A method of convergence acceleration of some continued fractions II

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.