Sequential Monte Carlo techniques for the solution of linear systems
Journal of Scientific Computing
Matrix computations (3rd ed.)
A new iterative Monte Carlo approach for inverse matrix problem
Journal of Computational and Applied Mathematics
Parallel resolvent Monte Carlo algorithms for linear algebra problems
Mathematics and Computers in Simulation - IMACS sponsored Special issue on the second IMACS seminar on Monte Carlo methods
Mixed Monte Carlo Parallel Algorithms for Matrix Computation
ICCS '02 Proceedings of the International Conference on Computational Science-Part II
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Finding the smallest eigenvalue of a given square matrix A of order n is computationally very intensive problem. The most popular method for this problem is the Inverse Power Method which uses LU-decomposition and forward and backward solving of the factored system at every iteration step. An alternative to this method is the Resolvent Monte Carlo method which uses representation of the resolvent matrix [I – qA]$^{\rm -{\it m}}$ as a series and then performs Monte Carlo iterations (random walks) on the elements of the matrix. This leads to great savings in computations, but the method has many restrictions and a very slow convergence. In this paper we propose a method that includes fast Monte Carlo procedure for finding the inverse matrix, refinement procedure to improve approximation of the inverse if necessary, and Monte Carlo power iterations to compute the smallest eigenvalue. We provide not only theoretical estimations about accuracy and convergence but also results from numerical tests performed on a number of test matrices.