Skip to main content
SpringerLink
Log in

Accelerating infinite products

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Slowly convergent infinite products \(\prod\nolimits_{n - 1}^\infty {b_n }\) are considered, where \(\left\{ {b_n } \right\}\) is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the “remainder” of the infinite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and efficient for some classes of sequences \(\left\{ {b_n } \right\}\). For complicated sequences \(\left\{ {b_n } \right\}\) it involves the solution of some linear systems, but it is still effective.

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

  2. A.M. Cohen, A divergent technique for the approximate solution of some simultaneous linear equations by iterative methods, J. Inst. Math. Appl. 19 (1977) 187–193.

    MATH  MathSciNet  Google Scholar 

  3. A.M. Cohen and I.M. Longman, The summation of power series having positive coefficients, Appl. Numer. Math. 4 (1988) 471–476.

    Article  MATH  MathSciNet  Google Scholar 

  4. D.A. Gismalla, L.D. Jenkins and A.M. Cohen, Acceleration of convergence of series with application to the evaluation of certain multiple integrals, Internat. J. Comput. Math. 24 (1988) 155–168.

    MATH  Google Scholar 

  5. D. Levin, Development of nonlinear transformations for improving convergence of sequences, Internat. J. Comput. Math. B 3 (1973) 371–388.

    MATH  Google Scholar 

  6. D. Levin, Ph.D. thesis, Tel Aviv University (1975).

  7. D. Levin and A. Sidi, Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series, Appl. Math. Comput. 9 (1981) 175–215.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Levin and A. Sidi, Rational approximations from the d-transformation, IMA J. Numer. Anal. (1982) 153–167.

  9. A. Sidi, Convergence properties of some nonlinear sequence transformations, Math. Comp. 33 (1979) 315–326.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Sidi, Analysis of convergence of the T-transformation for power series, Math. Comp. 35 (1980) 833–850.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. Alan M. Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cohen, A.M., Levin, D. Accelerating infinite products. Numerical Algorithms 22, 157–165 (1999). https://doi.org/10.1023/A:1019106823947

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019106823947

Search

Navigation