Skip to main content
SpringerLink
Log in

A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia’s sequence transformation via pfaffians

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

Abstract

In the literature, most known sequence transformations can be written as a ratio of two determinants. But, it is not always this case. One exception is that the sequence transformation proposed by Brezinski, Durbin, and Redivo-Zaglia cannot be expressed as a ratio of two determinants. Motivated by this, we will introduce a new algebraic tool—pfaffians, instead of determinants in the paper. It turns out that Brezinski–Durbin–Redivo-Zaglia’s transformation can be expressed as a ratio of two pfaffians. To the best of our knowledge, this is the first time to introduce pfaffians in the expressions of sequence transformations. Furthermore, an extended transformation of high order is presented in terms of pfaffians and a new convergence acceleration algorithm for implementing the transformation is constructed. Then, the Lax pair of the recursive algorithm is obtained which implies that the algorithm is integrable. Numerical examples with applications of the algorithm are also presented.

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

Similar content being viewed by others

References

  1. Aitken, A.C.: On Bernoullis numerical solution of algebraic equations. Proc. Roy. Soc. Edinb. 46, 289–305 (1926)

    Article  MATH  Google Scholar 

  2. Aitken, A.C.: Determinants and matrices. Oliver and Boyd, Edinburgh (1959)

    MATH  Google Scholar 

  3. Barbeau, E.J.: Euler subdues a very obstreperous series. Amer. Math. Monthly 86, 356–372 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benchiboun, M.D.: Etude de certaines généralisations du △2 d’Aitken et comparaison de procédés d’accélération de la convergence, thèse 3ème cycle, Université, de Lille I (1987)

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

    MATH  Google Scholar 

  6. Brezinski, C.: Conditions d’application et de convergence de procds d’extrapolation. Numer. Math. 20, 64–79 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezinski, C.: Quasi-linear extrapolation processes. In: Agarwal, R.P., et al. (eds.) Numerical Mathematics, Singapore 1988, ISNM, vol. 86, pp 61–78. Birk̈hauser-Verlag, Basel (1988)

  9. Brezinski, C.: A bibliography on continued fractions, Padé approximation sequence transformation and related subjects (Zaragoza, Spain: Prensas Universitarias de Zaragoza) (1991)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Brezinski, C.: Cross rules and non-Abelian lattice equations for the discrete and confluent non-scalar ε-algorithms. J. Phys. A: Math. Theor. 43, 5201–11 (2010)

    Article  MathSciNet  Google Scholar 

  13. Brezinski, C., Crouzeix, M.: Remarques sur le precédé △2 aitken d , Aitken. C. R. Acad. Sci. Paris 270 A, 896–898 (1970)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Brezinski, C., Redivo-Zaglia, M.: Shanks function transformations in a vector space. Appl. Numer. Math. 116, 57–63 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brezinski, C., He, Y., Hu, X.B., Sun, J.Q.: A generalization of the G-transformation and the related algorithms. Appl. Numer. Math. 60, 1221–1230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brezinski, C., He, Y., Hu, X.B., Sun, J.Q., Tam, H.W.: Confluent form of the multistep ε-algorithm, and the relevant integrable system. Stud. Appl. Math. 127, 191–209 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Brualdi, R.A., Schneider, H.: Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley. Linear Algebra Appl. 52/53, 769–791 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Caieniello, E.R.: Combinatorics and renormalization in quantum field theory, (Benjamin) (1973)

  22. Carstensen, C.: On a general epsilon algorithm Numerical and applied mathematics, Part II (Paris, 1988), IMACS Ann. Comput. Appl. Math. 1.2, pp 437–441. Baltzer, Basel (1989)

    Google Scholar 

  23. Ford, W.F., Sidi, A.: An algorithm for a generalization of the richardson extrapolation process. SIAM J. Numer. Anal. 24, 1212–1232 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  25. He, Y., Hu, X.B., Tam, H.W.: A q-difference version of the ε-algorithm. J. Phys. A: Math. Theor. 42, 095202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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, 1234–1245 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. He, Y., Hu, X.B., Tam, H.W., Zhang, Y.N.: A new method to generate non-autonomous discrete integrable systems via convergence acceleration algorithms. Eur. J. Appl. Math. 27, 194–212 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50, 3785–3791 (1981)

