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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2018; 55(6): 1529-1540

Online first article August 10, 2018      Printed November 1, 2018

https://doi.org/10.4134/JKMS.j170809

Copyright © The Korean Mathematical Society.

Newton Schulz method for solving nonlinear matrix equation $X^p + {A^*}XA=Q$

Hyun-Min Kim, Young-jin Kim, Jie Meng

Pusan National University, National Institute for Mathematical Sciences, Pusan National University

Abstract

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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

MSC numbers: 65H10

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