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Hermite Positive Definite Solutions And Perturbation Analysis Of Several Nonlinear Matrix Equations

Posted on:2010-10-20Degree:DoctorType:Dissertation
Country:ChinaCandidate:X Y YinFull Text:PDF
GTID:1100360302469350Subject:Applied Mathematics
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Matrix equation is an important field of matrix theory. It is widely used in mathematics itself and other natural sciences as well. Since nonlinear phenomena exists broadly in our life, the problem of solving nonlinear matrix equations has been one of the most important issues in the feild of numerical algebra and non-linear sciences for a long time. In recent years, a kind of symmetric nonlinear matrix equations X±A*X-nA=Q, n=1,2, where A is nonsingular and Q is positive definite, attracted much attention for its wide use in many fields such as dynamic programming, automatics control theory,statistic. There have been many results on Hermite positive definite solutions of these nonliear matrix equations in the literature. Moreover, scholars extended them to several more general forms:(1).X±A*X-nA=Q, where n is a positive interger; (2).Xs±A*X-tA=Q, where s, t are positive intergers; (3).X±A*X-qA=Q, where q>0, ect.This dissertation considers mainly these nonlinear matrix equations, including the existence, the fixed-point iteration, and the perturbation analysis of the Hermite positive definite solutions.1.For the nonlinear matrix equation X-A*X-2A=Q, it is already known that it always has an Hermite positive definite(HPD) solution. Based on this we give a sufficient condition for it to have a unique HPD solution:2‖A‖2‖Q-1‖3<1. Under this condition, a fixed-point iteration to solve the unique HPD solution is offered, the convergence of the iteration algorithm is proved as well. Moreover,based on the Schauder's fixed-point theorem, we discuss the perturbation analysis of the unique HPD solution. A perturbation bound of the unique HPD solution and an explicit expression of its condition number which is defined by Rice are given.2.For the nonlinear matrix equation X+A*X-nA=Q, we give a sufficient condition for it to have two different HPD solutions: Under this condition, we get the inclusion area of the HPD solutions and fixed-point iteration algorithms as well. Then we discuss the property and perturbation analysis of the maximal HPD solution. Using differential of the matrix-value function, we offer two new perturbation bounds for the maximal HPD solution. Furthermore, the Rice condition number of the maximal HPD solution is studied and its explicit expression is given. Theoretical analysis, as well as the numerical examples, shows that our perturbation bounds are supeior to the ones exsited.3.In chapter four we discuss the nonlinear matrix equations X±A*X-qA=Q where q is a positive real number. First for X-A*X-qA=Q, q≥1, we prove that it always has a HPD solution. and give the inclusion area of the HPD solutions. Two fixed-point iteration for the HPD solutions are offered and the convergence of the iterations are proved under certain conditions. Then for X+A*X-qA=Q, q≥1, we discuss the solvability of it and prove that if it has an HPD solution, it must has a minimum HPD solution. Iteration for the minimum solution is given and several numerical examples are presented to show the effectiveness of the iteration algorithm. Finally, for matrix equation X+A*X-qA=Q,0
Keywords/Search Tags:Nonlinear matrix equation, Hermite positive definite solution, Fixed-point theorem, Perturbation analysis, Condition number
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