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On the convergence behavior of continued fractions with real elements


Author: Walter Gautschi
Journal: Math. Comp. 40 (1983), 337-342
MSC: Primary 40A15; Secondary 33A15
DOI: https://doi.org/10.1090/S0025-5718-1983-0679450-2
MathSciNet review: 679450
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Abstract: We define the notion of transient (geometric) convergence rate for infinite series and continued fractions. For a class of continued fractions with real elements we prove a monotonicity property for such convergence rates which helps explain the effectiveness of certain continued fractions known to converge "only" sublinearly. This is illustrated in the case of Legendre's continued fraction for the incomplete gamma function.


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

  • [1] W. Gautschi, "A computational procedure for incomplete gamma functions," ACM Trans. Math. Software, v. 5, 1979, pp. 466-481. MR 547763 (81f:65015)
  • [2] W. Gautschi, "Algorithm 542--Incomplete gamma function," ACM Trans. Math. Software, v. 5, 1979, pp. 482-489.
  • [3] P. Henrici, Applied and Computational Complex Analysis, Vol. 2, Wiley, New York, 1977. MR 0453984 (56:12235)
  • [4] E. P. Merkes, "On truncation errors for continued fraction computations," SIAM J. Numer. Anal., v. 3, 1966, pp. 486-496. MR 0202283 (34:2156)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0679450-2
Keywords: Convergence of real continued fractions, Legendre's continued fraction for the incomplete gamma function
Article copyright: © Copyright 1983 American Mathematical Society

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