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A general extrapolation algorithm

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Summary

In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied.

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References

  1. Brezinski C (1971) Accélération de suites à convergence logarithmique. CR Acad Sc Paris, 273 A:727–730

    Google Scholar 

  2. Brezinski C (1975) Généralisations de la transformation de Shanks, de la table de Padé et de l'εalgorithme. Calcolo, 12:317–360

    Google Scholar 

  3. Brezinski C (1977) Accélération de la convergence en analyse numérique, (Lecture Notes in Mathematics vol 584). Springer, Berlin Heidelberg New York

    Google Scholar 

  4. Brezinski C (1978) Algorithmes d'accélération de la convergence. Etude numérique, Technip, Paris

    Google Scholar 

  5. Brezinski C (1979) Sur le calcul de certains rapports de déterminants. In: Wuytack L (ed) Padé approximation and its applications (Lecture Notes in Mathematics vol 765). Springer, Berlin Heidelberg New York

    Google Scholar 

  6. Cordellier F (1980) Analyse numérique des transformations de suites et de séries. Thèse, Université de Lille (in press)

  7. Gantmacher FR (1960) The theory of matrices. Chelsea Publications, New York

    Google Scholar 

  8. Germain-Bonne B (1978) Estimation de la limite de suites et formalisation de procédés d'accélération de convergence. Thèse, Université de Lille

  9. Gray HL, Atchison TA (1968) The generalizedG-transform. Math. Comput 22:595–606

    Google Scholar 

  10. Gray HL, Schucany WR (1969) Some limiting cases of theG-transformation. Math Comput 23:849–859

    Google Scholar 

  11. Håvie T (1979) Generalized Neville type extrapolation schemes. BIT 19:204–213

    Google Scholar 

  12. Laurent PJ (1964) Etude de procédés d'extrapolation en analyse numérique. Thèse, Université de Grenoble

  13. Levin D (1973) Development of non-linear transformations for improving convergence of sequences. Internat J Comput Math B3:371–388

    Google Scholar 

  14. Mühlbach G (1978) The general Neville-Aitken algorithm and some applications. Numer Math 31:97–110

    Google Scholar 

  15. Pye WC, Atchison TA (1973) An algorithm for the computation of higher orderG-transformations. SIAM J Numer Anal 10:1–7

    Google Scholar 

  16. Shanks D (1955) Nonlinear transformations of divergent and slowly convergent sequences. J Math Phys 34:1–42

    Google Scholar 

  17. Sidi A (1979) Some properties of a generalization of the Richardson extrapolation process. Technical report 142, Technion, Haifa, Israel

    Google Scholar 

  18. Smith DA, Ford WF (1979) Acceleration of linear and logarithmic convergence. SIAM J Numer Anal 16:223–240

    Google Scholar 

  19. Wynn P (1956) On a device for computing thee m (Sn) transformation. MTAC 10:91–96

    Google Scholar 

  20. Wynn P (1956) On a procrustean technique for the numerical transformation of slowly convergent sequences ans series. Proc Cambridge Phil Soc 52:663–671

    Google Scholar 

  21. Wynn P (1960) Confluent forms of certain nonlinear algorithms. Arch Math 11:223–234

    Google Scholar 

  22. Wynn P (1962) Acceleration technique in numerical analysis with particular reference to problems in one independant variable. Proc IFIP congress, North Holland Amsterdam, p 149

    Google Scholar 

  23. Wynn P (1966) On the convergence and stability of the epsilon algorithm. SIAM J Numer Anal 3:91–122

    Google Scholar 

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Brezinski, C. A general extrapolation algorithm. Numer. Math. 35, 175–187 (1980). https://doi.org/10.1007/BF01396314

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