The three-step alternating iteration scheme for finding an iterative solution of a singular (non-singular) linear systems in a faster way was introduced by Nandi {\it et al.} [Numer. Algorithms; 84 (2) (2020) 457-483], recently. The authors then provided its convergence criteria for a class of matrix splitting called proper G-weak regular splittings of type I. In this note, we analyze further the convergence criteria of the same scheme. In this aspect, we obtain sufficient conditions for the convergence of the same scheme for another class of matrix splittings called proper G-weak regular splittings of type II. We then show that this scheme converges faster than the two-step alternating and usual iteration schemes, even for this class of splittings. As a particular case, we also establish faster convergence criteria of three-step in a nonsingular matrix setting. This is shown that a large amount of computational time and memory are required in single-step and two-step alternating iterative methods to solve the nonsingular linear systems more efficiently than the three-step alternating iteration method. Finally, the semiconvergence of a three-step alternating iterative scheme is established. Its faster semiconvergence is demonstrated by considering a singular linear system arising from the Markov process.

翻译：为快速求解奇异（非奇异）线性系统的迭代解，Nandi等人[Numer. Algorithms; 84 (2) (2020) 457-483]近期提出了三步交替迭代格式。该文给出了此类格式在一类名为I型真G-弱正则分裂的矩阵分裂下的收敛准则。本文进一步分析了该格式的收敛性，针对另一类名为II型真G-弱正则分裂的矩阵分裂，给出了格式收敛的充分条件。结果表明，即使在这类分裂下，三步交替迭代格式的收敛速度仍快于两步交替迭代格式和常规迭代格式。作为特例，我们在非奇异矩阵框架下建立了三步格式的加速收敛准则，并指出：为更高效求解非奇异线性系统，单步和两步交替迭代法需要耗费大量计算时间与内存，而三步交替迭代法则更具优势。最后，本文建立了三步交替迭代格式的半收敛性，并通过马尔可夫过程导出的奇异线性系统验证了其更快的半收敛速度。