The use of attractive fixed points in accelerating the convergence of limit-periodic continued fractions

Author:
John Gill

Journal:
Proc. Amer. Math. Soc. **47** (1975), 119-126

MathSciNet review:
0352774

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Abstract | References | Additional Information

Abstract: A continued fraction can be interpreted as a composition of Möbius transformations. Frequently these transformations have powerful attractive fixed points which, under certain circumstances, can be used as converging factors for the continued fraction. The limit of a sequence of such fixed points can be employed as a constant converging factor.

**[1]**L. R. Ford,*Automorphic functions*, McGraw-Hill, New York, 1929.**[2]**John Gill,*Attractive fixed points and continued fractions*, Math. Scand.**33**(1973), 261–268 (1974). MR**0351043****[3]**John Gill,*Infinite compositions of Möbius transformations*, Trans. Amer. Math. Soc.**176**(1973), 479–487. MR**0316690**, 10.1090/S0002-9947-1973-0316690-6**[4]**T. L. Hayden,*Continued fraction approximation to functions*, Numer. Math.**7**(1965), 292–309. MR**0185798****[5]**A. N. Hovanskiĭ,*The application of continued fractions and their generalizations to problems in approximation theory*, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1963. MR**27**#6058.**[6]**Michael Mandell and Arne Magnus,*On convergence of sequences of linear fractional transformations*, Math. Z.**115**(1970), 11–17. MR**0258976****[7]**P. Wynn,*Converging factors for continued fractions. I, II*, Numer. Math.**1**(1959), 272–320. MR**0116158**

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0352774-1

Keywords:
Limit-periodic continued fractions,
converging factors,
circles of Appollonius

Article copyright:
© Copyright 1975
American Mathematical Society