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_− 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 (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.