Hybrid Monte Carlo Methods for Matrix Computation
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.
KeywordsMonte Carlo Method Markov Chain Matrix Inversion Solution of sytem of Linear Equations Matrix Decomposition Diagonal Dominant Matrices SPAI
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