The Newton-Shamanskii method for solving a quadratic matrix equation arising in quasi-birth-death problems

TL;DR

This paper introduces the Newton-Shamanskii method for efficiently solving quadratic matrix equations in quasi-birth-death Markov chains, demonstrating convergence and effectiveness through theoretical analysis and numerical experiments.

Contribution

It applies and analyzes the Newton-Shamanskii method for quadratic matrix equations, establishing convergence to the minimal nonnegative solution in this context.

Findings

The method produces a monotonically increasing sequence converging to the solution.

Numerical experiments confirm the method's effectiveness and efficiency.

The approach is suitable for computing stationary distributions in Markov chains.

Abstract

In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. We apply the Newton-Shamanskii method for solving the equation. We show that the sequence of matrices generated by the Newton-Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method.