Computation Of Matrix-vector Products Research Articles

Computing matrix–vector products with centrosymmetric and centrohermitian matrices

An algorithm proposed recently by Melman reduces the costs of computing the product Ax with a symmetric centrosymmetric matrix A as compared to the case of an arbitrary A. We show that the same result can be achieved by a simpler algorithm, which requires only that A be centrosymmetric. However, if A is hermitian or symmetric, this can be exploited to some extent. Also, we show that similar gains are possible when A is a skew-centrosymmetric or a centrohermitian matrix.

Large matrix–vector products on distributed bus networks with communication delays using the divisible load paradigm: performance analysis and simulation

We present a performance analysis and experimental simulation results on the problem of scheduling a divisible load on a bus network. In general, the computing requirement of a divisible load is CPU intensive and demands multiple processing nodes for efficient processing. We consider the problem of scheduling a very large matrix–vector product computation on a bus network consisting of a homogeneous set of processors. The experiment was conducted on a PC-based networking environment consisting of Pentium II machines arranged in a bus topology. We present a theoretical analysis and verify these findings on the experimental test-bed. We also developed a software support system with flexibility in terms of scalability of the network and the load size. We present a detailed discussion on the experimental results providing directions for possible future extensions of this work.

Classroom Note:Centrosymmetric Matrices

Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.

Preconditioned, Adaptive, Multipole-Accelerated Iterative Methods for Three-Dimensional First-Kind Integral Equations of Potential Theory

This paper presents a preconditioned, Krylov-subspace iterative algorithm, where a modified multipole algorithm with a novel adaptation scheme is used to compute the iterates for solving dense matrix problems generated by Galerkin or collocation schemes applied to three-dimensional, first-kind, integral equations that arise in potential theory. A proof is given that this adaptive algorithm reduces both matrix-vector product computation time and storage to order N, and experimental evidence is given to demonstrate that the combined preconditioned, adaptive, multipole-accelerated (PAMA) method is nearly order N in practice. Examples from engineering applications are given to demonstrate that the accelerated method is substantially faster than standard algorithms on practical problems.

A parallel multigrid method for history-dependent elastoplasticity computations

A multigrid method is described for solving history-dependent elastoplastic solid mechanics problems using the finite element method. A multigrid algorithm that solves linear matrix equations is combined with the Newton-Raphson iteration procedure. The method is implemented on shared memory parallel computers by employing an element-by-element approach to computing matrix-vector products. Some two- and three-dimensional fracture mechanics problems are used to study the behavior of the algorithm. The convergence of the linear multigrid solver deteriorates as plastic flow increases. This is a result of the effect of near incompressibility on the smoothing effect of the relaxation procedure employed; however, reasonable performance can still be achieved. The solution of some large-scale problems demonstrates the speed, storage requirements, and parallel performance of the algorithm. For example, a three-dimensional problem with 215 000 degrees-of-freedom and 10 load steps was solved in 69 100 CPU s with 108 Mbytes of storage on a single processor of a Convex C240. Maximum speed-ups of 2.96 and 4.13 were measured on a 4-processor Convex and an 8-processor Alliant, respectively, for the nonlinear algorithm.