
         Sqrt[2] (a + I b)) - α1^2 (-(a^2 + b^2)/2 + 
        1) - α1 α2 1/Sqrt[2] Conjugate[
       1/Sqrt[2] (a + I b)] (-(a^2 + b^2)/2 + 
        2) - α1 α4 1/(2 Sqrt[6]) Conjugate[
        1/Sqrt[2] (a + I b)]^3 (-(a^2 + b^2)/2 + 4) + 
     1 + α1 Conjugate[1/Sqrt[2] (a + I b)] + α2 1/
       Sqrt[2] Conjugate[
        1/Sqrt[2] (a + I b)]^2 + α4 1/(2 Sqrt[6]) Conjugate[
        1/Sqrt[2] (a + I b)]^4), {a, b}, {x, y}], {-1 < s < 1}]
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2
  • $\begingroup$ Is your α1 to α4 real or complex? $\endgroup$
    – Leo Fang
    May 21, 2014 at 22:34
  • $\begingroup$ @ Leo Fang, they are real. $\endgroup$
    – user0322
    May 22, 2014 at 11:06

2 Answers 2

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Check if CUDALink is supported on your system.

Needs["CUDALink`"]
CUDAQ[]

If return True use CUDAFourier and CUDAInverseFourier function.

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2
  • $\begingroup$ The OP is doing a symbolic Fourier transform with FourierTransform. CUDAFourier does a discrete numerical transform like Fourier. $\endgroup$
    – Simon Woods
    May 22, 2014 at 11:42
  • $\begingroup$ : CUDA was not able to find a valid CUDA driver in my system. @ Simon Woods you are right it finds the discrete Fourier transform $\endgroup$
    – user0322
    May 22, 2014 at 11:53
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It is sufficiently faster now by modifying

Clear[a, b];

Assuming[$-1<s<1$ && $\{a, b, \alpha1, \alpha 2,\alpha 4, x, y\}$$\in$Reals,

v = Simplify[the expression]

FourierTransform[v, {a, b}, {x, y}]

It is sufficiently faster now.

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