Newton's method for solving a quadratic matrix equation with special coefficient matrices

Abstract

We consider the iterative method for solving a quadratic matrix equation with special coefficient matrices which arises in the quasi-birth-death problem. In this paper, we show that the elementwise minimal positive solvents to quadratic matrix equations can be obtained using Newton's method. We also prove that the convergence rate of the Newton iteration is quadratic if the Fréchet derivative at the elementwise minimal positive solvent is nonsingular. However, if the Fréchet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.(This is summarized a paper which is to appear in Honam Mathematical Journal.) © Published under licence by IOP Publishing Ltd.