Polynomial integration on regions defined by a triangle and a conic

Details Polynomial integration on regions defined by a triangle and a conic Polynomial integration on regions defined by a triangle and a conic

2010-05-06

arxiv.org/abs/1005.0990v1

Computer Science

Symbolic Computation

8 pages, accepted by ISSAC 2010

Scientific paper

We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type \[\iint_{T \cap \{f\ge0\}} \phi_1\phi_2 \, dx\,dy\] for quadratic polynomials $f,\phi_1,\phi_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.

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