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Numerical stability in evaluating continued fractions


Authors: William B. Jones and W. J. Thron
Journal: Math. Comp. 28 (1974), 795-810
MSC: Primary 65G05
DOI: https://doi.org/10.1090/S0025-5718-1974-0373265-5
MathSciNet review: 0373265
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Abstract: A careful analysis of the backward recurrence algorithm for evaluating approximants of continued fractions provides rigorous bounds for the accumulated relative error due to rounding. Such errors are produced by machine operations which carry only a fixed number v of significant digits in the computations. The resulting error bounds are expressed in terms of the machine parameter v. The derivation uses a basic assumption about continued fractions, which has played a fundamental role in developing convergence criteria. Hence, its appearance in the present context is quite natural. For illustration, the new error bounds are applied to two large classes of continued fractions, which subsume many expansions of special functions of physics and engineering, including those represented by Stieltjes fractions. In many cases, the results insure numerical stability of the backward recurrence algorithm.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, With Formulas, Graphs and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D.C., 1964. MR 29 #4914. MR 0167642 (29:4914)
  • [2] G. Blanch, "Numerical evaluation of continued fractions," SIAM Rev., v. 6, 1964, pp. 383-421. MR 30 #1605. MR 0171374 (30:1605)
  • [3] W. Gautschi, "Computational aspects of three-term recurrence relations," SIAM Rev., v. 9, 1967, pp. 24-82. MR 35 #3927. MR 0213062 (35:3927)
  • [4] P. R. Graves-Morris (Editor), Padé Approximants and Their Applications, Proc. Conference (University of Kent, Canterbury, England, July 1972), Academic Press, New York, 1973. MR 0435681 (55:8639)
  • [5] W. B. Gragg, "The Padé table and its relation to certain algorithms of numerical analysis," SIAM Rev., v. 14, 1972, pp. 1-62. MR 46 #4693. MR 0305563 (46:4693)
  • [6] William B. Jones & W. J. Thron, "Convergence of continued fractions," Canad. J. Math., v. 20, 1968, pp. 1037-1055. MR 37 #6446. MR 0230888 (37:6446)
  • [7] William B. Jones & W. J. Thron (Editors), Proceedings of the International Conference on Padé Approximants, Continued Fractions and Related Topics, Special issue of the Rocky Mountain J. Math. (To appear.) MR 0327433 (48:5775)
  • [8] N. Macon & M. Baskervill, "On the generation of errors in the digital evaluation of continued fractions," J. Assoc. Comput. Mach., v. 3, 1956, pp. 199-202. MR 18, 337. MR 0080987 (18:337a)
  • [9] W. J. Thron, "A survey of recent convergence results for continued fractions," to appear in [7]. MR 0349974 (50:2467)
  • [10] H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, Princeton, N.J., 1948. MR 10, 32. MR 0025596 (10:32d)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1974-0373265-5
Keywords: Numerical stability, error analysis, continued fraction, special functions
Article copyright: © Copyright 1974 American Mathematical Society

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