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.