Matrix operations are the core of many linear systems. Efficient matrix multiplication is critical to many numerical applications, such as climate modeling, molecular dynamics, computational fluid dynamics and etc. Much research work has been done to improve the performance of matrix operations. However, the majority of these works is focused on two-dimensional ( 2 0 ) matrix. Very little resea...

We prove that counting copies of any graph F in another graph G can be achieved using basic matrix operations on the adjacency matrix of G. Moreover, the resulting algorithm is competitive for medium-sized F : our algorithm recovers the best known complexity for rooted 6-clique counting and improves on the best known for 9-cycle counting. Underpinning our proofs is the new result that, for a ge...

Concepts of the inner product and conjugate of matrix of complex numbers are defined here. Operations such as addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of the conjugate of a matrix of a complex field. Many equations for such operations consist like a case of the conjugate of matrix of a field and some operations on the set of s...

FlashMatrix is a matrix-oriented programming framework for general data analysis with high-level functional programming interface. It scales matrix operations beyond memory capacity by utilizing solid-state drives (SSDs) in non-uniform memory architecture (NUMA). It provides a small number of generalized matrix operations (GenOps) and reimplements a large number of matrix operations in the R fr...

For hierarchical matrices, approximations of the matrix-matrix sum and product can be computed in almost linear complexity, and using these matrix operations it is possible to construct the matrix inverse, efficient precondi-tioners based on approximate factorizations or solutions of certain matrix equations. H 2-matrices are a variant of hierarchical matrices which allow us to perform certain ...

We design and analyze new protocols to verify the correctness of various computations on matrices over ring F[x] univariate polynomials a field F. For sake efficiency, because many properties we are specific principal ideal domain, cannot simply rely previously-developed linear algebra for field. Our interactive, often randomized, feature constant number rounds communication between Prover Veri...

ÐArray operations are used in a large number of important scientific codes, such as molecular dynamics, finite element methods, climate modeling, etc. To implement these array operations efficiently, many methods have been proposed in the literature. However, the majority of these methods are focused on the two-dimensional arrays. When extended to higher dimensional arrays, these methods usuall...

In the presented work, we discuss the Movie-Based approach to design and represent of matrix algorithms including a lot of identical operations on matrix elements. These operations usually transform an initial matrix structure into a matrix with a given structure. Therefore, it is possible to represent a matrix algorithm as a series of frames or iterations reflecting the stages of this process....