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.
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机译：在本文中，我们指出了该期刊上发表的题为“用松弛蒙特卡洛方法求解线性方程组的系统”（Tan in J. Supercomput。22：113–123，2002）的局限性。我们认为，松弛蒙特卡罗方法在Sect中提出。仅在系数矩阵A必须是对角线主导的条件下，本文的7才是正确的。但是，对于非对角线占优势的情况；相应的Neumann级数可能会发散，这在模拟迭代Monte Carlo算法时将导致无限循环。在本文中，我们首先证明仅对于对角占优矩阵，相应的冯·诺依曼级数才能收敛，并且蒙特卡罗算法可以放宽。因此，无论松弛参数γ是单个值还是一组值，对于非对角占优矩阵都不成立。然后，我们提出并分析数值实验结果以验证我们的论点。
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