Fast solution of large N×N matrix equations in an MIMD–SIMD Hybrid System

Abstract

In this paper, we propose a new high-speed computation algorithm for solving a large N×N matrix system using the MIMD–SIMD Hybrid System. The MIMD–SIMD Hybrid System (also denoted as Hybrid System in this paper) is a new parallel architecture consisting of a combination of Cluster of Workstations (COWs) and SIMD systems working concurrently to produce an optimal parallel computation. We first introduce our prototype SIMD system and our Hybrid System setup before presenting how it can be implemented to find the unknowns in a large N×N linear matrix equation system using the Gauss–LU algorithm. This algorithm basically performs the ‘Divide and Conquer’ approach by breaking down the large N×N matrix system into a manageable 32 × 32 matrix for fast computation.

Introduction

Matrix computation has widely been used as a mathematical tool for solving large linear equations in image processing and visualization as well as in other scientific applications. In recent years, there has been an increased interest in parallel solutions of large-scale applications [4], [12] and it has become clear that software packages and library routines need to be developed to maintain pace with the demands. Consequently, new individual algorithms that allow for fast processing of linear solutions began to emerge. Library routines such as LINPACK [9], [10], which was later replaced by LAPACK [2], [3], were introduced. A scalable parallel version, SCALAPACK [7], was also developed and added to the list. Industry also uses these software library routines as benchmarking software to effectively measure the system performance of their individual computers or a parallel cluster of computers. However, parallel software library routines used for finding the unknowns in a large N×N matrix calculations were mainly written for the distributed-memory Multiple Instruction Multiple Data (MIMD) programming model, with so-called scalable distributed-memory hardware in mind. They are not suitable for the Hybrid System configuration, a combination of MIMD and Single Instruction Multiple Data (SIMD). As such, there is a need to develop or modify algorithms to make them suitable for solving large N×N matrices on the Hybrid System. This paper presents and evaluates suitably modified algorithms.