A tight lower bound for required I/O when computing a matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of phases needed to perform C := AB, where each phase is a series of operations involving S reads and writes to and from fast memory, and S is the size of fast memory. A lower bound on th...

Recursion’s removal improves the efficiency of recursive algorithms, especially algorithms with large formal parameters, such as fast matrix multiplication algorithms. In this article, a general method of breaking recursions in fast matrix multiplication algorithms is introduced, which is generalized from recursions removal of a specific fast matrix multiplication algorithm of Winograd.

Generalized sparse matrix-matrix multiplication is a key primitive for many high performance graph algorithms as well as some linear solvers such as multigrid. We present the first parallel algorithms that achieve increasing speedups for an unbounded number of processors. Our algorithms are based on two-dimensional block distribution of sparse matrices where serial sections use a novel hyperspa...

The current state of the art Matrix-Matrix-Multiplication (MMM) kernel is known as ATLAS, which generates the best performing MMM code by search. However, today’s computer architecture changes rapidly and it is hard to generate a high performance code without knowing how to use the new instruction sets. Since ATLAS does not make use of blocking for L2 cache, or SSE/SSE2 instruction, we are enco...

New techniques are presented forthe manipulation of sparse matrices on parallel MIMD computers. We consider the following problems: matrix addition, matrix multiplication, row and column permutation, matrix transpose, matrix vector multiplication, and Gaussian elimination.

The recent discovery that the exponent of matrix multiplication is determined by the rank of the symmetrized matrix multiplication tensor has invigorated interest in better understanding symmetrized matrix multiplication. I present an explicit rank 18 Waring decomposition of $sM_{\langle 3\rangle}$ and describe its symmetry group.

Sparse matrix multiplication is an important algorithm in a wide variety of problems, including graph algorithms, simulations and linear solving to name a few. Yet, there are but a few works related to acceleration of sparse matrix multiplication on a GPU. We present a fast, novel algorithm for sparse matrix multiplication, outperforming the previous algorithm on GPU up to 3× and CPU up to 30×....

We propose an accelerated density matrix purification scheme with error control. The method makes use of the scale-and-fold acceleration technique and screening submatrix products in block-sparse matrix-matrix multiplies to reduce computational cost. An bound a parameter sweep are combined select threshold value for screening, such that can be controlled. evaluate performance comparison without...

Efficient parallel multiplication of sparse matrices is key to enabling many large-scale calculations. This article presents the DBCSR (Distributed Block Compressed Sparse Row) library for scalable sparse matrix-matrix multiplication and its use in the CP2K program for linear-scaling quantum-chemical calculations. The library combines several approaches to implement sparse matrix multiplication...