# Comparison of three Monte Carlo methods for Matrix Inversion

## Abstract

Three Monte Carlo methods for Matrix Inversion (MI) are considered: with absorption, without absorption with uniform transition frequency function, and without absorption with almost optimal transition frequency function.

Recently Alexandrov, Megson and Dimov has shown that an *n*×*n* matrix can be inverted in 3*n*/2 + *N + T* steps on regular arrays with *O*(*n*^{2}*NT*) cells. A number of bounds on *N* and *T* have been established (*N* is the number of chains and *T* is the length of the chain in the stochastic process, which are independent of matrix size *n*), which show that these designs are faster than the existing designs for large values of *n*.

In this paper we take another implementation approach, we consider parallel Monte Carlo algorithms for MI on MIMD environment, i.e. running on a cluster of workstations under PVM. The Monte Carlo method with almost optimal frequency function performs best of the three methods as it needs about six-ten times less chains for the same precision.

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