Natural density distribution of Hermite normal forms of integer matrices

Fast Computation of Hermite Normal Forms of Random Integer Matrices

This paper is about how to compute the Hermite normal form of a random integer matrix in practice. We propose significant improvements to the algorithm by Micciancio and Warinschi, and extend these techniques to the computation of the saturation of a matrix. Tables of timings confirm the efficiency of this approach. To our knowledge, our implementation is the fastest implementation for computin...

Computing Hermite and Smith normal forms of triangular integer matrices

This paper considers the problem of transforming a triangular integer input matrix to canonical Hermite and Smith normal form. We provide algorithms and prove deterministic running times for both transformation problems that are optimal in the matrix dimension. The algorithms are easily implemented, assume standard integer arithmetic, and admit excellent performance in practice. The results pre...

Hermite and Smith Normal Forms ofTriangular Integer Matrices

This paper considers the problem of transforming a triangular integer input matrix to canonical Hermite and Smith normal form. We provide algorithms and prove deterministic running times for both transformation problems that are linear (hence optimal) in the matrix dimension. The algorithms are easily implemented, assume standard integer multiplication, and admit excellent performance in practi...

Asymptotically Fast Computation of Hermite Normal Forms of Integer

This paper presents a new algorithm for computing the Hermite normal form H of an A 2 Z Z nm of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O~(m ?1 nM(m log jjAjj)) bit operations to produce both H and a candidate for U. Here, jjAjj = maxij jAijj, M(t) bit operations are suucient to multiply two dte-bit integers, and is the exponent for mat...

Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix