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Baltus, Christopher; Jones, William B.

Truncation error bounds for modified continued fractions with applications to special functions. (English) Zbl 0675.65008

Numer. Math. 55, No. 3, 281-307 (1989).
Let \(K(a_ n,1;x_ 1)\) be a limit-periodic modified continued fraction with \(\lim_{n\to \infty}a_ n=a\in {\mathbb{C}}-(-\infty,1/4)\) and n-th approximant \[ g_ n=S_ n(x_ 1)=a_ 1/1+a_ 2/1+...+a_{n- 1}/1+a_ n/(1+x_ 1), \] where \(x_ 1\) denotes the smaller (in modulus) of the two fixed points of \(T(w)=a/(1+w).\) Further, let \(f_ n=S_ n(0)\) be the n-th ordinary reference continued fraction \(K(a_ n/1)\). The authors give truncation error bounds for both \(g_ n\) and \(f_ n\) and show that certain a posteriori bounds for \(g_ n\) are the best possible. The paper also includes results on the speed of convergence and applications to a number of special functions. Very interesting numerical examples indicate the sharpness of the error bounds.
Reviewer: L.Gatteschi

Cited in 6 Documents

MSC:

65D20 Computation of special functions and constants, construction of tables
30B70 Continued fractions; complex-analytic aspects
30E10 Approximation in the complex plane
41A20 Approximation by rational functions
40A15 Convergence and divergence of continued fractions

Keywords:

limit-periodic modified continued fraction; truncation error bounds; speed of convergence; numerical examples
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References:

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