Parallel computations of eigenvalues based on a Monte Carlo approach

This paper presents a parallel Monte Carlo algorithm for evaluating the eigenvalues of matrices. This algorithm is called Resolvent Monte Carlo algorithm (RMC) and uses the resolvent matrix iterations by the Monte Carlo method. The algorithm is suitable for estimating eigenvalues of large sparse symmetric matrices. The work described in this paper has the goal to reduce the computational time when parallel machine is used, and to reach good experimental results for a speed-up and parallel efficiency for large sparse matrices. Numerical tests are performed for a number of test matrices general sparse symmetric, band symmetric matrices and dense matrices on the parallel machine Intel PARAGON. Estimations for the speed-up as well as for the parallel efficiency are obtained. It is shown that the algorithm complexity is practically independent of the size of the matrix.