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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R.E. Bank and T.F. Chan, A composite step bi-conjugate gradient algorithm for solving nonsymmetric systems, Numer. Algorithms 7 (1994) 1–16.
C. Brezinski, A general extrapolation algorithm, Numer. Math. 35 (1980) 175–187.
C. Brezinski,Padé-type Approximation and General Orthogonal Polynomials, ISNM vol. 50 (Birkhäuser, Basel, 1980).
C. Brezinski,Biorthogonality and its Applications to Numerical Analysis (Marcel Dekker, New York, 1992).
C. Brezinski and M. Redivo-Zaglia,Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam, 1991).
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.
C. Brezinski and M. Redivo-Zaglia, Treatment of near-breakdown in the CGS algorithm. Numer. Algorithms 7 (1994), 33–73.
C. Brezinski, M. Redivo-Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math. 63 (1992) 29–38.
C. Brezinski, M. Redivo-Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms 1 (1991) 207–221.
C. Brezinski, M. Redivo-Zaglia and H. Sadok, Addendum to “Avoiding breakdown and nearbreakdown in Lanczos type algorithms”, Numer. Algorithms 2 (1992) 133–136.
C. Brezinski and H. Sadok, Lanczos type methods for solving systems of linear equations, Appl. Numer. Math. 11 (1993) 443–473.
C. Brezinski and H. Sadok, Avoiding breakdown in the CGS algorithm, Numer. Algorithms 1 (1991) 199–206.
T.F. Chan and T. Szeto, A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems, Numer. Algorithms 7 (1994) 17–32.
A. Draux,Polynômes Orthogonaux Formels. Applications, LNM 974 (Springer, Berlin, 1983).
A. Draux, Formal orthogonal polynomials revisited; applications, Numer. Algorithms 11 (1996), this issue.
Ö. Egecioglu and Ç. Koç, A fast algorthm for rational interpolation via orthogonal polynomials, Math. Comput. 53 (1989) 249–264.
R. Freund and H. Zha, A look-ahead algorithm for the solution of general gankel systems, Numer. Math. 64 (1993) 295–321.
L. Gemignani, Rational interpolation via orthogonal polynomials, Comput. Math. Appl. 26 (1993) 27–34.
W.B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972) 1–62.
W.B. Gragg, Matrix interpretations and applications of the continued fraction algorithm, Rocky Mt. J. Math. 4 (1974) 213–225.
W.B. Gragg and A. Lindquist, On the partial realization problem, Lin. Alg. Appl. 50 (1983) 277–319.
H.L. Gray, T.A. Atchison and G.V. McWilliams, Higher orderG-transformations, SIAM J. Numer. Anal. 8 (1971) 365–381.
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.
T. H⇘vie, Generalized Neville type extrapolation schemes, BIT 19 (1979) 204–213.
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.
P. Maroni, Prolégomènes à l'étude des polynômes orthogonaux, Ann. Mat. Pura ed Appl. 149 (1987) 165–184.
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.
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.
O. Perron,Die Lehre von den Kettenbrüchen (Teubner, Stuttgart, 1913).
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.
D. Shanks, Non linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) 1–42.
P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comp. 10 (1989) 36–52.
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.
H.S. Wall,Analytic Theory of Continued Fractions (Van Nostrand, Princeton, 1948).
P. Wynn, On a device for computing thee m(S n) transformation, MTAC 10 (1956) 91–96.
P. Wynn, A general system of orthogonal polynomials, Quart. J. Math. Oxford (2) 18 (1967) 81–96.
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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
- Orthogonal polynomials
- sequence transformations
- extrapolation methods
- numerical stability
AMS subject classification