Ever since the dawn of the computer age, researchers have been trying to find an optimal way of multiplying matrices, a fundamental operation that is a bottleneck for many important algorithms. Faster matrix multiplication would give more efficient algorithms for many standard linear algebra problems, such as inverting matrices, solving systems of linear equations, and finding determinants. Eve...

Scattered data interpolation by radial kernel functions leads to linear equation systems with large, fully populated, ill-conditioned interpolation matrices. A successful iterative solution of such a system requires an efficient matrix-vector multiplication as well as an efficient preconditioner. While multipole approaches provide a fast matrix-vector multiplication, they avoid the explicit set...

We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For k ≥ 4, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the ex...

By use of a simple identity, the product of two complex matrices can be formed with three real matrix multiplications and five real matrix additions, instead of the four real matrix multiplications and two real matrix additions required by the conventional approach. This alternative method reduces the number of arithmetic operations, even for small dimensions, achieving a saving of up to 25 per...

Recent changes in computational sciences force reevaluation of the role of dense matrix multiplication. Among others, this resulted in a proposal to consider generalized matrix multiplication, based on the theory of algebraic semirings. The aim of this note is to outline an initial object oriented model of the generalized matrix-multiply-add operation.

We show that the 3 multiplications in (a0, a1, ..., a3m−1)(b0, b1, ..., b3m−1) T can be converted to 2 ( 2m+5 5 ) multiplications. Thus when m = 100, 3 < 2 ( 2m+5 5 ) . This gives a θ(n) time algorithm for matrix multiplication.

We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s-rank of the matrix multiplication tensor imply upper bounds on the ordinary rank. In particular, if the “s-rank exponent of matrix multiplication” equals 2, then ω = 2. This connection between the s-rank exponent and the ordinary exponent enables us to significantly generalize the group-th...

Matrix Multiplication is one of the most commonly used algorithm in many application areas like Sonar Systems, Relational Database Management System and other applications like algebra etc. Matrix Multiplication is quite difficult when it tends to infinity. In this paper we study and evaluate the execution time of simple matrix multiplication and optimized matrix multiplication with OpenMP on m...

Many high performance computing applications require computing both sparse matrix-vector product (SMVP) and sparse matrix-transpose vector product (SMTVP) for better overall performance. Under such a circumstance, it is critical to maintain a similarly high throughput for these two computing patterns with the underlying sparse matrix encoded in a single storage format. The compressed sparse blo...