A new method of convergence acceleration is proposed for continued fractions \(b_0+K(a_n/b_n)\), where \(a_n\) and \(b_n\) are polynomials in \(n\) (\(\deg \,a_{n} = 2\), \(\deg \,b_{n} \leqslant 1\)) for \(n\) sufficiently large. It uses the fact that the modified approximant \(S_n(t_n')\) approaches the continued fraction value, if \(t_n'\) is sufficiently close to the \(n\)th tail \(t_n\). Presented method is of iterative character; in each step, by means of an approximation \(t_n'\), it produces a new better approximation \(t_n''\) of the \(n\)th tail \(t_n\). Formula for \(t_n''\) is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.
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Communicated by C. Brezinski
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Nowak, R. A method of convergence acceleration of some continued fractions. Numer Algor 41, 297–317 (2006). https://doi.org/10.1007/s11075-005-9013-3
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DOI: https://doi.org/10.1007/s11075-005-9013-3