A Fast Newton‐Shamanskii Iteration for a Matrix Equation Arising from M/G/1‐Type Markov Chains
Pei-Chang Guo · Mathematical Problems in Engineering · 2017
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.