Estimates of the speed of convergence of continued fraction expansions of functions

Field, David A.

doi:10.1090/S0025-5718-1977-0433830-6

Estimates of the speed of convergence of continued fraction expansions of functions
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Math. Comp. 31 (1977), 495-502 Request permission

Abstract:

The best recently published methods of obtaining a priori and a posteriori truncation error estimates of continued fractions are reviewed. The applicability of these methods is discussed and a numerical example with tables of their actual estimates is provided.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comp. 28 (1974), 239–251. MR 333287, DOI 10.1090/S0025-5718-1974-0333287-7
George A. Baker Jr., Best error bounds for Padé approximants to convergent series of Stieltjes, J. Mathematical Phys. 10 (1969), 814–820. MR 259080, DOI 10.1063/1.1664911
G. Blanch, Numerical evaluation of continued fractions, SIAM Rev. 6 (1964), 383–421. MR 171374, DOI 10.1137/1006092
David Elliott, Truncation errors in Padé approximations to certain functions: An alternative approach, Math. Comp. 21 (1967), 398–406. MR 227668, DOI 10.1090/S0025-5718-1967-0227668-6
David A. Field and William B. Jones, A priori estimates for truncation error of continued fractions $K(1/b_{n})$, Numer. Math. 19 (1972), 283–302. MR 314247, DOI 10.1007/BF01404877
William B. Gragg, Truncation error bounds for $g$-fractions, Numer. Math. 11 (1968), 370–379. MR 228158, DOI 10.1007/BF02161885
W. B. Gragg, Truncation error bounds for $\pi$-fractions, Bull. Amer. Math. Soc. 76 (1970), 1091–1094. MR 262723, DOI 10.1090/S0002-9904-1970-12574-5
T. L. Hayden, Continued fraction approximation to functions, Numer. Math. 7 (1965), 292–309. MR 185798, DOI 10.1007/BF01436523
Peter Henrici and Pia Pfluger, Truncation error estimates for Stieltjes fractions, Numer. Math. 9 (1966), 120–138. MR 212991, DOI 10.1007/BF02166031
Thomas H. Jefferson, Truncation error estimates for $T$-fractions, SIAM J. Numer. Anal. 6 (1969), 359–364. MR 260147, DOI 10.1137/0706033
William B. Jones and R. I. Snell, Truncation error bounds for continued fractions, SIAM J. Numer. Anal. 6 (1969), 210–221. MR 247737, DOI 10.1137/0706019
William B. Jones and W. J. Thron, A posteriori bounds for the truncation error of continued fractions, SIAM J. Numer. Anal. 8 (1971), 693–705. MR 295536, DOI 10.1137/0708063
Yudell L. Luke, The Padé table and the $\tau$-method, J. Math. and Phys. 37 (1958), 110–127. MR 99114, DOI 10.1002/sapm1958371110
E. P. Merkes, On truncation errors for continued fraction computations, SIAM J. Numer. Anal. 3 (1966), 486–496. MR 202283, DOI 10.1137/0703042
Morio Onoe, Tables of modified quotients of Bessel functions of the first kind for real and imaginary arguments, Columbia University Press, New York, 1958. MR 0095590
Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., New York, N. Y., 1950 (German). 2d ed. MR 0037384
W. B. Sweezy and W. J. Thron, Estimates of the speed of convergence of certain continued fractions, SIAM J. Numer. Anal. 4 (1967), 254–270. MR 216734, DOI 10.1137/0704024
W. J. Thron, On parabolic convergence regions for continued fractions, Math. Z. 69 (1958), 173–182. MR 96064, DOI 10.1007/BF01187398
W. J. Thron, A survey of recent convergence results for continued fractions, Rocky Mountain J. Math. 4 (1974), 273–282. MR 349974, DOI 10.1216/RMJ-1974-4-2-273
H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746