Skip to main content
SpringerLink
Log in

Generalizations of Shanks transformation and corresponding convergence acceleration algorithms via pfaffians

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Firstly, a new sequence transformation that can be expressed in terms of a ratio of two pfaffians is derived based on a special kernel. It can be regarded as a direct generalization of Aitken’s Δ2 process from the point of view of pfaffians and then the corresponding convergence acceleration algorithm is constructed. Numerical examples with applications of this algorithm are also presented. Secondly, we find a way to generalize the Shanks transformation via pfaffians so that a larger class of new sequence transformations are derived. The corresponding recursive algorithms are also proposed.

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

Fig. 1

Similar content being viewed by others

References

  1. Aitken, A.C.: On Bernoulli’s numerical solution of algebraic equations. Proc. R. Soc. Edinburgh 46, 289–305 (1927)

    Article  Google Scholar 

  2. Barbeau, E.J.: Euler subdues a very obstreperous series. Am. Math. Mon. 86(5), 356–372 (1979)

    Article  MathSciNet  Google Scholar 

  3. Barbeau, E.J., Leah, P.J.: Euler’s 1760 paper on divergent series. Hist. Math. 3(2), 141–160 (1976)

    Article  MathSciNet  Google Scholar 

  4. Benchiboun, M.D.: Etude de certaines généralisations du Δ2 d’Aitken et comparaison de procédés d’accélération de la convergence. Université, de Lille I, Thèse 3ème cycle (1987)

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Brezinski, C.: Quasi-linear extrapolation processes. In: Numerical Mathematics Singapore 1988, pp 61–78. Springer (1988)

  8. Brezinski, C.: A bibliography on continued fractions, padé approximation, sequence transformation and related subjets Prensas Universitarias de Zaragoza (1991)

  9. Brezinski, C.: Extrapolation algorithms and padé approximations: a historical survey. Appl. Numer. Math. 20(3), 299–318 (1996)

    Article  MathSciNet  Google Scholar 

  10. Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122(1-2), 1–21 (2000)

    Article  MathSciNet  Google Scholar 

  11. Brezinski, C., He, Y., Hu, X.B., Redivo-Zaglia, M., Sun, J.Q.: Multistep ε–algorithm, Shanks transformation, and the Lotka–Volterra system by Hirota’s method. Math. Comput. 81(279), 1527–1549 (2012)

    Article  MathSciNet  Google Scholar 

  12. Brezinski, C., Redivo-Zaglia, M.: Extrapolation methods: Theory and practice. North-Holland, Amsterdam (1991)

  13. Brezinski, C., Redivo-Zaglia, M.: Generalizations of Aitken’s process for accelerating the convergence of sequences. Comput. Appl. Math. 26(2), 171–189 (2007)

    Article  MathSciNet  Google Scholar 

  14. Brezinski, C., Redivo-Zaglia, M.: The genesis and early developments of Aitken’s process, Shanks transformation, the ε–algorithm, and related fixed point methods. Numer. Alg. 80(1), 11–133 (2019)

    Article  MathSciNet  Google Scholar 

  15. Brezinski, C., Redivo-Zaglia, M.: Extrapolation and rational approximation: The works of the main contributors. Springer Nature, Switzerland (2020)

    Book  Google Scholar 

  16. Bromwich, T.J.I.: An introduction to the theory of infinite series, 3rd edn. Chelsea. Originally published by Macmillan (London, 1908 and 1926) New York (1991)

  17. Caianiello, E.R.: Combinatorics and renormalization in quantum field theory. W.A Benjamin Inc, (1973)

  18. Carstensen, C.: On a general epsilon algorithm. Numerical and Applied Mathematics, Baltzer, Basel 437–441 (1989)

  19. Chang, X.K., He, Y., Hu, X.B., Li, S.H.: A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia’s sequence transformation via pfaffians. Numer. Alg. 78(1), 87–106 (2018)

    Article  MathSciNet  Google Scholar 

  20. Chang, X.K., He, Y., Hu, X.B., Sun, J.Q., Weniger, E.J.: Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order. Numer. Alg. 83(2), 593–627 (2020)

