A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains

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 or 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.