Numerical Algorithms

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

A method of convergence acceleration of some continued fractions

  • Rafał Nowak

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.


convergence acceleration continued fraction tail modified approximant 

AMS subject classification

30B70 40A15 65B99 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

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