Skip to main content
SpringerLink
Log in

On the calculation of the poles of multivariate meromorphic functions using the symbolic-numeric two-point qd-algorithm

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The so-called quotient-difference algorithm, or qd-algorithm, is used for determining the poles of a meromorphic function from its Taylor coefficients. A generalization of this algorithm to the univariate and multivariate two-point cases applied to a power series (positive or negative exponents) is presented. We describe also the symbolic-numeric two-point qd-algorithm to compute the poles of multivariate meromorphic functions in a given domain from its series expansion coefficients. This algorithm can be regarded as computing the parametrized eigenvalues for a tridiagonal matrix. Numerical examples are furnished to illustrate our results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Abouir, J., Benouahmane, B.: Multivariate homogeneous two-point Padé approximants. Jaen J. Approx. 10(1-2), 29–48 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Abouir, J., Cuyt, A.: Stable multi-dimensional model reduction of two-dimensional digital filters. Int. J. Comput. Sci. Math. 1(1), 16–27 (2007)

    Article  MathSciNet  Google Scholar 

  3. Allouche, H., Cuyt, A.: Reliable root detection with the qd-algorithm: when Bernoulli, Hadamard and Rutishauser cooperate. Appl. Numer. Math. 60, 1188–1208 (2010)

    Article  MathSciNet  Google Scholar 

  4. Benouahmane, B., Cuyt, A.: Properties of multivariate homogeneous orthogonal polynomials. J. Approx. Theory 113, 1–20 (2001)

    Article  MathSciNet  Google Scholar 

  5. Benouahmane, B., Cuyt, A., Verdonk, B.: On the solution of parameterized (smallest, largest or multiple) eigenvalue problems. Technical report, University of Antwerp (UIA) (2002)

  6. Brezinski, C.: Padé-type Approximation and General Orthogonal Polynomials, ISNM 50 pp, vol. 67–105. Basel, Birkhäuser (1980)

    Book  Google Scholar 

  7. Cuyt, A.: On the convergence of the multivariate homogeneous qd-algorithm. BIT. 34, 535–545 (1994)

    Article  MathSciNet  Google Scholar 

  8. Cuyt, A.: Symbolic-Numeric qd-algorithms with Applications in Function Theory and Linear Algebra, Symbolic Algebraic Methods and Verification Methods. Springer Wien GmbH, pp. 35–54 (2001)

  9. Cuyt, A., Verdonk, B.: Extending the qd-algorithm to tackle multivariate problems. In: Papamichael, N., Ruscheweyh, S., Saff, B (eds.) Proceedings of the Third CMFT Conference ”Computational Methods and Function Theory 1997”, pp 135–159. World Scientific, Nicosia (1999)

  10. Diaz-Mendoza, C., Gonzalez-Vera, P., Orive, R.: On the convergence of two-point Padé-type approximants. Numer. Math. 72, 295–312 (1996)

    Article  MathSciNet  Google Scholar 

  11. Draux, A., Sadik, M.: Generalized qd algorithm for block and matrices. Numer. Algo. 61(3), 377–396 (2012)

    Article  MathSciNet  Google Scholar 

  12. Draux, A., Sadik, M.: Qd block algorithm. Appl. Numer. Math. 60(12), 1300–1308 (2010)

    Article  MathSciNet  Google Scholar 

  13. Draux, A.: On two-point Padé-type and two-point Padé approximants. Annali di Matematica Pura ed Applicata. 158(1), 99–150 (1991)

    Article  MathSciNet  Google Scholar 

  14. Dudgeon, D., Mersereau, R.: Multidimentional Digital Signal Processing. Prentice-Hall, Englewood Cliffs (1984)

    MATH  Google Scholar 

  15. Gonzalez-Vera, P.: On certain application of two-point Padé approximants with single pole. J. Comp. Appl. Math. 19, 151–160 (1987)

    Article  Google Scholar 

  16. Henrici, P.: Applied and Computational Complex Analysis. Power Series, Integration Conformal Mapping and Location of Zeros, vol. 1. Wiley, New York (1974)

    MATH  Google Scholar 

  17. O’Connor, B.T., Huang, T.S.: Stability of two-dimentional recursive digital filters. IEEE Trans. Acoust. Speech Signal Process 26, 550–560 (1978)

    Article  MathSciNet  Google Scholar 

  18. Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 5, 233–251 (1954)

    Article  MathSciNet  Google Scholar 

  19. Rutishauser, H.: Lectures on Numerical Mathematics. Birkhäuser, Boston (1990)

    Book  Google Scholar 

Download references

Funding

The work of the first author is supported by a CNRST Scientific Scholarship.

Author information

Authors and Affiliations

  1. Faculty of Sciences and Technology, Laboratory LMCMAN, Hassan II University of Casablanca, BP 146, 20650, Mohammedia, Morocco

    A. Elidrissi, J. Abouir & B. Benouahmane

Authors
  1. A. Elidrissi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Abouir
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Benouahmane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Abouir.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elidrissi, A., Abouir, J. & Benouahmane, B. On the calculation of the poles of multivariate meromorphic functions using the symbolic-numeric two-point qd-algorithm. Numer Algor 84, 1443–1458 (2020). https://doi.org/10.1007/s11075-020-00887-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00887-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation