NEWTON'S METHOD FOR SOLVING A QUADRATIC MATRIX EQUATION WITH SPECIAL COEFFICIENT MATRICES
Sang-Hyup Seo, Jong-Hyun Seo, Hyun-Min Kim · Honam Mathematical Journal · 2013
We consider the iterative solution of a quadratic matrix equation with special coefficient matrices which arises in the quasibirth and death problem. In this paper, we show that the elementwise minimal positive solvent of the quadratic matrix equations can be obtained using Newton's method if there exists a positive solvent and the convergence rate of the Newton iteration is quadratic if the Fr $\acute{e}$ chet derivative at the elementwise minimal positive solvent is nonsingular. Although the Fr $\acute{e}$ chet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.