Parallel Algorithms for Matrix Normal Forms

Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the input matrix are computed, and the algorithm is probabilistic of Las Vegas type, i.e., always finds the correct answer. The Smith form algorithm is also a good sequential algorithm. Our algorithm reduces the problem of Smith form computation to two Hermite form computations. Thus the Smith form problem has complexity asymptotically that of the Hermite form problem. We also construct fast parallel algorithms for Jordan normal form and testing similarity of matrices. Both the similarity and non-similarity problems are in the complexity class RNC for the usual coefficient fields, i.e., they can be probabilistically decided in poly-logarithmic time using polynomially many processors.