Computing the invariant structure of integer matrices: fast algorithms into practice

We present a new heuristic algorithm for computing the determinant of a nonsingular n x n integer matrix. Extensive empirical results from a highly optimized implementation show the running time grows approximately as n3 log n, even for input matrices with a highly nontrivial Smith invariant structure. We extend the algorithm to compute the Hermite form of the input matrix. Both the determinant and Hermite form algorithm certify correctness of the computed results.