摘要
约束矩阵方程问题是指在一定的约束矩阵集合中求矩阵方程(组)的解.其研究是近年来数值代数研究领域的重要课题.本文研究以下几类特殊约束矩阵方程问题的理论与计算.
1.两类线性约束矩阵方程问题及其最佳逼近问题的迭代算法
提出了求线性矩阵方程组：A1XB1=C1,A2XB2=C2的(最小二乘)双对称解的迭代算法；从算子角度,将十余种常见的矩阵结构约束(如对称、中心对称、自反等)划归为一类特殊的算子约束.针对一般形式的线性矩阵方程组,提出了求这一类特定算子约束(最小二乘)解的迭代算法.在不计舍入误差的前提下,所提出的算法均可在有限步内获得上述线性矩阵方程(组)相应的约束(最小二乘)解,并可解决其最佳逼近问题.
2.非线性矩阵方程：Xs+A*X-1A=Q的Hermitian正定解
深入研究了非线性矩阵方程：Xs+A*X-tA=Q(s,t为正整数)的定解理论和数值算法.利用矩阵分解原理给出了方程存在Hermitian正定解的两个充分必要条件.给出了方程仅有两个解的充分条件及解的计算公式.研究了AQ(?)=Q(?)A情形下,方程可解的必要条件和解的特性.分析了固定点迭代算法的收敛性,给出了单调收敛条件.此外还考虑了s≥1,0 3.非线性矩阵方程：Xs-A*X-1A=Q的Hermitian正定解
研究了非线性矩阵方程：Xs-A*X-1A=Q (s,t为正整数)的Hermitian正定解.证明了解的存在性.给出了方程存在唯一解的充分条件.获得了解范围的最新估计.进行了解的扰动分析,导出了一般解和唯一解的扰动界.
4.非对称代数Riccati方程的极小非负解
分析了当非对称代数Riccati方程的四个系数矩阵构成一个非奇异M-矩阵或奇异不可约M-矩阵时,方程极小非负解的敏感性.基于不变子空间的扰动性质,导出了极小非负解在任意酉不变范数意义下的扰动界,并获得了条件数的显式表达式.
5.TLS问题和LS问题解的相关量比较
在TLS问题和LS问题解残量的比较基础上,在更一般情形下,对TLS问题和LS问题解的加权残量进行了比较.导出了TLS解、改进的LS解及普通LS解加权残量之间的误差界,进一步完善了已有的相关结果.
The constrained matrix equation problem is to find solutions of a matrix equation (or a system of matrix equations) in a constrained matrix set. The research of it has been an important topic in the field of numerical algebra in recent years. In this thesis, theory and computation of some special constrained matrix equation problems are studied.
1. Iterative algorithms for solving two classes of constrained matrix equation problems and associated optimal approximation problems
An Iterative algorithms for the (least squares) bisymmetric solutions of the matrix equations A1XB1=C1,A2XB2=C2 is proposed. In the sight of operator, more than ten kinds of common constraints on the structure of matrices (such as symmetric constraint, centrosymmetric constraint, reflexive constraint and so on) are reduced to a kind of special operator constraint. Then an iterative method is constructed to find the (least squares) solutions of the general system of linear matrix equations with this operator constraint. By the proposed iterative algorithms, the constrained solutions above can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the associated optimal approximation problems can also be solved.
2.Hermitian positive definite solutions of the nonlinear matrix equation Xs+ A*X-tA=Q
The solvability and numerical algorithms for the nonlinear matrix equation Xs+ A*X-tA=Q are investigated deeply, where s and t are positive integers. Two necessary and sufficient conditions for the existence of a Hermitian positive definite solution are derived by using matrix decomposition principle. Necessary conditions for the existence of the Hermitian positive definite solutions of the matrix equation Xs+A*X-tA=Q with the case AQ1/2=Q1/2A are studied. Based on the convergence analysis of a fixed-point iteration, some monotonically convergent conditions of the iteration are given. Besides, the matrix equation with two cases:s≥1,0 3. Hermitian positive definite solutions of the nonlinear matrix equation Xs-A*X-tA=Q
The Hermitian positive definite solutions of the nonlinear matrix equation Xs-A*X-tA=Q are studied, where s and t are positive integers. The existence of a Her-mitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some new estimates of the Her-mitian positive definite solutions are obtained. Moreover, two perturbation bounds for a Hermitian positive definite solution and the unique solution of the matrix equation are derived respectively.
4. Perturbation analysis of the minimal nonnegative solution of the nonsymmet-ric algebraic Riccati equation
The sensitivity analysis of the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix is considered. Based on perturbation properties of invariant subspace, some sharp perturbation bounds for the minimal nonnegative solution of the matrix equation for any unitary invariant norm are derived. In addition, explicit expressions of the condition number for the minimal nonnegative solution are obtained.
5. Comparison of the squared residuals of the TLS and the LS problems
Based on the comparison of the squared residuals of the Total Least Squares (TLS) and the Least Squares (LS) problems, the weighted squared residuals of the TLS and the LS problems are compared. Bounds of difference between the weighted squared residuals of the TLS, modified LS, and ordinary LS solutions are derived, which extend the existing related results.
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