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Breakdowns In The Computation Of Orthogonal Polynomials

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Nonlinear Numerical Methods and Rational Approximation II

Part of the book series: Mathematics and Its Applications ((MAIA,volume 296))

Abstract

When computing recursively the members of a family of formal orthogonal polynomials, a division by zero can occur, thus producing a breakdown in the algorithm which has to be stopped. In this paper, such breakdowns are analyzed in detail and classified. It is also showed how to avoid them in some particular cases. Applications to Padé approximation, extrapolation methods and Lanczos method for solving systems of linear equations are discussed.

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References

  1. H. Allouche, A. Cuyt, Well-defined determinant representations for non-normal multivariate rational interpolants, Numerical Algorithms, to appear.

    Google Scholar 

  2. C. Baheux, Algorithmes d’Implémentation de la Méthode de Lanczos,Thèse, Université des Sciences et Technologies de Lille, 1994.

    Google Scholar 

  3. C. Brezinski, Padé-type Approximation and General Orthogonal Polynomials.,ISNM vol. 50, Birkhäuser, Basel, 1980.

    MATH  Google Scholar 

  4. C. Brezinski, The Mühlbach-Neville-Aitken algorithm and some extensions, BIT, 20 (1980) 444–451.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Brezinski, Other manifestations of the Schur complement, Linear Algebra Appl.,111 (1988) 231–247.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Brezinski, Algebraic properties of the E-transformation, in Numerical Analysis and Mathematical Modelling, Banach Center Publications, vol. 24, PWN, Warsaw, 1990, pp. 85–90.

    Google Scholar 

  7. C. Brezinski, M.Morandi Cecchi, M. Redivo-Zaglia, The reverse bordering method, SIAM J. Matrix Anal. Appl., to appear.

    Google Scholar 

  8. C. Brezinski, M. Redivo-Zaglia, Extrapolation Methods. Theory and Practice, North-Holland, Amsterdam, 1991.

    MATH  Google Scholar 

  9. C. Brezinski, M. Redivo-Zaglia, A new presentation of orthogonal polynomials with applications to their computation, Numerical Algorithms, 1 (1991) 207–222.

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Brezinski, M. Redivo-Zaglia, Treatment of near-breakdown in the CGS algorithm, Numerical Algorithms, to appear.

    Google Scholar 

  11. C. Brezinski, M. Redivo-Zaglia, Look-ahead in Bi-CGSTAB and other methods for linear systems, to appear.

    Google Scholar 

  12. C. Brezinski, M. Redivo-Zaglia, H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math., 63 (1992) 29–38.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Brezinski, M. Redivo-Zaglia, H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numerical Algorithms, 1 (1991) 261–284.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Brezinski, M. Redivo-Zaglia, H. Sadok, Addendum to “Avoiding breakdown and near-breakdown in Lanczos type algorithms”, Numerical Algorithms, 2 (1992) 133–136.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Brezinski, M. Redivo-Zaglia, H. Sadok, Breakdowns in the implementation of the Lánczos method for solving linear systems, Inter. J. Comp. Math. with Applies., to appear.

    Google Scholar 

  16. C. Brezinski, H. Sadok, Lanczos type algorithms for solving systems of linear equations, Appl. Numer. Math.,11 (1993) 443–473.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Brezinski, H. Sadok, Avoiding breakdown in the CGS algorithm, Numerical Algorithms, 1 (1991) 199–206.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Draux, Polynômes Orthogonaux Formels. Applications, LNM 974, Springer—Verlag, Berlin, 1983.

    MATH  Google Scholar 

  19. M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms Part I, SIAM J. Matrix Anal. Appl.,13 (1992) 594–639.

    Article  MathSciNet  MATH  Google Scholar 

  20. M.R. Hestenes, The conjugate—gradient method for solving linear systems, in Proceedings of the Sixth Symposium on Applied Mathematics, J. Curtiss ed., Amer. Math. Soc., Providence, 1956, pp.83–102.

    Google Scholar 

  21. W.D. Joubert Generalized Conjugate Gradient and Lanczos Methods for the Solution of Nonsymmetric Systems of Linear Equations, Ph.D. Thesis, The University of Texas at Austin, 1983.

    Google Scholar 

  22. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand., 45 (1950) 255–282.

    Article  MathSciNet  Google Scholar 

  23. C. Lanczos, Solution of systems of linear equations by minimized iterations, J. Res. Natl. Bur. Stand.,49 (1952) 33–53.

    Article  MathSciNet  Google Scholar 

  24. P. Sonneveld, CGS, a fast Lanczos—type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comp.,10 (1989) 36–52.

    Article  MathSciNet  MATH  Google Scholar 

  25. H.A. Van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 13 (1992) 631–644.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’Analyse Numérique et d’Optimisation UFR IEEA - M3, Université des Sciences et Technologies de Lille, 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
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  2. M. Redivo-Zaglia
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Editor information

Editors and Affiliations

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

    Annie Cuyt

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© 1994 Springer Science+Business Media Dordrecht

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Brezinski, C., Redivo-Zaglia, M. (1994). Breakdowns In The Computation Of Orthogonal Polynomials. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_5

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  • DOI: https://doi.org/10.1007/978-94-011-0970-3_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4420-2

  • Online ISBN: 978-94-011-0970-3

  • eBook Packages: Springer Book Archive

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