Condition for relaxed Monte Carlo method of solving systems of linear equations

Authors:
Guoming Lai;Xiaola Lin
Affiliations:
School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China 510275 and Department of Math and Information Technology, Hanshan Normal University, Chaozhou, China 521041;School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China 510275
Venue:
The Journal of Supercomputing
Year:
2011

Citing 2
Cited 0

A new iterative Monte Carlo approach for inverse matrix problem

Journal of Computational and Applied Mathematics
Solving Systems of Linear Equations with Relaxed Monte Carlo Method

The Journal of Supercomputing - Special issue on computational issues in fluid dynamics optimization and simulation

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Abstract

In this paper, we point out the limitation of the paper entitled "Solving Systems of Linear Equations with Relaxed Monte Carlo Method" published in this journal (Tan in J. Supercomput. 22:113---123, 2002). We argue that the relaxed Monte Carlo method presented in Sect. 7 of the paper is only correct under the condition that the coefficient matrix A must be diagonal dominate. However, for nondiagonal dominate case; the corresponding Neumann series may diverge, which would lead to infinite loop when simulating the iterative Monte Carlo algorithm. In this paper, we first prove that only for the diagonal dominate matrix, the corresponding von Neumann series can converge, and the Monte Carlo algorithm can be relaxed. Therefore, it is not true for nondiagonal dominate matrix, no matter the relaxed parameter 驴 is a single value or a set of values. We then present and analyze the numerical experiment results to verify our arguments.