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

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...

A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix

Given a square, nonsingular matrix of univariate polynomials F ∈ K[x] over a field K, we give a fast, deterministic algorithm for finding the Hermite normal form of F with complexity O (nωd) where d is the degree of F. Here soft-O notation is Big-O with log factors removed and ω is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal entries ...

Computing the Hermite Form of a Matrix of Ore Polynomials

Let F[∂;σ, δ] be the ring of Ore polynomials over a field (or skew field) F, where σ is a automorphism of F and δ is a σ-derivation. Given a matrix A ∈ F[∂;σ, δ], we show how to compute the Hermite form H of A and a unimodular matrix U such that UA = H. The algorithm requires a polynomial number of operations in F in terms of n and the degrees (in ∂) of the entries in A. When F = k(z) for some ...

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

Asymptotically Fast Computation of the Hermite Normal Form of an Integer Matrix

This paper presents a new algorithm for computing the Hermite normal form H of an A ∈ Z n×m of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O (̃mnM(m log ||A||)) bit operations to produce both H and a candidate for U . Here, ||A|| = maxij |Aij |, M(t) bit operations are sufficient to multiply two dte-bit integers, and θ is the exponent for ma...