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Which methods are known to calculate double integrals like

$$\displaystyle \int_{0}^{1}\int_{0}^{1} \frac {1}{x^y+y^x} dy dx$$

numerical ?

Amzoti
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Peter
  • 84,454
  • You might have to remove the singularities (have not tried to see if it is possible) when using composite Simpson's or Gaussian quadrature. However, for this one you're likely best bet will be Monte Carlo methods. – Amzoti Oct 29 '13 at 00:41

1 Answers1

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In a case like this with singularities at the the boundaries, you may want to try an adaptive Monte Carlo method such as the VEGAS or MISER algorithms.

Nathaniel Bubis
  • 33,018
  • The singularities should not make problems because for x>0 and y>0 the function behaves well. The monte-carlo method has a quite low accuracity. The two-dimensional gauss-legendre-quadrature seems to be the proper method. I could calculate the double integral with PARI to more than 100 digits. – Peter Nov 03 '13 at 19:04