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.
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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 *
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Do a www search for "hakmem" which has some notes by Gosper on
continued fraction arithmetic.
Dr A.C. Norman"
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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)
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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
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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
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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
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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
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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
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--
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/