Numerical Algorithms

, Volume 41, Issue 3, pp 297–317 | Cite as

A method of convergence acceleration of some continued fractions

Article

A new method of convergence acceleration is proposed for continued fractions \(b_0+K(a_n/b_n)\), where \(a_n\) and \(b_n\) are polynomials in \(n\) (\(\deg \,a_{n} = 2\), \(\deg \,b_{n} \leqslant 1\)) for \(n\) sufficiently large. It uses the fact that the modified approximant \(S_n(t_n')\) approaches the continued fraction value, if \(t_n'\) is sufficiently close to the \(n\)th tail \(t_n\). Presented method is of iterative character; in each step, by means of an approximation \(t_n'\), it produces a new better approximation \(t_n''\) of the \(n\)th tail \(t_n\). Formula for \(t_n''\) is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.

Keywords

convergence acceleration continued fraction tail modified approximant 

AMS subject classification

30B70 40A15 65B99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. Brezinski, Succesive modifications of limit periodic continued fractions, J. Comput. Appl. Math. 19 (1987) 67–74.MATHMathSciNetGoogle Scholar
  2. [2]
    L. Lorentzen, Computation of limit periodic continued fractions. A survey, Numer. Algorithms 10 (1995) 69–111.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    L. Lorentzen and H. Waadeland, Continued Fractions with Applications (North-Holland, Amsterdam, 1992).MATHGoogle Scholar
  4. [4]
    S. Paszkowski, Convergence acceleration of continued fractions with tails of square-irrational expansion (in preparation).Google Scholar
  5. [5]
    S. Paszkowski, Convergence acceleration of some continued fractions, Numer. Algorithms 32 (2003) 193–247.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    O. Perron, Die Lehre von den Kettenbrüchen, Bd II. Analytisch-funktionentheoretische Kettenbrüche, 3. Aufl. (B.G. Teubner, Stuttgart, 1957).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

Personalised recommendations