## 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 ${S_n}({f^{(n)β}})$, where $K({aβ_n}/1)$ 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 MΓΆbius transformations*, Trans. Amer. Math. Soc.**176**(1973), 479β487. 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), 119β126. MR**352774**, DOI 10.1090/S0002-9939-1975-0352774-1
β, - T. L. Hayden,
*Continued fraction approximation to functions*, Numer. Math.**7**(1965), 292β309. 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), 93β119. 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**
A. Pringsheim, - W. J. Thron,
*On parabolic convergence regions for continued fractions*, Math. Z.**69**(1958), 173β182. MR**96064**, DOI 10.1007/BF01187398 - W. J. Thron,
*A survey of recent convergence results for continued fractions*, Rocky Mountain J. Math.**4**(1974), 273β282. 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.Β 2, 155β170. MR**566679**, DOI 10.1007/BF01396057 - P. Wynn,
*Converging factors for continued fractions. I, II*, Numer. Math.**1**(1959), 272β320. MR**116158**, DOI 10.1007/BF01386391

*Modifying factors for sequences of linear fractional transformations*, Norske Vid. Selsk. Skr. (Trondheim) (3) (1978). J. W. L. Glaisher,

*On the transformation of continued products into continued fractions*, Proc. London Math. Soc.

**5**(1873/74).

*Vorlesungen ΓΌber Zahlenlehre*, Band 1/2, Leipzig, 1916. Walter M. Reid,

*Uniform convergence and truncation error estimates of continued fractions*$K({a_n}/1)$, Ph.D. Thesis, University of Colorado, Boulder, 1978.

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**275**(1983), 265-285 - MSC: Primary 40A15; Secondary 30B70
- DOI: https://doi.org/10.1090/S0002-9947-1983-0678349-1
- MathSciNet review: 678349