For the nonlinear matrix equations arising in the analysis of M/G/1-type and GI/M/1-type Markov chains, the minimal nonnegative solution G or R can be found by Newton-like methods. We prove monotone convergence results for the Newton-Shamanskii iteration for this class of equations. Starting with zero initial guess or some other suitable initial guess, the Newton-Shamanskii iteration provides a monotonically increasing sequence of nonnegative matrices converging to the minimal nonnegative solution. A Schur decomposition method is used to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.
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机译：对于在分析M / G / 1型和GI / M / 1型Markov链中产生的非线性矩阵方程，最小的非负溶液G或R可以通过牛顿的方法找到。我们证明了这类方程式的牛顿萨满迭代的单调融合结果。从零初步猜测或其他一些合适的初始猜测开始，牛顿 - 萨满迭代提供了一个单调增加的非负矩阵序列，它们会聚到最小的非负解。 SCUR分解方法用于加速牛顿 - 姆斯库利迭代。数值示例说明了牛顿 - 姆斯库利迭代的有效性。
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