A fast parallel algorithm for special linear systems of equations using processor arrays with reconfigurable bus systems

Abstract

A parallel algorithm using Processor Arrays with Reconfigurable Bus Systems has been designed to solve dense Symmetric Positive Definite (SPD) systems of equations Ax = b. The key content of this report is the parallelisation of the algorithm by Delosme & Ipson [8]. In order to design a parallel algorithm for PARBS, many procedures involved in [8] are handled in a slightly different way. The parallel time and processor’s complexity of each step of the algorithm is calculated. The parallel time complexity is O(n) using number of Processing Elements. Introduction: There are many direct and iterative methods for solving system of equations with SPD matrix [13,14,25,28,30]. One of the numerically stable approaches to solve a system of equations is LU decomposition[13]. Besides factorization this method involves a forward elimination and a back substitution process which cannot be parallelised easily. In order to construct a parallel algorithm for this problem, we need to investigate a method which does not involve these steps. One such method was proposed by Delosme and Ipson. Other than these two parallel steps the algorithm presented in [8] includes many steps which can be exploited for parallelism to reduce the complexity of the algorithm. Such work has been carried out and a parallel algorithm based on the PRAM model of computation has been published in [38]. Implementing this method on processor arrays with a reconfigurable bus system requires a special technique with modifications to the existing algorithm. In Section 3 this implementation has been discussed. The report is designed in the following way: In Sec tion 1 we give some important results from [8], in Section 2 we present a brief description of the architecture used for designing the problem. In Section 3 we give a PARBS solution of the problem and discuss the time complexity of each of the steps. 1. This work is supported by the German Academic Exchange Services (DAAD) under the “Sandwich Model” fellowship with the author who is permanently working in the Department of Computer Science, North Maharashtra University, Jalgaon (MS), India. 2n 2n × 5n ×