Efficient parallelizations of Hermite and Smith normal form algorithms

Abstract

Hermite and Smith normal form are important forms of matrices used in linear algebra. These terms have many applications in group 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 Hermite and Smith normal form of large dense matrices. The main problems of the computation are the large execution times and the memory requirements which might exceed the memory of one processor. To avoid these problems, we develop parallelizations of Hermite and Smith normal form algorithms. These are the first parallelizations of algorithms for computing the normal forms with corresponding transformation matrices, both over the rings Z and F [ x ] . We show that our parallel versions have good efficiency, i.e., by doubling the processes, the execution time is nearly halved. Furthermore, they succeed in computing normal forms of dense large example matrices over the rings Q [ x ] , F 3 [ x ] , and F 5 [ x ] .