Existence of a limiting distribution for the binary GCD algorithm
In this article, we prove the existence and uniqueness of a certain distribution function on the unit interval. This distribution appears in Brent's model of the analysis of the binary gcd algorithm. The existence and uniqueness of such a function was conjectured by Richard Brent in his origina...
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| Published in: | Journal of discrete algorithms (Amsterdam, Netherlands) Vol. 5; no. 1; pp. 176 - 186 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
01.03.2007
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| Subjects: | |
| ISSN: | 1570-8667, 1570-8675 |
| Online Access: | Get full text |
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| Summary: | In this article, we prove the existence and uniqueness of a certain distribution function on the unit interval. This distribution appears in Brent's model of the analysis of the binary gcd algorithm. The existence and uniqueness of such a function was conjectured by Richard Brent in his original paper [R.P. Brent, Analysis of the binary Euclidean algorithm, in: J.F. Traub (Ed.), New Directions and Recent Results in Algorithms and Complexity, Academic Press, New York, 1976, pp. 321–355]. Donald Knuth also supposes its existence in [D.E. Knuth, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, third ed., Addison-Wesley, Reading, MA, 1997] where developments of its properties lead to very good estimates in relation to the algorithm. We settle here the question of existence, giving a basis to these results, and study the relationship between this limiting function and the
binary Euclidean operator
B
2
, proving rigorously that its derivative is a fixed point of
B
2
. |
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| ISSN: | 1570-8667 1570-8675 |
| DOI: | 10.1016/j.jda.2006.03.013 |