Skip to main content
SpringerLink
Log in

A new presentation of orthogonal polynomials with applications to their computation

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper a new presentation of orthogonal polynomials is given. It is based on the introduction of two auxiliary sequences of arbitrary monic polynomials and it leads to a very simple derivation of the usual determinantal formulae for orthogonal polynomials and of their recurrence relations either in the definite or in the indefinite case. New expressions for the coefficients of these recurrence relations are obtained and they are compared to the usual ones from the point of view of their numerical stability. The qd-algorithm is also recovered very easily.

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

Similar content being viewed by others

References

  1. C. Brezinski,Padé-type Approximation and General orthogonal Polynomials, ISNM Vol. 50, (Birkhäuser, Basel, 1980).

    Google Scholar 

  2. C. Brezinski, Bordering methods and progressive forms for sequence transformations Zastosow. Mat. 20 (1990) 435–443.

    Google Scholar 

  3. C. Brezinski and M. Redivo Zaglia,Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam) to appear.

  4. C. Brezinski, M. Redivo Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math., to appear.

  5. C. Brezinski and H. Sadok, Lanczos type methods for systems of linear equations, submitted.

  6. A. Draux,Polynômes Orthogonaux Formels. Applications, LNM 974 (Springer-Verlag, Berlin, 1983).

    Google Scholar 

  7. V. N. Faddeeva,Computational Methods of Linear Algebra (Dover, New-York, 1959).

    Google Scholar 

  8. R. Fletcher, Conjugate gradient methods for indefinite systems, in:Numerical Analysis, ed. G.A. Watson, LNM 506 (Springer-Verlag, Berlin, 1976) pp. 73–89.

    Google Scholar 

  9. G.H. Golub and M.H. Gutknecht, Modified moments for indefinite weight functions, Numer. Math. 57 (1990) 607–624.

    Google Scholar 

  10. M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. Parts I, II-s moreo, SIAm J. Matrix Anal. Appl., to appear.

  11. P. Henrici,Applied and Computational Complex Analysis, Vol. 1, Wiley, New-York, 1974.

    Google Scholar 

  12. H. Rutishauser, Der Quotienten-Differenzen-Algorithms, Z. Angew. Math. Physik, 5 (1954) 233–251.

    Google Scholar 

  13. G. Szegö,Orthogonal Polynomials (American Mathematical Society, Providence, 1939).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d'Analyse Numérique et d'Optimisation, UFR IEEA-M3, Université des Sciences et Techniques de Lille Flandres-Artois, F-59655, Villeneuve d'Ascq Cedex, France

    C. Brezinski

  2. Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, via Gradenigo 6/a, I-35131, Padova, Italy

    M. Redivo Zaglia

Authors
  1. C. Brezinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Redivo Zaglia
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brezinski, C., Redivo Zaglia, M. A new presentation of orthogonal polynomials with applications to their computation. Numer Algor 1, 207–221 (1991). https://doi.org/10.1007/BF02142322

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02142322

Subject classifications

Keywords

Search

Navigation