A new method of convergence acceleration is proposed for continued fractions \documentclass[12pt]{minimal}
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\begin{document}$$b_0+K(a_n/b_n)$$\end{document}, where \documentclass[12pt]{minimal}
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\begin{document}$$a_n$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$b_n$$\end{document} are polynomials in \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document} (\documentclass[12pt]{minimal}
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\begin{document}$$\deg \,a_{n} = 2$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$$\deg \,b_{n} \leqslant 1$$\end{document}) for \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document} sufficiently large. It uses the fact that the modified approximant \documentclass[12pt]{minimal}
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\begin{document}$$S_n(t_n')$$\end{document} approaches the continued fraction value, if \documentclass[12pt]{minimal}
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\begin{document}$$t_n'$$\end{document} is sufficiently close to the \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}th tail \documentclass[12pt]{minimal}
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\begin{document}$$t_n$$\end{document}. Presented method is of iterative character; in each step, by means of an approximation \documentclass[12pt]{minimal}
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\begin{document}$$t_n'$$\end{document}, it produces a new better approximation \documentclass[12pt]{minimal}
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\begin{document}$$t_n''$$\end{document} of the \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}th tail \documentclass[12pt]{minimal}
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\begin{document}$$t_n$$\end{document}. Formula for \documentclass[12pt]{minimal}
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\begin{document}$$t_n''$$\end{document} is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.
