A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/l-Type Markov Chains

Guo Pei-Chang

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A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/l-Type Markov Chains

机译：由M / G / l型马尔可夫链产生的矩阵方程的快速Newton-Shamanskii迭代

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For the nonlinearmatrix equations arising in the analysis of M/G/1-type and GI/M/1-typeMarkov 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 nonnegativematrices converging to the minimal nonnegative solution. A Schur decomposition method is used to accelerate theNewton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.

机译：对于分析M / G / 1型和GI / M / 1型马尔可夫链的非线性矩阵方程，可通过类牛顿法求出最小非负解G或R。我们证明了这类方程组的牛顿-沙曼斯基迭代的单调收敛结果。从零个初始猜测或一些其他合适的初始猜测开始，牛顿-Shamanskii迭代提供了非负矩阵的单调递增序列，并收敛到最小非负解。 Schur分解方法用于加速Newton-Shamanskii迭代。数值示例说明了Newton-Shamanskii迭代的有效性。

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《Mathematical Problems in Engineering》 |2017年第10期|4018239.1-4018239.8|共8页
作者
Guo Pei-Chang;
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China Univ Geosci, Sch Sci, Beijing 100083, Peoples R China;

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1. The modified Newton-Shamanskii method for the solution of a quadratic vector equation arising in Markovian binary trees [J] . Guo Pei-Chang, Xu Shu-Fang Calcolo . 2015,第3期

机译：修正的Newton-Shamanskii方法用于求解马尔可夫二叉树中的二次矢量方程
2. Equicontinuity and Stability Properties of Markov Chains Arising from Iterated Function Systems on Polish Spaces [J] . DAWID CZAPLA, KATARZYNA HORBACZ Stochastic Analysis and Applications . 2014,第1期

机译：波兰空间上迭代函数系统引起的马尔可夫链的等连续性和稳定性
3. A CLASS OF ITERATION METHODS BASED ON THE MOSER FORMULA FOR NONLINEAR EQUATIONS IN MARKOV CHAINS [J] . Bai ZZ. Linear Algebra and its Applications . 1997,第0期

机译：马氏链中基于非线性方程的Moser公式的一类迭代方法
4. MNM-a novel technique for the iterative solution of matrix equations arising in the method of moments formulation [C] . Mittra R., Prakash V.V.S., Ma J.-F., Improving Supply Security on 11kV Overhead Networks . 1988

机译：MNM-矩量表示法中矩阵方程迭代求解的一种新技术
5. Wiener-Hopf Factorization for Time-inhomogeneous Markov Chains and Statistical Inference for Stochastic Partial Differential Equations [D] . Huang, Yicong. 2018

机译：随机偏微分方程的时源性马尔可夫链和随机偏微分方程的统计推断维纳 - HOPF分解
6. RANDOM EVOLUTIONS MARKOV CHAINS AND SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS [O] . R. J. Griego, R. Hersh 1969

机译：随机演化马尔可夫链和偏微分方程组
7. A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains [O] . Pei-Chang Guo 2017

机译：用于M / G / 1型马尔可夫链产生的矩阵方程的快速牛顿 - 萨满迭代

A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/l-Type Markov Chains

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