    Article  MathSciNet  Google Scholar 

  21. Delahaye, J.P.: Sequence Transformations. Springer-Verlag, Berlin (1988)

    Book  Google Scholar 

  22. Delahaye, J.P., Germain-Bonne, B.: Résultats négatifs en accélération de la convergence. Numer. Math. 35(4), 443–457 (1980)

    Article  MathSciNet  Google Scholar 

  23. Hardy, G.H.: Divergent Series. Clarendon Press, Oxford (1949)

    MATH  Google Scholar 

  24. He, Y., Hu, X.B., Sun, J.Q., Weniger, E.J.: Convergence acceleration algorithm via an equation related to the lattice Boussinesq equation. SIAM J. Sci. Comput. 33(3), 1234–1245 (2011)

    Article  MathSciNet  Google Scholar 

  25. Hirota, R.: The direct method in soliton theory. Cambridge University Press, Translated by Nagai, A., Nimmo, J. and Gilson, C (2004)

  26. Miki, H., Goda, H., Tsujimoto, S.: Discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems. SIGMA Symmetry, Integrability and Geometry: Methods and Applications 8, 008 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Ohta, Y.: Special solutions of discrete integrable systems. In: Discrete Integrable Systems, pp 57–83. Springer (2004)

  28. Reed, M., Simon, B.: Methods of modern mathematical physics iv: Analysis of operators (1978)

  29. Shanks, D.: An analogy between transient and mathematical sequences and some nonlinear sequence-to-sequence transforms suggested by it, Part I, Memorandum 9994 (1949)

  30. Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J Math Phys 34(1-4), 1–42 (1955)

    Article  MathSciNet  Google Scholar 

  31. Sidi, A.: Practical extrapolation methods: Theory and applications. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  32. Smith, D.A., Ford, W.F.: Acceleration of linear and logarithmic convergence. SIAM J. Numer. Anal. 16(2), 223–240 (1979)

    Article  MathSciNet  Google Scholar 

  33. Sun, J.Q., Chang, X.K., He, Y., Hu, X.B.: An extended multistep Shanks transformation and convergence acceleration algorithm with their convergence and stability analysis. Numer. Math. 125, 785–809 (2013)

    Article  MathSciNet  Google Scholar 

  34. Walz, G.: Asymptotics and Extrapolation. Akademie Verlag, Berlin (1996)

    MATH  Google Scholar 

  35. Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10(5-6), 189–371 (1989)

    Article  Google Scholar 

  36. Wimp, J.: Sequence Transformations and Their Applications. Academic Press, New York (1981)

    MATH  Google Scholar 

  37. Wynn, P.: On a device for computing the em(sn) transformation. Mathematical Tables and Other Aids to Computation, 91–96 (1956)

Download references

Acknowledgements

We thank Dr. S.H. Li for his useful conversations.

Funding

X.K. Chang was supported in part by the National Natural Science Foundation of China (Grant Nos. 11688101, 11731014, 11701550) and the Youth Innovation Promotion Association CAS. Y. He was supported in part by the National Natural Science Foundation of China (Grant No. 11971473), the Youth Innovation Promotion Association CAS, and the National Key Research and Development Program of China (No. 2020YFA0714200). X.B. Hu was supported in part by the National Natural Science Foundation of China (Grant Nos. 11931017, 11871336 and 12071447).

Author information

Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PO Box 2719, Beijing, 100190, People’s Republic of China

    Ya-Jie Liu, Xiang-Ke Chang & Xing-Biao Hu

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Ya-Jie Liu, Xiang-Ke Chang & Xing-Biao Hu

  3. Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Yi He

  4. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Yi He

Authors
  1. Ya-Jie Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang-Ke Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi He
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xing-Biao Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiang-Ke Chang.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, YJ., Chang, XK., He, Y. et al. Generalizations of Shanks transformation and corresponding convergence acceleration algorithms via pfaffians. Numer Algor 88, 1733–1756 (2021). https://doi.org/10.1007/s11075-021-01092-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01092-y

Keywords

Search

Navigation