求解非线性矩阵方程xp + A * XA = Q的牛顿-舒尔茨方法

Pub Date : 2018-01-01 DOI:10.4134/JKMS.J170809

Hyun Min Kim, Young Jin Kim, Jie Meng

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引用次数: 1

摘要

在一些研究中，研究了矩阵方程Xp + A * XA = Q的正定解。本文考虑了求矩阵p次根的不动点迭代和牛顿法。从这两个方面考虑，我们将使用牛顿-舒尔茨算法(n.s.a.)。我们将证明不动点迭代的残差关系和局部收敛性。局部收敛性保证了nsa算法的收敛性，并通过数值实验验证了nsa算法显著降低了cpu时间。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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

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.

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