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Numerical Algorithms

, Volume 52, Issue 1, pp 47–68 | Cite as

A matricial computation of rational quadrature formulas on the unit circle

  • Adhemar Bultheel
  • Maria-José Cantero
Original Paper

Abstract

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.

Keywords

Orthogonal rational functions Para-orthogonal rational functions Szegő quadrature formulas Möbius transformations 

Mathematics Subject Classification (2000)

42C05 

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Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of Computer ScienceK.U. LeuvenLeuvenBelgium
  2. 2.Department of Applied MathematicsUniversity of ZaragozaZaragozaSpain

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