| 矩阵方程X-A~*X~(-1)A=Q的Hermite正定解及其扰动分析 |
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| 引用本文： | 李静,张玉海.矩阵方程X-A~*X~(-1)A=Q的Hermite正定解及其扰动分析[J].计算数学,2008,30(2):129-142. |
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| 作者姓名： | 李静 张玉海 |
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| 作者单位： | 李静（山东大学威海分校应用数学系,山东,威海,264209）;张玉海（山东大学数学与系统科学学院,济南,250100） |
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| 基金项目： | 山东大学校科研和教改项目 |
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| 摘 要： | 考虑非线性矩阵方程X-A*X-1A=Q,其中A是n阶复矩阵,Q是n阶Hermite正定解,A*是矩阵A的共轭转置.本文证明了此方程存在唯一的正定解,并推导出此正定解的扰动边界和条件数的显式表达式.以上结果用数值例子加以说明.
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| 关 键 词： | 矩阵方程 正定解 扰动边界 条件数 |
| 修稿时间： | 2006年9月24日 |
THE HERMITIAN POSITIVE DEFINITE SOLUTION AND ITS PERTURBATION ANALYSIS FOR THE MATRIX EQUATION X-A~*X~(-1)A=Q |
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| Li Jing,Zhang Yuhai.THE HERMITIAN POSITIVE DEFINITE SOLUTION AND ITS PERTURBATION ANALYSIS FOR THE MATRIX EQUATION X-A~*X~(-1)A=Q[J].Mathematica Numerica Sinica,2008,30(2):129-142. |
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| Authors: | Li Jing Zhang Yuhai |
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| Institution: | Li Jing (Department of Applied Mathematics,Shandong University At Weihai,Weihai 264209,Shandong,China) Zhang Yuhai (School of Mathematics and System Sciences,Shandong University,Jinan 250100,China) |
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| Abstract: | Consider the nonlinear matrix equation X-A~*X~(-1)A=Q,where A,Q are n×n complex matrices with Q Hermitian positive definite and A~* denotes the conjugate transpose of a matrix A.This paper shows there exists a unique positive definite solution to the equation. The perturbation bounds for the Hermitian positive definite solution to the matrix equation are derived,explicit expressions of the condition number for the Hermitian positive definite solution are obtained and the backward error of an approximate solution to the Hermitian positive definite solution is evaluated.The results are illustrated by numerical examples. |
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