SpringerLink
Log in

A generalized eigenvalue problem for quasi-orthogonal rational functions

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among \({\{\alpha_1,\ldots,\alpha_n\}\subset(\mathbb{C}_0\cup\{\infty\})}\), are not all real (unless \({\alpha_n}\) is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF or a so-called para-ORF are used instead. These zeros depend on one single parameter \({\tau\in(\mathbb{C}\cup\{\infty\})}\), which can always be chosen in such a way that the zeros are all real and simple. In this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi-ORF and the corresponding weights in the RGQ. First, we study the connection between quasi-ORFs, para-ORFs and ORFs. Next, a condition is given for the parameter τ so that the zeros are all real and simple. Finally, some illustrative and numerical examples are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Thailand)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baltensperger, R., Berrut, J.-P., Dubey, Y.: The linear rational pseudospectral method with preassigned poles. Numer. Algorithms 33(1–4), 53–63 (2003) [International Conference on Numerical Algorithms, vol. I. Marrakesh (2001)]

    Google Scholar 

  2. Berrut, J.-P., Baltensperger, R.: The linear rational pseudospectral method for boundary value problems. BIT 41(Suppl. 5), 868–879 (2001) (BIT 40th Anniversary Meeting)

    Google Scholar 

  3. Berrut J.-P., Mittelmann H.D.: Optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems. J. Comput. Phys. 204(1), 292–301 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bultheel A., Cantero M.J.: A matrical computation of rational quadrature formulas on the unit circle. Numer. Algorithms 52(1), 47–68 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bultheel A., Cruz-Barroso R., Deckers K., González-Vera P.: Rational Szegő quadratures associated with Chebyshev weight functions. Math. Comput. 78(266), 1031–1059 (2009)

    MATH  Google Scholar 

  6. Bultheel A., González Vera P., Hendriksen E., Njåstad O.: Orthogonal rational functions. In: Cambridge Monographs on Applied and Computational Mathematics, vol. 5. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  7. Bultheel A., González-Vera P., Hendriksen E., Njåstad O.: Orthogonal rational functions and quadrature on the real half line. J. Complex. 19(3), 212–230 (2003)

    Article  MATH  Google Scholar 

  8. Bultheel A., González-Vera P., Hendriksen E., Njåstad O.: Orthogonal rational functions on the real half line with poles in [−∞, 0]. J. Comput. Appl. Math. 179(1–2), 121–155 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bultheel A., González-Vera P., Hendriksen E., Njåstad O.: Quadrature and orthogonal rational functions. J. Comput. Appl. Math. 127(1–2), 67–91 (2001) (Invited paper)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cantero M.J., Moral L., Velázquez L.: Measures and paraorthogonal polynomials on the unit circle. East J. Approx. 8(4), 447–464 (2002)

    MATH  MathSciNet  Google Scholar 

  11. Chihara T.S.: On quasi-orthogonal polynomials. Proc. Am. Math. Soc. 8, 765–767 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  12. Day D., Romero L.: Roots of polynomials expressed in terms of orthogonal polynomials. SIAM J. Numer. Anal. 43(5), 1969–1987 (2005) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  13. Deckers, K.: Orthogonal Rational Functions: Quadrature, Recurrence and Rational Krylov, p. 270 + lxiv. PhD thesis, Department of Computer Science, Katholieke Universiteit Leuven, Leuven (2009)

  14. Deckers K., Bultheel A.: Orthogonal rational functions with complex poles: The Favard theorem. J. Math. Anal. Appl. 356(2), 764–768 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Deckers K., Bultheel A.: Recurrence and asymptotics for orthogonal rational functions on an interval. IMA J. Numer. Anal. 29(1), 1–23 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Deckers K., Van Deun J., Bultheel A.: An extended relation between orthogonal rational functions on the unit circle and the interval [−1, 1]. J. Math. Anal. Appl. 334(2), 1260–1275 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Deckers K., Van Deun J., Bultheel A.: Computing rational Gauss-Chebyshev quadrature formulas with complex poles: The algorithm. Adv. Eng. Softw. 40(8), 707–717 (2009)

    Article  MATH  Google Scholar 

  18. Deckers K., Van Deun J., Bultheel A.: Rational Gauss–Chebyshev quadrature formulas for complex poles outside [−1, 1]. Math. Comput. 77(262), 967–983 (2008)

    MATH  MathSciNet  Google Scholar 

  19. Džrbašian, M.M.: A survey on the theory of orthogonal systems and some open problems. In: Nevai, P. (ed.) Orthogonal Polynomials: Theory and Practice. NATO-ASI Series C: Mathematical and Physical Sciences, vol. 294, pp. 135–146, Kluwer Academic Publishers, Boston (1990)

