Computing Hermite and Smith normal forms of triangular integer matrices

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...

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

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...

Near Optimal Algorithms for Computing Smith Normal Forms of Integer

We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d. For the case of matrices over Z Z d, we present an algorithm that computes the Smith form S of an A 2 Z Z nm d in only O(n ?1 m) operations from Z Z d. Here, is the exponent for matrix multiplication over rings: two n n matrices over a ring R can be multiplied in O(n) operati...