Accelerating convergence of limit periodic continued fractionsK(a n /1)

Abstract

It is shown that the convergence of limit periodic continued fractions K(a,/1) with lima,=a can be substantially accelerated by replacing the sequence of approximations {S,(0)} by the sequence {S.(Xl)}, where x 1 = -1/2 +lf]~+ a. Specific estimates of the improvement are derived. Subject Classifications: AMS (MOS): 30B70, 40A15, 40D15, 65B99.

A method of convergence acceleration of some continued fractions II

“…, 10) yield the following values of acc(f n ): On the other hand, from (39) we have that a n + 1 4 = O(n −3 ). By virtue of theorem of Thron and Waadeland [15], we obtain that modified approximants S n (−1/2) are convergent to the value V faster than classical approximants S n (0), and so 'fixed point' method works. However, it is not very efficient.…”

“…One can verify that conditions given by Thron and Waadeland in [15] are not satisfied, so we cannot use their results to accelerate the convergence. Indeed, 'fixed point' method fails, i.e.…”

“…In fact, (45) does not satisfy the assumptions of Thron and Waadeland [15]. Namely, applying 'fixed point' method to CF K(ã n /1), equivalent to (44), seems to be worthless, since This example shows that convergence of considered CF cannot be accelerated using any of the classic methods.…”

Solution formulas for linear difference equations with applications to continued fractions

“…When transforming (7) and (9) into a first-order system and when using the results of Section 2 explicit solution formulas (28) for (7) are obtained which have a structure similar to (17). It turned out that in the special case aj(n)e C, hi(n)= 0 for 1 _< j < q and all n, an explicit solution formula, of course in different notation, had already been derived by D. Andr6 [-7, Section 178].…”

“…The method of modified continued fractions was developed and investigated in the work of J. Gill [2], [3], W. Thron and H. Waadeland [17]- [19], and by L. Jacobsen [4], [5]. In these papers mainly special types of limit periodic analytic continued fractions, such as J-fractions and T-fractions, are investigated.…”