A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such AV=US, with S normal form of A. The expected running time about same as required multiply together two dimension size entries Explicit bounds are in both multipliers. main tool used by massager, relaxed version V, specifying column operations computation. From perspective efficiency, tools fast linear system solving partial linearization matrices. As an application algorithm, find fractional part inverse input matrix.