Convergence acceleration for continued fractions 𝐾(𝑎_{𝑛}/1)

Jacobsen, Lisa

doi:10.1090/S0002-9947-1983-0678349-1

Convergence acceleration for continued fractions $K(a_{n}/1)$
HTML articles powered by AMS MathViewer

by Lisa Jacobsen PDF

Trans. Amer. Math. Soc. 275 (1983), 265-285 Request permission

Abstract:

A known method for convergence acceleration of limit periodic continued fractions $K({a_n}/1),{a_n} \to a$, is to replace the approximants ${S_n}(0)$ by "modified approximants" ${S_n}({f^{\ast }})$, where $f^{\ast } = K(a/1)$. The present paper extends this idea to a larger class of converging continued fractions. The "modified approximants" will then be ${S_n}({f^{(n)’}})$, where $K({a’_n}/1)$ is a converging continued fraction whose tails ${f^{(n)\prime }}$ are all known, and where ${a_n} - a_n^\prime \to 0$. As a measure for the improvement obtained by this method, upper bounds for the ratio of the two truncation errors are found.

References

John Gill, Infinite compositions of Möbius transformations, Trans. Amer. Math. Soc. 176 (1973), 479–487. MR 316690, DOI 10.1090/S0002-9947-1973-0316690-6
John Gill, The use of attractive fixed points in accelerating the convergence of limit-periodic continued fractions, Proc. Amer. Math. Soc. 47 (1975), 119–126. MR 352774, DOI 10.1090/S0002-9939-1975-0352774-1

Modifying factors for sequences of linear fractional transformations

On the transformation of continued products into continued fractions

5

T. L. Hayden, Continued fraction approximation to functions, Numer. Math. 7 (1965), 292–309. MR 185798, DOI 10.1007/BF01436523
William B. Jones and W. J. Thron, Twin-convergence regions for continued fractions $K(a_{n}/1)$, Trans. Amer. Math. Soc. 150 (1970), 93–119. MR 264043, DOI 10.1090/S0002-9947-1970-0264043-9
William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864

Vorlesungen über Zahlenlehre

Uniform convergence and truncation error estimates of continued fractions

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
W. J. Thron and H. Waadeland, Accelerating convergence of limit periodic continued fractions $K(a_{n}/1)$, Numer. Math. 34 (1980), no. 2, 155–170. MR 566679, DOI 10.1007/BF01396057
P. Wynn, Converging factors for continued fractions. I, II, Numer. Math. 1 (1959), 272–320. MR 116158, DOI 10.1007/BF01386391