Skip to main content

Advertisement

SpringerLink
A method of convergence acceleration of some continued fractions II
Download PDF
Download PDF
  • Original Paper
  • Open Access
  • Published: 19 September 2012

A method of convergence acceleration of some continued fractions II

  • Rafał Nowak1 

Numerical Algorithms volume 63, pages 573–600 (2013)Cite this article

  • 618 Accesses

  • Metrics details

Abstract

Most of the methods for convergence acceleration of continued fractions K(a m /b m ) are based on the use of modified approximants S m (ω m ) in place of the classical ones S m (0), where ω m are close to the tails f (m) of the continued fraction. Recently (Nowak, Numer Algorithms 41(3):297–317, 2006), the author proposed an iterative method producing tail approximations whose asymptotic expansion accuracies are being improved in each step. This method can be successfully applied to a convergent continued fraction K(a m /b m ), where \(a_m = \alpha_{-2} m^2 + \alpha_{-1} m + \ldots\), b m  = β  − 1 m + β 0 + ... (α  − 2 ≠ 0, \(|\beta_{-1}|^2+|\beta_{0}|^2\neq 0\), i.e. \(\deg a_m=2\), \(\deg b_m\in\{0,1\}\)). The purpose of this paper is to extend this idea to the class of two-variant continued fractions K (a n /b n  + a n ′/b n ′) with a n , a n ′, b n , b n ′ being rational in n and \(\deg a_n=\deg a_n'\), \(\deg b_n=\deg b_n'\). We give examples involving continued fraction expansions of some elementary and special mathematical functions.

Download to read the full article text

Working on a manuscript?

Avoid the most common mistakes and prepare your manuscript for journal editors.

Learn more

References

  1. Birkhoff, G.D.: Formal theory of irregular linear difference equations. Acta Math. 54, 205–246 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  2. Birkhoff, G.D., Trjitzinsky, W.J.: Analytic theory of singular difference equations. Acta Math. 60, 1–89 (1933)

    Article  MathSciNet  Google Scholar 

  3. Cuyt, A., Petersen, V.B., Verdonk, B., Waadeland, H., Jones, W.B.: Handbook of Continued Fractions for Special Functions. Springer, Dordrecht (2008)

    MATH  Google Scholar 

  4. Hautot, A.: Convergence acceleration of continued fractions of Poincaré type. Appl. Numer. Math. 4(2–4), 309–322 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jacobsen, L., Waadeland, H.: Convergence acceleration of limit periodic continued fractions under asymptotic side conditions. Numer. Math. 53(3), 285–298 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jones, W.B., Thron, W.J.: Continued Fractions. Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, vol. 11. Addison-Wesley, London (1980) (distributed by Cambridge University Press, New York)

    Google Scholar 

  7. Lorentzen, L.: Computation of limit periodic continued fractions. A survey. Numer. Algorithms 10(1–2), 69–111 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  9. Lorentzen, L., Waadeland, H.: Continued Fractions. Convergence Theory, vol. 1, 2nd edn. Atlantis Studies in Mathematics for Engineering and Science 1. World Scientific, Hackensack, NJ (2008)

    Book  Google Scholar 

  10. Nowak, R.: A method of convergence acceleration of some continued fractions. Numer. Algorithms 41(3), 297–317 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Paszkowski, S.: Convergence acceleration of continued fractions of Poincaré’s type 1. Numer. Algorithms 2(2), 155–170 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Paszkowski, S.: Convergence acceleration of some continued fractions. Numer. Algorithms 32(2–4), 193–247 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Perron, O.: Die Lehre von den Kettenbrüchen. Bd II. Analytisch-funktionentheoretische Kettenbrüche. 3. Aufl. B.G. Teubner, Stuttgart (1957)

  14. Pincherle, S.: Delle funzioni ipergeometriche e di varie questioni ad esse attinenti. Giorn. Mat. Battaglini 32, 209–291 (1894)

    MATH  Google Scholar 

  15. Thron, W., Waadeland, H.: Accelerating convergence of limit periodic continued fractions K(a n /1). Numer. Math. 34, 155–170 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wall, H.S.: Analytic Theory of Continued Fractions. Van Nostrand, New York (1948)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Wrocław, ul. Joliot-Curie 15, 50-383, Wrocław, Poland

    Rafał Nowak

Authors
  1. Rafał Nowak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafał Nowak.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Nowak, R. A method of convergence acceleration of some continued fractions II. Numer Algor 63, 573–600 (2013). https://doi.org/10.1007/s11075-012-9642-2

Download citation

  • Received: 11 March 2012

  • Accepted: 27 August 2012

  • Published: 19 September 2012

  • Issue Date: August 2013

  • DOI: https://doi.org/10.1007/s11075-012-9642-2

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Convergence acceleration
  • Continued fraction
  • Tail
  • Modified approximant

Mathematics Subject Classifications (2010)

  • 65B99
  • 33F05
Download PDF

Working on a manuscript?

Avoid the most common mistakes and prepare your manuscript for journal editors.

Learn more

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • California Privacy Statement
  • How we use cookies
  • Manage cookies/Do not sell my data
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.