A Posteriori Bounds for the Truncation Error of Continued Fractions

• Published in: SIAM journal on numerical analysis Vol. 8; no. 4; pp. 693 - 705
• Main Authors: Jones, William B., Thron, W. J.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01-12-1971
• Subjects: A posteriori knowledge; Complex numbers; Continued fractions; Error bounds; Hypergeometric functions; Mathematical constants; Mathematics; Truncation errors
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0708063

Truncation error bounds for continued fractions are obtained in terms of general conditions which ensure that the approximants $\{w_n \}$ form a simple sequence; i.e., that |wn + m - wn| ≤ c|wn - wn - 1|, where c is a constant, independent of n ≥ 1 and m ≥ 1. The method is based on establishing the existence of a nested sequence { Ωn } of bounded, convex regions with the following two properties: (a) wn + m ∈ Ωn for all n ≥ 1 and m ≥ -1, and (b) the diameter of Ωn + 1 is bounded above by c|wn - wn - 1|. Applications are considered for several classes of continued fraction expansions including the continued fractions of Gauss (hypergeometric functions), S-fractions, J-fractions and T-fractions.