    Article  MathSciNet  Google Scholar 

  29. Hirota, R.: The direct method in soliton theory, translated from the 1992 Japanese Original and Edited by Nagai, A., Nimmo, J. and Gilson, C. With a Foreword by Hietarinta, J. and Nimmo, J. in Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  30. Iwasaki, M., Nakamura, Y.: On the convergence of a solution of the discrete Lotka–Volterra system. Inverse Prob. 18, 1569–1578 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Iwasaki, M., Nakamura, Y.: An application of the discrete Lotka–Volterra system with variable step-size to singular value computation. Inverse Prob. 20, 553–563 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Miki, M., Goda, H., Tjujimoto, S.: Discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems. SIGMA 8, 008 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Nagai, A., Satsuma, J.: Discrete soliton equations and convergence acceleration algorithms. Phys. Lett. A 209, 305–312 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nagai, A., Tokihiro, T., Satsuma, J.: The Toda molecule equation and the ε-algorithm. Math. Comp. 67, 1565–1575 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nakamura, Y.: Calculating laplace transforms in terms of the toda molecule. SIAM J. Sci. Comput. 20, 306–317 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nakamura, Y. (ed.): Applied integrable systems. Tokyo, Syokabo (2000). (in Japanese)

  37. Ohta, Y.: Special solutions of discrete integrable systems. In: Grammaticos, B., Tamizhmani, T., Kosmann-Schwarzbach, Y. (eds.) Discrete Integrable Systems, Lecture Notes in Phys, vol. 644, pp 57–83. Springer, Berlin (2004)

  38. Papageorgiou, V., Grammaticos, B., Ramani, A.: Integrable lattices and convergence acceleration algorithms. Phys. Lett. A 179, 111–115 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  39. Papageorgiou, V., Grammaticos, B., Ramani, A.: Integrable dicerence equations and numerical analysis algorithms. In: Levi, D., et al. (eds.) Symmetries and integrability of dicerence equations, CRM proceedings and lecture notes, vol. 9, pp 269–279. AMS, Providence (1996)

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Reed, M., Simon, B.: Methods of modern Mathematical physics IV: analysis of operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  42. Rutishauser, H.: Der quotienten-differenzen-algorithmus Z. Angew. Math. Physik 5, 233–251 (1954)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Sun, J.Q., He, Y., Hu, X.B., Tam, H.W.: Q-difference and confluent forms of the lattice boussinesq equation and the relevant convergence acceleration algorithms. J. Math. Phys. 52, 023522 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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  MATH  Google Scholar 

  46. Symes, W.W.: The QR algorithm and scattering for the finite nonperiodic toda lattice. Phys. D 4, 275–280 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tsujimoto, S., Nakamura, Y., Iwasaki, M.: The discrete Lotka-Volterra system computes singular values. Inverse Prob. 17, 53–58 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comp. Phys. Reports 10, 189–371 (1989)

    Article  Google Scholar 

  49. Wimp, J.: Sequence transformations and their applications. Academic Press, New York (1981)

    MATH  Google Scholar 

  50. Wynn, P.: On a device for computing the e m (s n ) transformation. Math. Tables Aids Comput. 10, 91–96 (1956)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11331008, Grant No. 11371251, Grant no. 11201469, Grant No. 11571358).

Author information

Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100090, People’s Republic of China

    Xiang-Ke Chang, Xing-Biao Hu & Shi-Hao Li

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

    Yi He

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

    Xing-Biao Hu & Shi-Hao Li

Authors
  1. Xiang-Ke Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xing-Biao Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shi-Hao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shi-Hao Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, XK., He, Y., Hu, XB. et al. A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia’s sequence transformation via pfaffians. Numer Algor 78, 87–106 (2018). https://doi.org/10.1007/s11075-017-0368-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0368-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation