Parallel Algorithms for Computing the Smith Normal Form of Large Matrices

Abstract

Smith normal form computation has many applications in group theory, module theory and number theory. As the entries of the matrix and of its corresponding transformation matrices can explode during the computation, it is a very difficult problem to compute the Smith normal form of large dense matrices. The computation has two main problems: the high execution time and the memory requirements, which might exceed the memory of one processor. To avoid these problems, we develop two parallel Smith normal form algorithms using MPI. These are the first algorithms computing the Smith normal form with corresponding transformation matrices, both over the rings ℤ and $\mathbb{F}$[x]. We show that our parallel algorithms both have a good efficiency, i.e. by doubling the processes, the execution time is nearly halved, and succeed in computing the Smith normal form of dense example matrices over the rings ℤ and $\mathbb{F}_2$[x] with more than thousand rows and columns.