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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ammar, G., Gragg, W.B., Reichel, L.: On the eigenproblem for orthogonal matrices. In: Proceedings of the 25th Conference on Decision and Control Athens, pp. 1963–1966. IEEE, Piscataway (1986)
Bultheel, A., Cruz-Barroso, R., Deckers, K., González-Vera, P.: Rational Szegő quadratures associated with Chebyshev weight functions. Math. Comput. (2008, in press)
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: A Szegő theory for rational functions. Technical Report TW131, Department of Computer Science. K.U. Leuven (1990)
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: Orthogonal rational functions and quadrature on the unit circle. Numer. Algorithms 3, 105–116 (1992)
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: Quadrature formulas on the unit circle based on rational functions. J. Comput. Appl. Math. 50, 159–170 (1994)
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: Orthogonal rational functions and interpolatory product rules on the unit circle. II. Quadrature and convergence. Analysis 18, 185–200 (1998)
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: Orthogonal Rational Functions. Volume 5 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1999)
Bultheel, A., Van Barel, M., Van gucht, P.: Orthogonal bases in discrete least squares rational approximation. J. Comput. Appl. Math. 164–165, 175–194 (2004)
Cantero, M.J., Cruz-Barroso, R., González-Vera, P.: A matrix approach to the computation of quadrature formulas on the interval. Appl. Numer. Math. 58(3), 296–318 (2008)
Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)
Cruz-Barroso, R., Delvaux, S.: Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations. J. Approx. Theory (2008). arxiv.org/abs/0712.2738v1
Deckers, K., Bultheel, A.: Orthogonal rational functions and rational modifications of a measure on the unit circle. J. Approx. Theory (2008, in press). doi:10.1016/j.jat.2008.04.017
Fasino, D., Gemignani, L.: Structured eigenvalue problems for rational gauss quadrature. Numer. Algorithms 45(1–4), 195–204 (2007)
Freud, G.: Orthogonal Polynomials. Pergamon, Oxford (1971)
Gautschi, W.: On the construction of Gaussian quadrature rules from modified moments. Math. Comput. 24, 245–260 (1970)
Gautschi, W.: A survey of Gauss-Christoffel quadrature formulae. In: Butzer, P.L., Fehér, F., Christoffel, E.B. (eds.) The influence of his work on mathematical and physical sciences, pp. 72–147. Birkhäuser, Basel (1981)
Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Statist. Comput. 3, 289–317 (1982)
Gemignani, L.: Quasiseparable structures of companion pencils under the qz-algorithm. Calcolo 42(3–4), 215–226 (2005)
Geronimus, Ya.: Orthogonal Polynomials. Consultants Bureau, New York (1961)
Jones, W.B., Njåstad, O., Thron, W.J.: Moment theory, orthogonal polynomials, quadrature and continued fractions associated with the unit circle. Bull. Lond. Math. Soc. 21, 113–152 (1989)
Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1: Classical theory, Volume 54 of Colloquium Publications. AMS, New York (2005)
Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 2: Spectral theory, Volume 54 of Colloquium Publications. AMS, New York (2005)
Simon, B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120–154 (2007)
Thron, W.J.: L-polynomials orthogonal on the unit circle. In: Cuyt, A.M. (ed.) Nonlinear Numerical Methods and Rational Approximation, pp. 271–278. Kluwer, Dordrecht (1988)
Van Barel, M., Fasino, D., Gemignani, L., Mastronardi, N.: Orthogonal rational functions and structured matrices. SIAM J. Matrix Anal. Appl. 26(3), 810–829 (2005)
Vanberghen, Y., Vandebril, R., Van Barel, M.: A qz-algorithm for semiseparable matrices. J. Comput. Appl. Math. 218(2), 482–491 (2008)
Velázquez, L.: Spectral methods for orthogonal rational functions. J. Comput. Appl. Math. 254(4), 954–986 (2008). arXiv e-print 0704.3456v1
Watkins, D.S.: Some perspectives on the eigenvalues problem. SIAM Rev. 35, 430–471 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bultheel, A., Cantero, MJ. A matricial computation of rational quadrature formulas on the unit circle. Numer Algor 52, 47–68 (2009). https://doi.org/10.1007/s11075-008-9257-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-008-9257-9
Keywords
- Orthogonal rational functions
- Para-orthogonal rational functions
- Szegő quadrature formulas
- Möbius transformations