| Matrix equation is an important field in matrix theory, and the research on linear and nonlinear matrix equation is always one of the important problems which people concentrate on. Matrix equation has very wide applications in mathematics itself and other natural sciences. With the development of modern natural science and Engineering Technology, the nonlinear problems appear in many fields. Nonlinear matrix equation has been used widely in transport theory, dynamic programming, trapezoidal mesh, statistics, engineering computing and so on. So the research on nonlinear matrix equation has been one of the hot topics which people pays close attention to in computational mathematics.In this paper, we mainly study Newton’s method for solving nonlinear matrix equations X=Q+AH(?)X-C)-1A and X=Q-AH(I(?)X-C)-1A. And for matrix equation Xs+AH f(X)A=Q, we discuss the conditions for the existence of the positive definite solutions, iterative methods, the perturbation analysis and so on.In Chapter Two, under the condition I(?)Q-C>0, we mainly study matrix equation where Q is an n x n Hermitian positive definite matrix, C is an mn×mn Hermitian positive semi-definite matrix, A is an mn x n complex matrix, and I is the identity matrix of order m.First we derive an equivalent form of the equation,where And we prove these two equations are equivalent for the existence of the solutions. Then we discuss the properties of the solutions to the equivalent equattion,and we derive the following theorem.We denote B=(B1T…BmT)T,Bi∈Cmn,i=1,…,m.Theorem1If B1,…,Bm are simultaneously unitary diagonalizable,then the above equivalent equation has at least2mn solutions,all these solutions and all Bi(i=1,…,m)are also simultaneously unitary diagonalizable.Moreover,in these solutions, there are one positive definite solution and one negative semi-definite solution,and the others are indefinite solutions.Next we focus on studying Newton’s method for solving the equivalent equation. We obtain the following two results:Theorem2Assume that there is a unitary matrix U,such that where Ai=diag(λ1(i),…,λmn(i)). And assume that Re(∑i=1mλk(i)λl(i))≥0,k,l=1,2,…,mn.With Y0=αI,α>0,Newton’s iterative method is quadratically convergent to the positive definite solution of the equivalent equation, here ymax**=max{y1**,…,ymm**),yj**(j=1,…,mn)is the negative root of the equation y2-y-∑i=1m|λj(i)|2=0,R+denotes the set of positive numbers;and with Y0=αI,α∈(ymax**,0),Newton’s iterative method is quadratically convergent to the negative definite solution of the equivalent equation. σ*is the smallest positive real root of the equation (1-σ)3-σ2=0, and the positivenumber δ satisfiesthen for the initial matrix Yo=I, the equivalent equation has a solution Y in S(Y0,δ)and Newton’s iterationconverges to Y, moreover, it holds thatwhereFinally we give two examples, compare Newton’s method with the existing method, and illustrate the efficiency of Newton’s method.In Chapter Three, we consider the matrix equationwhere Q is an n x n Hermitian positive definite matrix, C is an mn x mn Hermitian positive semi-definite matrix, A is an mn x n complex matrix, and I is the identity matrix of order m.First we give an equivalent form of the equation, where Y=I(?)X-C, P=I(?)Q-C>0,B=I(?)A.Then we obtain Newton’s iterative method for solving the equivalent equation. We prove that under the condition Newton iterative sequence{Yk} is monotone decreasing, and quadratically convergent to the maximal solution YL of the equivalent equation, moreover, we haveDenote the set of all n x n positive definite matrices by P(n). In Chapter Four, we consider the matrix equation where A is nonsingular, Q is an Hermitian positive definite matrix, s is a positive real number,f is a continuous map from P(n) into P{n), and f is either monotone (meaning that0≤X≤Y implies that f(X)≤f(Y)) or anti-monotone(meaning that0≤X≤Y implies that f{X)≥f(Y)).First we derive a necessary and sufficient condition for the existence of positive definite solutions.Theorem4Assume f is a continuous map from P(n) into HP(n), where HP(n) denote the set of all n x n Hermitian positive definite matrices. Then the equation has an Hermitian positive definite solution if and only if there is a nonsingular matrix W, such that WHW=WWH, and A={f1/2(WHW))-1ZQ1/2, whereIn this case, the equation has an Hermitian positive definite solution X=WHW.In order to discuss an iterative method for solving the equation, we assume that for a given matrix B, the equation f(X)=B always has a positive definite solution and its solution is easy to obtain. We consider the following iterative method Here we assume that A,Q,f in the equation satisfy f-1{A-HQA-1)<Q1/s. We have the following two conclusions:Theorem5Suppose f-1exist and f is anti-monotone. Let s=1. The equation has a positive definite solution in the interval (0, λmax(Q)I) if and only if there is a number χ∈(0,λmax(Q)),such that Xk<χI for all k. Moreover, in this case, the iteration with Xo=0converges to the smallest positive definite solution of the equation.Theorem6Suppose f-1exist and f is anti-monotone, and suppose f, f-1are differentiable at any point of Ω1=[f-1(A-H QA-1), Q1/s], and Ω2=[f(Q1/s), A-HQA-1], respectively. Let and a=M1M2‖A-1‖2‖A‖2.(i)If the equation with s>1has an Hermitian positive definite solution X and a<1, then X is the unique solution of the equation.(ii) Assume that there is a closed set Ω∈Ω1satisfying that g:Ωâ†'Ω and g(X)=f-1(A-H(Q-Xs)A-1), if a<1, then the equation with s>1has a unique solution X in Ω. Further, we consider the iterative method with Xo∈Ω.The sequence {Xk} converges to the unique solution X, moreover Last we discuss the perturbation analysis of the equation. Let be the perturbation equation. Let A, A be nonsingular matrices and Q,Q be positive definite, and0<s<1. Suppos f-1:P(n)â†'P(n) exist and f is anti-monotone. We obtain the perturbation theorem as follows:Theorem7Let X, X be the positive definite solutions of the equation and its perturbation equation, respectively. Map f-1is differentiable at any point of Ω4with Y∈,f([f-1(A-HQA-1),Q1/s])}.Letwhere... |
PDF Full Text Request