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
Parallel Monte Carlo Algorithms for Sparse SLAE Using MPI
Proceedings of the 6th European PVM/MPI Users' Group Meeting on Recent Advances in Parallel Virtual Machine and Message Passing Interface
Coarse Grained Parallel Monte Carlo Algorithms for Solving SLAE Using PVM
Proceedings of the 5th European PVM/MPI Users' Group Meeting on Recent Advances in Parallel Virtual Machine and Message Passing Interface
Mixed Monte Carlo Parallel Algorithms for Matrix Computation
ICCS '02 Proceedings of the International Conference on Computational Science-Part II
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In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D - C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D-1C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Experimental results with dense and sparse matrices are presented.