Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm

Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm

Let B be the maximum of the entries of GG. The main issue is whether we can estimate the entries of b in terms of B, m, n during the algorithm. The entries of A can then be estimated through A = bG. As they do not affect b, we may remove from G all columns that do not contribute a pivot to the Hermite normal form. Once that is done, A has as many columns as its rank, and at the end of the algor...

Numerical Experiments with the LLL-Based Hermite Normal Form Algorithm for Solving Linear Diophantine Systems

In this paper we are concerned with the practical performance of the LLL-based Hermite normal form algorithm, proposed by Havas [13] (LLLHMM) when applied to solve linear Diophantine systems. We first compare the efficiency of the LLLHMM with the algorithm, named LDSSBR, based on the Rosser’s idea. The numerical results show that for small problems, the bit length of the longest integer occurri...

Hermite Normal Form

The Hermite Normal Form is a canonical matrix analogue of Reduced Echelon Form, but involving matrices over more general rings. In this work we formalise an algorithm to compute the Hermite Normal Form of a matrix by means of elementary row operations, taking advantage of the Echelon Form AFP entry. We have proven the correctness of such an algorithm and refined it to immutable arrays. Furtherm...

A Linear Space Algorithm for Computing the Hermite Normal Form

Computing the Hermite Normal Form of an n n integer matrix using the best urrent algorithms typi ally requires O(n3 logM) spa e, whereM is a bound on the entries of the input matrix. Although polynomial in the input size (whi h is O(n2 logM)), this spa e blow-up an easily be ome a serious issue in pra ti e when working on big integer matri es. In this paper we present a new algorithm for omputi...

Bricklaying and the Hermite Normal Form