A fast Newton-Shamanskii iteration for M/G/1-type and GI/M/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 $G$ or $R$ can be found by Newton-like methods. Recently a fast Newton's iteration is proposed in \cite{Houdt2}. We apply the Newton-Shamanskii iteration to the 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. We use the technique in \cite{houdt2} to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.