### Dan Piponi

Shared publicly -Gaussian quadrature is a numerical integration method where you compute an integral approximately by performing a weighted sum of the value of the integrand at some predefined set of points (called nodes).

Several years ago I asked a question on math overflow about how to do Gauss quadrature on a disk [1] with more than 21 points. I never really got an answer that told me how to do this in full generality, just pointers to specific cases that have already been solved.

I recently came across a paper [2] that characterises sets of nodes and weights for Gauss quadrature as a convex optimisation. What's more, this heuristic allows you to generalise to higher dimensions and interesting shapes. For example, the attached picture shows the weights and nodes for a kidney shaped region. One obvious intuition about Gauss quadrature is that you don't want nodes to be close to each other because then the integrand values you compute at these points will be close to each other and carry less independent information about the function. For rectangles and disks the points are well spread out. But surprisingly, in this kidney shape there are three pairs of nodes that are close to each other. So the intuition is incorrect.

[1] http://mathoverflow.net/questions/28669/numerical-integration-over-2d-disk

[2] http://stanford.edu/~boyd/papers/gauss_quad_lp.html

*Gaussian*quadrature, in particular, is when you cleverly choose the nodes so that you can exactly integrate higher order polynomials that you might have expected from the number of nodes meaning you can calculate integrals accurately with the minimum of computation.Several years ago I asked a question on math overflow about how to do Gauss quadrature on a disk [1] with more than 21 points. I never really got an answer that told me how to do this in full generality, just pointers to specific cases that have already been solved.

I recently came across a paper [2] that characterises sets of nodes and weights for Gauss quadrature as a convex optimisation. What's more, this heuristic allows you to generalise to higher dimensions and interesting shapes. For example, the attached picture shows the weights and nodes for a kidney shaped region. One obvious intuition about Gauss quadrature is that you don't want nodes to be close to each other because then the integrand values you compute at these points will be close to each other and carry less independent information about the function. For rectangles and disks the points are well spread out. But surprisingly, in this kidney shape there are three pairs of nodes that are close to each other. So the intuition is incorrect.

[1] http://mathoverflow.net/questions/28669/numerical-integration-over-2d-disk

[2] http://stanford.edu/~boyd/papers/gauss_quad_lp.html

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+Jacques Carette Always important that. Linking to the right paper.

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