Computation of limit periodic continued fractions. A survey

A method of convergence acceleration of some continued fractions II

“…Consider a two-variant CF of the form (2), belonging to the class D. In some cases, one may approximate its value using the following classic methods: (A) Computing classical approximants S n (0) for sufficiently large n, (B) Applying the methods of [7] to even (odd) part of CF, (C) Applying the methods of [10] to even (odd) part of CF, (D) Using equivalent transformation in order to obtain limit 2-periodic CF, and using the techniques given in [7], (E) Using the recent methods of Paszkowski [12]; for more details on numerical computation of CFs see, e.g., [9,Section 5].…”

“…In the case of the classic methods (B), the most efficient one is 'improvement machine' of Lorentzen and Waadeland (see [5], [7,Section 6] or [9, p. 227]). However their idea can be applied only to CFs of loxodromic type.…”

“…Similar concept was used in 'improvement machine'-method of Lorentzen and Waadeland [5], [7,Section 6] (see also [9, p. 227] for more details), and later by Paszkowski [11]. However, 'improvement machine' cannot be applied to all the subclasses in D and, as numerical examples show, is less efficient than ours.…”

“…[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.…”

“…[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. 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.…”