A Cubic Algorithm for Computing the Hermite Normal Form of a Nonsingular Integer Matrix-Reference-Cited by-同舟云学术

A Cubic Algorithm for Computing the Hermite Normal Form of a Nonsingular Integer Matrix

Published:2023-10-14 Issue:4 Volume:19 Page:1-36
ISSN:1549-6325
Container-title:ACM Transactions on Algorithms
language:en
Short-container-title:ACM Trans. Algorithms

Author:

Birmpilis Stavros¹^ORCID,Labahn George¹^ORCID,Storjohann Arne¹^ORCID

Affiliation:

1. University of Waterloo, Canada

Abstract

A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix A of dimension n . The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by O(n 3 (log n + log ||A||) 2 (log n ) 2 ) bit operations, where || A ||= max ij | A ij | denotes the largest entry of A in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time (n 3 log || A ||) 1+ o (1) bit operations, where the exponent “ + o (1)” captures additional factors c 1 (log n ) c2 (loglog|| A ||) c3 for positive real constants c 1 ,c 2 ,c 3 .

Publisher

Association for Computing Machinery (ACM)

Subject

Mathematics (miscellaneous)

Link

https://dl.acm.org/doi/pdf/10.1145/3617996

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