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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Minkowski reduction of integral matrices
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by John L. Donaldson PDF
Math. Comp. 33 (1979), 201-216 Request permission

Abstract:

In 1905 Hermann Minkowski introduced his theory of reduction of positive definite quadratic forms. Recently, Hans J. Zassenhaus has suggested that this theory can be applied to the problem of row reduction of matrices of integers. Computational investigations have shown that for matrices with more columns than rows, the number of steps required for reduction decreases drastically. In this paper it is proved that as the number of columns increases, the probability that a matrix is Minkowski reduced approaches one. This fact is the motivation behind the introduction of a modified version of Minkowski reduction, resulting in a reduction procedure more suitable for computation.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Math. Comp. 33 (1979), 201-216
  • MSC: Primary 10E25; Secondary 15A36, 68C05
  • DOI: https://doi.org/10.1090/S0025-5718-1979-0514819-7
  • MathSciNet review: 514819