Numerical Algorithms

, Volume 63, Issue 4, pp 573–600 | Cite as

A method of convergence acceleration of some continued fractions II

Open Access
Original Paper


Most of the methods for convergence acceleration of continued fractions K(am/bm) are based on the use of modified approximants Sm(ωm) in place of the classical ones Sm(0), where ωm 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(am/bm), where \(a_m = \alpha_{-2} m^2 + \alpha_{-1} m + \ldots\), bm = β − 1m + β0 + ... (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 (an/bn + an′/bn′) with an, an′, bn, bn′ 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.


Convergence acceleration Continued fraction Tail Modified approximant 

Mathematics Subject Classifications (2010)

65B99 33F05 


  1. 1.
    Birkhoff, G.D.: Formal theory of irregular linear difference equations. Acta Math. 54, 205–246 (1930)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Birkhoff, G.D., Trjitzinsky, W.J.: Analytic theory of singular difference equations. Acta Math. 60, 1–89 (1933)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cuyt, A., Petersen, V.B., Verdonk, B., Waadeland, H., Jones, W.B.: Handbook of Continued Fractions for Special Functions. Springer, Dordrecht (2008)MATHGoogle Scholar
  4. 4.
    Hautot, A.: Convergence acceleration of continued fractions of Poincaré type. Appl. Numer. Math. 4(2–4), 309–322 (1988)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Jacobsen, L., Waadeland, H.: Convergence acceleration of limit periodic continued fractions under asymptotic side conditions. Numer. Math. 53(3), 285–298 (1988)MathSciNetMATHCrossRefGoogle Scholar
  6. 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. 7.
    Lorentzen, L.: Computation of limit periodic continued fractions. A survey. Numer. Algorithms 10(1–2), 69–111 (1995)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. North-Holland, Amsterdam (1992)MATHGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Nowak, R.: A method of convergence acceleration of some continued fractions. Numer. Algorithms 41(3), 297–317 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Paszkowski, S.: Convergence acceleration of continued fractions of Poincaré’s type 1. Numer. Algorithms 2(2), 155–170 (1992)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Paszkowski, S.: Convergence acceleration of some continued fractions. Numer. Algorithms 32(2–4), 193–247 (2003)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Perron, O.: Die Lehre von den Kettenbrüchen. Bd II. Analytisch-funktionentheoretische Kettenbrüche. 3. Aufl. B.G. Teubner, Stuttgart (1957)Google Scholar
  14. 14.
    Pincherle, S.: Delle funzioni ipergeometriche e di varie questioni ad esse attinenti. Giorn. Mat. Battaglini 32, 209–291 (1894)MATHGoogle Scholar
  15. 15.
    Thron, W., Waadeland, H.: Accelerating convergence of limit periodic continued fractions K(a n/1). Numer. Math. 34, 155–170 (1980)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wall, H.S.: Analytic Theory of Continued Fractions. Van Nostrand, New York (1948)MATHGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

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