  20. Fritzsche B., Kirstein B., Lasarow A.: Orthogonal rational matrix-valued functions on the unit circle. Math. Nachrichten 278(5), 525–553 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Fritzsche B., Kirstein B., Lasarow A.: Orthogonal rational matrix-valued functions on the unit circle: Recurrence relations and a Favard-type theorem. Math. Nachrichten 279(5–6), 513–542 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gautschi W., Gori L., Lo Cascio M.L.: Quadrature rules for rational functions. Numer. Math. 86(4), 617–633 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Golinskii L.B.: Quadrature formula and zeros of para-orthogonal polynomials on the unit circle. Acta Math. Hungarica 96(3), 169–186 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. 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)

    Article  MATH  Google Scholar 

  25. Müller, K., Bultheel, A.: Orthogonal systems of rational functions on the unit circle (Translation of the Russian paper by M.M. Džrbašian). Technical Report TW253, Department of Computer Science, K.U.Leuven (1997)

  26. Riesz M.: Sur le problème des moments, troisième note. Arkiv for Matematik, Astronomi och Fysik 17(16), 52 (1923)

    Google Scholar 

  27. Simon B.: Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloquium Series American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  28. Simon B.: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory. AMS Colloquium Series American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  29. Simon B.: Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle. J. Math. Anal. Appl. 329(1), 376–382 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tee T.W., Trefethen L.N.: A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28(5), 1798–1811 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Van Assche W., Vanherwegen I.: Quadrature formulas based on rational interpolation. Math. Comput. 61(204), 765–783 (1993)

    MATH  MathSciNet  Google Scholar 

  32. Van Deun J.: Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles. Numer. Algorithms 45(1–4), 89–99 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Van Deun J., Bultheel A.: A quadrature formula based on Chebyshev rational functions. IMA J. Numer. Anal. 26(4), 641–656 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Van Deun J., Bultheel A.: An interpolation algorithm for orthogonal rational functions. J. Comput. Appl. Math. 164/165, 749–762 (2004)

    Article  MathSciNet  Google Scholar 

  35. Van Deun, J., Bultheel, A.: Computing orthonormal rational functions on a halfline. Rendiconti del Circolo di Palermo Serie II. In: Proceedings of the 5th International Conference on Functional Analysis and Approximation Theory (FAAT), 16–22 June, vol. 76, pp. 621–634, Maratea (2005)

  36. Van Deun J., Bultheel A.: Modified moments and orthogonal rational functions. Appl. Numer. Anal. Comput. Math. 1(2), 455–468 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Van Deun J., Bultheel A.: Orthogonal rational functions and quadrature on an interval. J. Comput. Appl. Math. 153(1-2), 487–495 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  38. Van Deun J., Bultheel A., González-Vera P.: Computing orthogonal rational functions with poles near the boundary. Comput. Math. Appl. 53(9), 1421–1428 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Van Deun J., Bultheel A., González-Vera P.: On computing rational Gauss–Chebyshev quadrature formulas. Math. Comput. 75(253), 307–326 (2006)

    MATH  Google Scholar 

  40. Van Deun J., Deckers K., Bultheel A., Weideman J.A.C.: Algorithm 882: Near best fixed pole rational interpolation with applications in spectral methods. ACM Trans. Math. Softw. 35(2), 14:1–14:21 (2008)

    Article  MathSciNet  Google Scholar 

  41. Van gucht P., Bultheel A.: Using orthogonal rational functions for system identification. IEEE Trans. Autom. Control 48(4), 705–709 (2003)

    Article  MathSciNet  Google Scholar 

  42. Weideman, J.A.C.: Spectral methods based on nonclassical orthogonal polynomials. In: Applications and Computation of Orthogonal Polynomials (Oberwolfach, 1998). International Series of Numerical Mathematics, vol. 131, pp. 239–251. Birkhäuser. Basel (1999)

  43. Weideman J.A.C., Laurie D.P.: Quadrature rules based on partial fraction expansions. Numer. Algorithms 24, 159–178 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, Belgium

    K. Deckers & A. Bultheel

  2. Department of Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium

    J. Van Deun

Authors
  1. K. Deckers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Bultheel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Van Deun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Deckers.

Additional information

This work is partially supported by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.

K. Deckers is a Postdoctoral Fellow of the Research Foundation, Flanders (FWO).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deckers, K., Bultheel, A. & Van Deun, J. A generalized eigenvalue problem for quasi-orthogonal rational functions. Numer. Math. 117, 463–506 (2011). https://doi.org/10.1007/s00211-010-0356-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-010-0356-x

Mathematics Subject Classification (2000)

Search

Navigation