Efficient parallel Monte Carlo methods for matrix computations

Three Monte Carlo methods for matrix inversion (MI) and finding a solution vector of a system of linear algebraic equations (SLAE) 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 have shown that an n×n matrix can be inverted in 3n/2+N+T steps on a regular array with O(n2NT) cells. Alexandrov and Megson have also shown that a solution vector of SLAE can be found in n+N+T steps on a regular array with the same number of 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; these are independent of n), which show that these designs are faster than existing designs for large values of n. In this paper we take another implementation approach; we consider parallel Monte Carlo algorithms for MI and solving SLAE in MIMD environment, e.g. running on a cluster of workstations under PVM. The Monte Carlo method with almost optimal frequency function performs best of the three methods; it needs about six to ten times fewer chains for the same precision.