From: Olivier Biberstein Newsgroups: sci.math,sci.math.symbolic Subject: Continued Fractions (Replies) Date: Tue, 24 Nov 1998 12:33:27 +0000 Dear all, Couple of weeks ago I posted the following question: > I'm interested in continued fractions, especially algorithms for > addition, multiplication, etc. > > Does anybody have some pointers, papers, or references about this topic First of all I would like to thank very much everybody who replied. Here follows the list of all the answers I received. Cheers, Olivier. ------------ Try: H. S. Wall, "Analytic Theory of Continued Fractions," Chelsea Publishing Co., 1967. If it is not in a nearby library, Chelsea may still have it for sale. Clyde Davenport cmdaven@usit.net ------------ Lorentzen and Waadeland Continued fractions with applications Studies in Computl Maths 3 North-Holland isbn 0-444-89265-6 * Dr W B Stewart phone +44 1865 279628 * * Exeter College fax +44 1865 279630 * * Oxford * * OX1 3DP * * UK home 760629 * ------------ Do a www search for "hakmem" which has some notes by Gosper on continued fraction arithmetic. Dr A.C. Norman" ------------ Have a look at http:\\www.inwap.com\pdp10\hbaker\hakmem\cf.html Items 101a and 101b might help. R. Burge r3769@aol.com (R3769) ------------ Try: H. S. Wall, "Analytic Theory of Continued Fractions," Chelsea Publishing Co., 1967. If it is not in a nearby library, Chelsea may still have it for sale. Clyde Davenport cmdaven@usit.net ------------ Continued fractions have many uses in number theory. Yahoo has a good numner theory site with some good links. Also, you can't go wrong with a number theory book by Hardy & Wright. They're the best. (Ribbenboim and Robbins are good also) jericho jericho ------------ Some details on implementing continued fraction arithmetic in Mathematica are given in: Ilan Vardi, Code and Pseudo Code, The Mathematica Journal, Volume 6 Issue 2, pp. 66--71. The article attributes the algorithm for addition and multiplication of continued fractions to a preprint of R. W. Gosper circa 1976, though I have so far been unable to trace that reference. Mark Sofroniou, Wolfram Research. Mark Sofroniou ------------ You might find one of my webpages http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html useful (as well several links). Steven Finch sfinch@mathsoft.com MathSoft, Inc. Favorite Mathematical Constants 101 Main St. Unsolved Mathematics Problems Cambridge, MA 02142 MathSoft Math Puzzle Page USA http://www.mathsoft.com/asolve/sfinch.html ------------ Very interesting details and references may be found at : http://www.astro.virginia.edu/~eww6n/math/ContinuedFraction.html and http://www.calvin.edu/academic/math/confrac/index.html Raymond Manzoni ------------ You are probably thinking of Gosper's algorithms for arithmetic (addition, multiplication) of reals in terms of their continued fraction representations. I have never seen these; they were never published and no-one I know seems to know anything about them, but everyone has heard of them. Very frustrating, particularly Gosper has published other algorithms which work and have proven important :-/ See Erk's pages for some information about simpler arithmetic operations (e.g. taking the reciprocal and negative) of a real number using its continued fraction representation. > Does anybody have some pointers, papers, or references about this topic It's a huge topic. You seem to be interested in arithemetic operations, and I can't provide references on those. But there are some interesting theories arising from continued fractions, such as a hierarchy of irrationality, and "limiting properties" of the sequence of partial fractions. For these topics, see e.g. Author: Olds, C. D. (Carl Douglas), 1912-. Title: Continued fractions. Pub. Info.: [New York] Random House [1963]. LC Subject: Continued-fractions. (very gentle introduction) Author: Khinchin, Aleksandr IAkovlevich, 1894-1959. Title: Continued fractions. [Translated from the Russian by Scripta Technica, inc. English translation edited by Herbert Eagle. Pub. Info.: Chicago, University of Chicago Press [1964]. LC Subject: Continued-fractions. (recently reprinted by Dover books!-- very readable) Author: Rockett, Andrew Mansfield. Title: Continued fractions / Andrew M. Rockett, Peter Szusz. Pub. Info.: Singapore ; New Jersey : World Scientific, 1992. LC Subject: Continued-fractions. Processes-Infinite. Author: Lorentzen, Lisa. Title: Continued fractions with applications / Lisa Lorentzen, Haakon Waadeland. Pub. Info.: Amsterdam ; New York : North-Holland ; New York, N.Y. : Distributors for the U.S. and Canada, Elsevier Science Pub Co., 1992. LC Subject: Continued-fractions. Instead of continued fractions with numerical coefficients, you can generalize to coefficients which are themselves functions; see for instance Author: Wall, H. S. (Hubert Stanley), 1902-. Title: Analytic theory of continued fractions. Pub. Info.: New York, D. Van Nostrand Co., 1948. LC Subject: Continued-fractions. Author: Jones, William B., 1939-. Title: Continued fractions : analytic theory and applications / William B. Jones and W. J. Thron Pub. Info.: Reading, Mass. : Addison-Wesley Pub. Co., 1980. LC Subject: Continued-fractions. There has also been much work on multidimensional continued fractions (I can give you a long list of papers), for which a starting point might be Author: Brentjes, A. J. Title: Multi-dimensional continued fraction algorithms / A.J Brentjes. Pub. Info.: Amsterdam : Mathematisch Centrum, 1981. LC Subject: Continued-fractions. Diophantine-analysis. Algorithms. Author: Schweiger, Fritz. Title: Ergodic theory of fibred systems and metric number theory / Fritz Schweiger. Pub. Info.: Oxford : Clarendon Press ; New York : Oxford University Press, 1995. LC Subject: Ergodic-theory. Number-theory. Differentiable-dynamical-systems. See also the current discussion is sci.physics.research on connections between simple continued fractions, moduli space of elliptic curves, rational tangles, and possible links to string theory and quantum gravity. Chris Hillman http://www.math.washington.edu/~hillman/personal.html ------------ -- Olivier BIBERSTEIN Centre for Software Reliability (CSR) Bedson Building tel: (+44 191) 222-8058 University of Newcastle fax: (+44 191) 222-8788 Newcastle upon Tyne NE1 7RU mailto:Olivier.Biberstein@ncl.ac.uk United Kingdom http://www.cs.ncl.ac.uk/~olivier.biberstein/