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NEWTON SCHULZ METHOD FOR SOLVING NONLINEAR MATRIX EQUATION Xp + AXA = Q

  • Kim, Hyun-Min ;
  • Kim, Young-jin ;
  • Meng, Jie
  • Received : 2017.12.23
  • Accepted : 2018.06.01
  • Published : 2018.11.01

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