Matrix Multiplication with Diagonals: Structured Sparse Matrices and Beyond.

Matrix-matrix multiplication is a fundamental computation in a wide range of scientific problems. In this paper, we investigate algorithmic aspects of this important operation whereby the underlying arithmetic calculations are ordered in a way that is different from the standard triple-loop framework. We introduce the notion of “compact diagonal storage” which builds upon the previously developed diagonal storage, an orientation-independent uniform scheme to store nonzero elements of dense as well as structured matrices such as symmetric, triangular, regular, and arbitrarily distributed banded matrices. The results from an extensive set of numerical experiments with structured sparse matrices demonstrate significant performance gain of both serial and shared-memory parallel calculations of our matrix-matrix and matrix-transposed matrix multiplication when compared with an optimized matrix-matrix multiplication code.