Numerical Algorithms

, Volume 63, Issue 4, pp 573–600

A method of convergence acceleration of some continued fractions II

• Rafał Nowak
Open Access
Original Paper

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.

Keywords

Convergence acceleration Continued fraction Tail Modified approximant

65B99 33F05

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