Efficient Vector and Parallel Manipulation of Tensor Products

Abstract

We present efficient vector and parallel methods for manipulating tensor products of matrices. We consider both computing the matrix-vector product (A1 ⊗ ⋯ ⊗ AK)x and solving the system of linear equations (A1 ⊗ ⋯ ⊗ AK)x = b. The methods described are independent of K. We accompany this article with a companion algorithm which describes an implementation of a complete set of tensor product routines based on LAPACK and the Level 2 and 3 Basic Linear Algebra Subprograms (BLAS) which provide vectorization and parallelization.