A method of convergence acceleration of some continued fractions

Abstract

A new method of convergence acceleration is proposed for continued fractions b 0 þ Kða n =b n Þ, where a n and b n are polynomials in n (deg a n ¼ 2, deg b n 1) for n sufficiently large. It uses the fact that the modified approximant S n ðt 0 n Þ approaches the continued fraction value, if t 0 n is sufficiently close to the nth tail t n . Presented method is of iterative character; in each step, by means of an approximation t 0 n , it produces a new better approximation t 00 n of the nth tail t n . Formula for t 00 n is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.

A method of convergence acceleration of some continued fractions II

“…In this paper, we extend the results in [10] to the class of so-called twovariant CFs ∈ D kl , i.e. of the form (2), where a n = p −k n k + p 1−k n k−1 + .…”

“…In the sequel, we investigate the asymptotic behaviour of the product I n considering both the solutions of the recurrence (10). The following analysis derives a suitable choice of the coefficients τ j , such that…”

“…For example, the good approximations of the tails of CF in (1) can be obtained using so-called 'improvement machine'-the method of Lorentzen and Waadeland, since its can be transformed equivalently into a limit periodic CF of loxodromic type; see [7] or [9, p. 227] for more details. One can also verify that the even part of CF in (1) can be effectively computed using 'plus method'-one of the methods given in [10], where the convergence acceleration of CFs ∈ C 20 ∪ C 21 was proposed; C kl means the class In the case of CFs of the form (2), the performance of the methods in [10] can be increased, if one not contract the CF into even or odd part, but take into account the two-variant character of CF.…”

“…[4,5,7,8,[10][11][12]. For example, the good approximations of the tails of CF in (1) can be obtained using so-called 'improvement machine'-the method of Lorentzen and Waadeland, since its can be transformed equivalently into a limit periodic CF of loxodromic type; see [7] or [9, p. 227] for more details.…”

“…He proposed the iterative method producing modifying factors which accuracy increases in each step. The idea of the method in [10] is quite similar, but there were not used calculations of square roots. In this paper we continue this idea and deal with two-variant CFs (2).…”