Numerical Algorithms

, Volume 41, Issue 3, pp 297–317

# A method of convergence acceleration of some continued fractions

Article

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.

### Keywords

convergence acceleration continued fraction tail modified approximant

### AMS subject classification

30B70 40A15 65B99

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### References

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