NEWTON SCHULZ METHOD FOR SOLVING NONLINEAR MATRIX EQUATION X p + A ⁎ XA = Q

Abstract

The matrix equation Xp + A∗XA = Q has been studied to find the positive definite solution in several researches. In this paper, we consider fixed-point iteration and Newton’s method for finding the matrix p-th root. From these two considerations, we will use the NewtonSchulz algorithm (N.S.A). We will show the residual relation and the local convergence of the fixed-point iteration. The local convergence guarantees the convergence of N.S.A. We also show numerical experiments and easily check that the N.S. algorithm reduce the CPU-time significantly.