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

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 γ is a single value or a set of values. We then present and analyze the numerical experiment results to verify our arguments.