矩阵方程$X A^*X^{-q}A=Q~(q\geq 1)$的Hermite正定解

In this paper, Hermitian positive definite solutions of the nonlinear matrix equation X ＋ A^＊X^-qA = Q （q≥1） are studied. Some new necessary and sufficient conditions for the existence of solutions are obtained. Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions, and the convergence analysis is also given. The theoretical results are illustrated by numerical examples.     

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

矩阵方程X+A*X-nA=I的正定解

In this paper we give some sufficient conditions and some necessary conditions under which the matrix equation X A^*X^-nA=I has a positive definite solution. An iterative method which converges to a positive definite solution of this equation is constructed. And an error estimate formula on this iterative method is also derived.     

关于矩阵方程Xs+A*X-qA=I*（英文）

In Theorem 2.4 of the paper[Yang Yueting,The iterative method for solving nonlinear matrix equation X^s + A^*X^-tA = Q,Appl.Math.Comput., 188（2007）46-53],Yang showed that this equation has the maximal solution X_L. In this paper,we point out that Yang’s result is doubtful,study the existence of the Hermitian positive definite solutions,the maximal solution and the minimal solution for the case that Q = I,and give the algorithms for finding these special solutions.     

ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES

The Hermitian and skew-Hermitian splitting （HSS） method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.     

关于矩阵方程AXB=E的加权最小二乘Hermite解

In this paper, we discuss the matrix equation AXB = C. We obtain a terse expression of weighted least-squares solution by applying the canonical correlation decomposition(CCD), we discuss the sufficient and necessary conditions for existence of Hermitian solutions, and derive a general form of the Hermite solutions.We also derive the expression of the optimal approximation solution in the set of Hermite soluti6ns to given matrix.     

ON SOLUTIONS OF MATRIX EQUATION AXAT ＋ BYBT=C

By making use of the quotient singular value decomposition (QSVD) of a matrix pair,this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the general solutions of the linear matrix equation AXA^T BYB^T=C with the unknown X and Y, which may be both symmetric, skew-symmetric, nonnegativede finite, positive definite or some cross combinations respectively. Also, the solutions of some optimal problems are derived.     

关于一类二阶矩阵方程的敏度分析

In this paper, the sensitivity of the solution for a class of quadratic matrix equation which arises in the analysis of structural systems and vibration problems is discussed. With Brouwer fixed piont theory, the perturbation of the quadratic matrix equation is analyzed and two computational perturbation bounds are derived. Then a Rice condition number of some kind of solutions is given using the analytic expansion method. Two examples are presented in the last part.     

中心代数上一矩阵方程的中心对称与中心斜对称解

Let Ω be a finite dimensional central algebra and chart Ω≠2 .The matrix equation AXB-CXD=E over Ω is considered.Necessary and sufficient conditions for the existence of centro(skew)symmetric solutions of the matrix equation are given.As a particular case ,the matrix equation X-AXB=C over Ω is also considered.     

关于矩阵方程X+A*X-1A=P的解及其扰动分析

考虑非线性矩阵方程X＋A^＊（X^-1）A=P其中A是n阶非奇异复矩阵,P是n阶Hermite正定矩阵．本文给出了Hermite正定解和最大解的存在性以及获得最大解的一阶扰动界,改进了文[5,6]中的部分结论．     

矩阵方程X-A~*X~(-1)A=Q的Hermite正定解及其扰动分析

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.     

矩阵方程X-A~*X~qA=Q(q>0)的Hermite正定解

本文讨论了矩阵方程X-A*XqA=Q(q>0)的Hermite正定解,给出了q>1时解存在的必要条件,存在区间,以及迭代求解的方法．证明了0相似文献   

矩阵方程X-A*X-2A=Q的正定解及其扰动分析

研究非线性矩阵方程X-A*X-2A=Q(Q>0)的Hermite正定解及其扰动问题.给出了该方程存在唯-Hermite正定解的充分条件及解的迭代计算公式.在此条件下,给出了该唯一解的扰动界及正定解条件数的一种表达式,并用数值例子对所得结果进行了说明.     

矩阵方程X+A*X-qA=Q(q≥1)的Hermite正定解

In this paper,Hermitian positive definite solutions of the nonlinear matrix equation X + A*X-qA = Q (q ≥ 1) are studied.Some new necessary and sufficient conditions for the existence of solutions are obtained.Two iterative methods are presented to compute the smallest and the quasi largest positive definite solutions,and the convergence analysis is also given.The theoretical results are illustrated by numerical examples.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

矩阵方程X+A^{*}X^{-q}A=Q(q\geq 1)的Hermitian正定解

本文研究矩阵方程X A~*X~(-q)A=Q(q≥1)的Hermitian正定解,给出了存在正定解的充分条件和必要条件,构造了求解的迭代方法.最后还用数值例子验证了迭代方法的可行性和有效性.