Numerical Algorithms

, Volume 63, Issue 4, pp 573–600

# A method of convergence acceleration of some continued fractions II

Open Access
Original Paper

## Abstract

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.

### Keywords

Convergence acceleration Continued fraction Tail Modified approximant

65B99 33F05

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