The Iteration Method For The Least-squares Solutions And Optimal Approximation To Some Class Of Constrained Matrix Equations

Least-squares problems and the related optimal approximation problems of the con-strained linear matrix equations have been of interest in variety applications in manyfields. For example, electricity, molecular spectroscopy, vibration theory, structural de-sign, solid mechanics, finite elements, parametre identification, automatic control theory,biology, linear optimal control and so on.This master thesis is mainly concerned with the problem how to get the followingtypes of constraint least-squares solutions of the linear matrix equations and its optimalapproximation by applying iteration method systematically. The problems are as follow:Problem I. Given A∈R^m×n,B∈R^n×s and D∈R^m×s, find X|ˉ∈T such thatwhere T = CSR^n×n or T = CASR^n×n.Problem II. Given A∈R^m×n,B∈R^n×s,C∈Rm×k,D∈R^k×s and E∈R^m×s,find [ X|ˉ, Y|ˉ]∈T such thatwhere T = SR^n×n×SR^k×k or T = ASR^n×n×ASR^k×k.Problem III. Given A∈R^m×n,B∈R^n×q,C∈R^m×q,G∈R^l×n,H∈R^n×t andD∈R^l×t, find X|ˉ∈T such thatwhere T = SR^n×n or T = ASR^n×n.Problem IV. Let S denotes the solution set of Problem I or Problem III, givenX|ˉ∈R^n×n, find X|ˉ∈S such thatProblem V. Let S denotes the solution set of Problem II, given X?∈R^n×n,Y ?∈Rk×k, find [ X|ˉ, Y|ˉ]∈S such thathere CSR^n×n,CASR^n×n,SR^n×n,ASR^n×n respectively denotes centre-symmetric matrixwith order n, centre-Antisymmetric matrix with order n, symmetric matrix with ordern and Antisymmetric matrix with order n, SR^n×n×SRk×k,ASR^n×n×ASRk×kdenotes linear subspace on real number field {[A,B]|A∈SR^n×n,B∈SRk×k} and {[A,B]|A∈ASR^n×n,B∈ASRk×k},·denotes the Frobenius norm, and·R denotes a new normof the linear space Rm×q×Rl×t.We can get the following results in this thesis:1. For Problems I～III, an iterative method with short recurrences is presented tosolve the least-squares problems associated with the above mentioned matrix equations.For any initial matrix, a least-squares solution of these matrix equations over given matrixset can be determined within finite iteration steps in the absence of round-o? errors. Andthe corresponding minimum norm least-squares solution can be also obtained by choosinga kind of special initial matrices.2. For Problems IV～V, we can convert it to the problem of finding the minimumnorm least-squares solution of a new matrix equations. We can apply the iterative methodto get the minimum norm least-squares solutions first, and then get the solutions ofProblems IV～V.3. We analyze the theoretical properties of this iterative method. we can provethat the iteration will not stop before getting the least-squares solution and the approx-imate solution generated by this iterative method minimizes the Frobenius norm of theresidual sequence over a special a?ne subspace, which means that the Frobenius norm ofthe residual sequence is strictly monotone decreasing. Similar to the classical conjugategradient method and krylov-subspace method, we derive a rough error bound by meansof the minimization property of this iterative method. Finally, we give several numericalexamples to verify the obtained theoretical results.