求解非线性矩阵方程xp + A * XA = Q的牛顿-舒尔茨方法

IF 0.7 4区数学 Q2 MATHEMATICS

Journal of the Korean Mathematical Society Pub Date : 2018-01-01 DOI:10.4134/JKMS.J170809

Hyun Min Kim, Young Jin Kim, Jie Meng

{"title":"求解非线性矩阵方程xp + A * XA = Q的牛顿-舒尔茨方法","authors":"Hyun Min Kim, Young Jin Kim, Jie Meng","doi":"10.4134/JKMS.J170809","DOIUrl":null,"url":null,"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.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"55 1","pages":"1529-1540"},"PeriodicalIF":0.7000,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"NEWTON SCHULZ METHOD FOR SOLVING NONLINEAR MATRIX EQUATION X p + A ⁎ XA = Q\",\"authors\":\"Hyun Min Kim, Young Jin Kim, Jie Meng\",\"doi\":\"10.4134/JKMS.J170809\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":49993,\"journal\":{\"name\":\"Journal of the Korean Mathematical Society\",\"volume\":\"55 1\",\"pages\":\"1529-1540\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2018-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the Korean Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4134/JKMS.J170809\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Korean Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4134/JKMS.J170809","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

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。

查看原文本刊更多论文

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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of the Korean Mathematical Society 数学-数学

CiteScore

1.20

自引率

16.70%

发文量

审稿时长

6-12 weeks

期刊介绍： This journal endeavors to publish significant research of broad interests in pure and applied mathematics. One volume is published each year, and each volume consists of six issues (January, March, May, July, September, November).