# NEWTON SCHULZ METHOD FOR SOLVING NONLINEAR MATRIX EQUATION Xp + A⁎XA = Q

• Kim, Hyun-Min ;
• Kim, Young-jin ;
• Meng, Jie
• Accepted : 2018.06.01
• Published : 2018.11.01
• 173 17

#### Abstract

The matrix equation $X^p+A^*XA=Q$ 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 Newton-Schulz 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.

#### Keywords

fixed-point iteration;Newton's method;Newton-Schulz algorithm;local convergence