CECURA research project


CECURA: Computacion efectiva de formulas de cuadratura racionales (1-1-2008 to 31-12-2012) Ministerio de educacion y ciencias (Spain), project MTM2008-2012

Short description

This is a joint project between the department of mathematical analysis of the University of La Laguna, Canary Islands, Tenerife, Spain, and the research group NALAG at the department of computer science of the K.U.Leuven, Belgium.

The aim of the project is to continue the research on numerical quadrature formulas and the analysis of orthogonal rational functions and their relation to rational approximation and interpolation, moment problems, etc.

Let us assume that the integral ∫K f(z)dμ(z) , μ being a positive Borel measure and f a function with singularities close to the support K of the measure, is required to be approximately calculated by mean of a quadrature formula like In(f) = ∑{j=1..n} λj f(zj), where the mutually distinct nodes {zj,j=1..n} are in K and the weights {λj,j=1..k} are chosen so that In(f) exactly integrates certain rational functions up to a degree as large as possible and with poles simulating the singularities of f. The case where K is an interval of the real line and all the poles are located at infinity gives rise to the so-called Gaussian formulas. Meanwhile, if we take K as the unit circle and all the poles at the origin and infinity the Szego quadratures appear. Both situations have been deeply studied from a theoretical and numerical point of view for the Gaussian formulas and only theoretically the Szego formulas. The aim of the research project is, on the one hand, to complete certain theoretical aspects and oh the other to develop algorithms that enable us to compute quadrature formulas exactly integrating rational functions with prescribed poles arbitrarily distributed outside the support of the measure, both for K = T = {z ∈ C: |z|=1} and K = [0,∞).

Researchers

Sponsored by

Ministry of Education and Science, Spain
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