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IMA Journal of Numerical Analysis Advance Access published online on August 21, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn073
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

A. Bultheel{dagger}

Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium

P. González-Vera

Department of Mathematical Analysis, La Laguna University, Tenerife, Spain

E. Hendriksen

Beerstratenlaan 23, 2421GN Nieuwkoop, The Netherlands

O. Njåstad

Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

{dagger} Corresponding author. Email: adhemar.bultheel{at}cs.kuleuven.be

Received on 27 September 2007. Accepted for publication 21 October 2008.


  Abstract

This paper is concerned with rational Szego quadrature formulas to approximate integrals of the form Iµ (f) = {int} Formula f (ei{theta})dµ ({theta}) by a formula such as In (f) = {sum}Formula{lambda}kf (zk), where the weights {lambda}k are positive and the nodes zk are carefully chosen on the complex unit circle. It will be shown that, for a given set of poles, the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension 2n. Also, the problem where one node (Radau) or two nodes (Lobatto) are prefixed will be analysed and the corresponding rational Szego–Radau and rational Szego–Lobatto quadrature formulas will be characterized.

Key Words: rational Szego–Radau quadrature; rational Szego–Lobatto quadrature; maximal domain of validity


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