On Two Iteration Methods for the Quadratic Matrix Equations

Abstract

By simply transforming the quadratic matrix equation into an equivalent fixed-point equation, we construct a successive approximation method and a Newton's method based on this fixed-point equation. Under suitable conditions, we prove the local convergence of these two methods, as well as the linear convergence speed of the successive approximation method and the quadratic convergence speed of the Newton's method. Numerical results show that these new methods are accurate and effective when they are used to solve the quadratic matrix equation.