Relaxed Monte Carlo Linear Solver Chih Jeng Kenneth Tan 1 and Vassil Alexandrov 2 1 School of Computer Science The Queen’s University of Belfast Belfast BT7 1NN Northern Ireland United Kingdom cjtan@acm.org 2 Department of Computer Science The University of Reading Reading RG6 6AY United Kingdom v.n.alexandrov@rdg.ac.uk Abstract. The problem of solving systems of linear algebraic equations by parallel Monte Carlo numerical methods is considered. A parallel Monte Carlo method with relaxation is presented. This is a report of a research in progress, showing the effectiveness of this algorithm. Theo- retical justification of this algorithm and numerical experiments are pre- sented. The algorithms were implemented on a cluster of workstations using MPI. Keyword: Monte Carlo method, Linear solver, Systems of linear alge- braic equations, Parallel algorithms. 1 Introduction One of the more common numerical computation task is that of solving large systems of linear algebraic equations Ax = b (1) where A ∈ IR n×n and x, b ∈ IR n . A great multitude of algorithms exist for solving Equation 1. They typically fall under one of the following classes: di- rect methods, iterative methods, and Monte Carlo methods. Direct methods are particularly favorable for dense A with relatively small n. When A is sparse, iterative methods are preferred when the desired precision is high and n is rela- tively small. When n is large and the required precision is relatively low, Monte Carlo methods have been proven to be very useful [6,4,15,1]. As a rule, Monte Carlo methods are not competitive with classical numer- ical methods for solving systems of linear algebraic equations, if the required precision is high [13]. In Monte Carlo methods, statistical estimates for the components of the solu- tion vector x are obtained by performing random sampling of a certain random V.N. Alexandrov et al. (Eds.): ICCS 2001, LNCS 2073, pp. 1289–1297, 2001. c  Springer-Verlag Berlin Heidelberg 2001