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A multidimensional continued fraction based on a high-order recurrence relation

Authors: Yves Tourigny and Nigel P. Smart
Journal: Math. Comp. 76 (2007), 1995-2022
MSC (2000): Primary 11J70
Published electronically: May 11, 2007
MathSciNet review: 2336278
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Abstract: The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call ``naive rounding'' leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.

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Additional Information

Yves Tourigny
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Nigel P. Smart
Affiliation: Department of Computer Science, University of Bristol, Bristol, United Kingdom

Keywords: Multidimensional continued fraction, recurrence relation
Received by editor(s): August 22, 2006
Published electronically: May 11, 2007
Additional Notes: The first author acknowledges the support of the Engineering and Physical Sciences Research Council (United Kingdom) under a Discipline Hopping Award
Article copyright: © Copyright 2007 American Mathematical Society

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