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I recently implemented a Tanh-Sinh quadrature integrator for a two dimensional integral, simply by integration first over the the first variable and then over the second. My question is whether or not there is a generalized formula for two (or higher) dimension of the form: $$\int{f(x)}dx\approx \sum_{k_x=-N}^N\sum_{k_y=-N}^N{f(g(k_xh,k_yh))g'(k_xh,k_yh)}$$

Where $g'$ is some (unknown) function of the partial derivatives of $g$. Note that I'm not after a specific form, I'm just looking for a generalization to 2D.

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a) What's $g'(k_xh,k_yh)$? b) You probably want $\approx$ instead of $=$? –  joriki Apr 18 '12 at 23:31
    
@joriki - in tanh sinh quadrature the function $g(kh)$ is $tanh(sinh(kh))$. I'm wondering if there exist similar functions in two dimensions. –  nbubis Apr 18 '12 at 23:33
    
Yes, I understood that from the question. That doesn't answer my question. What's the derivative of a function of two arguments? –  joriki Apr 18 '12 at 23:35
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I think then you should indicate in the question that you don't really have a precise form that you want, just a more general idea that there should be a product involving $f(g)$ and some derivative(s) of $g$? –  joriki Apr 18 '12 at 23:50
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I don't believe people have done better than using a Cartesian product of one-dimensional double exponential quadratures. I would suggest contacting Takuya Ooura to ask if there have been multidimensional versions of the algorithm developed. –  J. M. Apr 19 '12 at 2:07

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