Convergence acceleration of continued fractions of Poincaré's type 1

A method of convergence acceleration of some continued fractions II

“…This process is rather technical and requires to study all the considered CFs subclasses, separately. One can also verify that for all considered subclasses in D except the subclass D = 10 , the coefficients τ 2 j−1 vanish, and so the asymptotic expansion (12) contains only the powers of n −1 . We have prepared the Maple™ program computing symbolically the coefficients τ j of (12), in the cases of all six subclasses in D; download the file tvcf-bilequ.mw (or its HTML and PDF version) from the page http://www.rafalnowak.pl/publications/tvcf.…”

“…However, the bilinear equation (10) has two solutions of the form (11). According to Paszkowski [12,p.…”

“…In Theorem 3.2, we give this coefficient explicitly. Since the roots of (13) may be zero, we do not call τ −ν the leading coefficient of the expansion (12). Once we have the initial coefficient τ −ν , we can easily find the next.…”

“…[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.…”

“…Then, a standard sequence transformation, such as Brezinki's θ or u-Levin algorithm, was applied to {S m (ω m )}. Paszkowski [11] showed that such classic sequence transformations may be unsuitable to the character of approximants of CF. He proposed the iterative method producing modifying factors which accuracy increases in each step.…”