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A look-ahead strategy for the implementation of some old and new extrapolation methods

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Abstract

Sequence transformations, used for accelerating the convergence, are related to biorthogonal polynomials. In the particular cases of theG-transformation and the Shanks transformation (that is the ε-algorithm of Wynn), there is a connection with formal orthogonal polynomials. In this paper, this connection is exploited in order to propose a look-ahead strategy for the implementation of these two transformations. This strategy, which is quite similar to the strategy used for treating the same type of problems in Lanczos-based methods for solving systems of linear equations, consists in jumping over the polynomials which do not exist, thus avoiding a division by zero (breakdown) in the algorithms, and over those which could be badly computed (near-breakdown) thus leading to a better numerical stability. Numerical examples illustrate the procedure.

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References

  1. R.E. Bank and T.F. Chan, A composite step bi-conjugate gradient algorithm for solving nonsymmetric systems, Numer. Algorithms 7 (1994) 1–16.

    Article  Google Scholar 

  2. C. Brezinski, A general extrapolation algorithm, Numer. Math. 35 (1980) 175–187.

    Article  Google Scholar 

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

    Google Scholar 

  4. C. Brezinski,Biorthogonality and its Applications to Numerical Analysis (Marcel Dekker, New York, 1992).

    Google Scholar 

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

    Google Scholar 

  6. C. Brezinski and M. Redivo-Zaglia, Breakdowns in the computation of orthogonal polynomials, inNonlinear Numerical Methods and Rational Approximation, II, ed. A. Cuyt (Kluwer, Dordrecht, 1994) pp.49–59.

    Google Scholar 

  7. C. Brezinski and M. Redivo-Zaglia, Treatment of near-breakdown in the CGS algorithm. Numer. Algorithms 7 (1994), 33–73.

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. C. Brezinski, M. Redivo-Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms 1 (1991) 207–221.

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. T.F. Chan and T. Szeto, A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems, Numer. Algorithms 7 (1994) 17–32.

    Article  Google Scholar 

  14. A. Draux,Polynômes Orthogonaux Formels. Applications, LNM 974 (Springer, Berlin, 1983).

    Google Scholar 

  15. A. Draux, Formal orthogonal polynomials revisited; applications, Numer. Algorithms 11 (1996), this issue.

  16. Ö. Egecioglu and Ç. Koç, A fast algorthm for rational interpolation via orthogonal polynomials, Math. Comput. 53 (1989) 249–264.

    Google Scholar 

  17. R. Freund and H. Zha, A look-ahead algorithm for the solution of general gankel systems, Numer. Math. 64 (1993) 295–321.

    Article  Google Scholar 

  18. L. Gemignani, Rational interpolation via orthogonal polynomials, Comput. Math. Appl. 26 (1993) 27–34.

    Article  Google Scholar 

  19. W.B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972) 1–62.

    Article  Google Scholar 

  20. W.B. Gragg, Matrix interpretations and applications of the continued fraction algorithm, Rocky Mt. J. Math. 4 (1974) 213–225.

    Google Scholar 

  21. W.B. Gragg and A. Lindquist, On the partial realization problem, Lin. Alg. Appl. 50 (1983) 277–319.

    Article  Google Scholar 

  22. H.L. Gray, T.A. Atchison and G.V. McWilliams, Higher orderG-transformations, SIAM J. Numer. Anal. 8 (1971) 365–381.

    Article  Google Scholar 

  23. 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; Part II, idem, M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. Part II, SIAM J. Matrix Anal. Appl. 15 (1994) 15–58.

    Article  Google Scholar 

  24. T. H⇘vie, Generalized Neville type extrapolation schemes, BIT 19 (1979) 204–213.

    Article  Google Scholar 

  25. P. Maroni, Sur quelques espaces de distributions qui sont des formes linéaires sur l'espace vectoriel des polynômes, in:Polynômes Orthogonaux et Applications, eds. C. Brezinski et al., LNM vol.1171 (Springer, Berlin, 1985) pp. 184–194.

    Google Scholar 

  26. P. Maroni, Prolégomènes à l'étude des polynômes orthogonaux, Ann. Mat. Pura ed Appl. 149 (1987) 165–184.

    Article  Google Scholar 

  27. P. Maroni, Le calcul des formes linéaires et les polynômes orthogonaux semi-classiques, in:Orthogonal Polynomials and their Applications, eds. M. Alfaro et al., LNM vol. 1329 (Springer, Berlin, 1988) pp. 279–288.

    Google Scholar 

  28. P. Maroni, Une théorie algébrique des polynômes orthogonaux, applications aux polynômes semi-classiques, in:Orthogonal Polynomials and their Applications, eds. C.Brezinski et al. (J.C. Baltzer, Basel, 1991) pp. 95–130.

    Google Scholar 

  29. O. Perron,Die Lehre von den Kettenbrüchen (Teubner, Stuttgart, 1913).

    Google Scholar 

  30. W.C. Pye and T.A. Atchison, An algorithm for the computation of the higher orderG-transformation, SIAM J. Numer. Anal. 10 (1973) 1–7.

    Google Scholar 

  31. D. Shanks, Non linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) 1–42.

    Google Scholar 

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

    Article  Google Scholar 

  33. E.L. Stiefel, Kernel polynomials in linear algebra and their numerical applications, in:Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, NBS Appl. Math. Series, vol. 49 (1958), pp. 1–22.

  34. H.S. Wall,Analytic Theory of Continued Fractions (Van Nostrand, Princeton, 1948).

    Google Scholar 

  35. P. Wynn, On a device for computing thee m(S n) transformation, MTAC 10 (1956) 91–96.

    Google Scholar 

  36. P. Wynn, A general system of orthogonal polynomials, Quart. J. Math. Oxford (2) 18 (1967) 81–96.

    Google Scholar 

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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

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  1. C. Brezinski
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  2. M. Redivo-Zaglia
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Brezinski, C., Redivo-Zaglia, M. A look-ahead strategy for the implementation of some old and new extrapolation methods. Numer Algor 11, 35–55 (1996). https://doi.org/10.1007/BF02142487

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