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Mathematics of Computation

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A generalization of the simple continued fraction algorithm

Author: Theresa P. Vaughan
Journal: Math. Comp. 32 (1978), 537-558
MSC: Primary 10F20; Secondary 10A30
MathSciNet review: 0480367
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Abstract: In this paper we present a generalization of the continued fraction algorithm, based on a geometric and matrix-theoretic approach.

We first give a geometric representation in the plane $ {R^2}$, of the simple continued fraction algorithm, described in terms of geometric and arithmetic properties of $ 2 \times 2$ matrices with nonnegative integer entries and determinant 1. The algorithm of this paper is then derived as a natural generalization of the situation in $ {R^2}$. We describe a computational procedure for our algorithm, and give several examples.

References [Enhancements On Off] (What's this?)

  • [1] Leon Bernstein, The Jacobi-Perron algorithm—Its theory and application, Lecture Notes in Mathematics, Vol. 207, Springer-Verlag, Berlin-New York, 1971. MR 0285478
  • [2] C. D. OLDS, Continued Fractions, Math. Assoc. Amer., New Mathematical Library, 1963.
  • [3] HARRIS HANCOCK, Development of the Minkowski Geometry of Numbers, Dover, New York, 1964.

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Article copyright: © Copyright 1978 American Mathematical Society