The Research Of The Invariant Subspace Method For Solving The Continue-time Algebraic Riccati Equation

There are a lot of problems in physics and projects such as linear optimal control,Kalman filter, H∞ control, the total least squares problem,the deuce border value problem, which boildown to solve a sort of special equation-algebraic Riccati equation, the invariant subspace method is an effect means, which comes down to numerically solve the matrix eigenproblems. In this paper, the Hamiltonian matrix eigenproblem is the emphasis we discuss and research, it is important and valuable to compute the real or complex stable matrix radius, calculate the H∞ -norm of the transfer matrix and seek part of eigenvalues and their eigenvectors accordingto the maximum model in linear response theory of the computational chemistry. In allusion to this problem, it is hang in doubt in numerical domain to seek an effect algorithm that is numerical stability and preserve the Hamiltonian structure. Here we present several available algorithms for computing exact or approximate eigenvalues of Hamiltonian matrix.At first we introduce the origin of the Hamiltonian matrix eigenproblem and some basis methods about this problem along with their advantage and disadvantage, thereby choose it as my researchful theme. Then we introduce some conceptions about Symplectic geometry, and export the conception of Symplectic matrix through analyzing the algebraic structure of Symplectic geometry. It is in reason to use Symplectic matrix as tool to research the Hamiltonian matrix eigenproblem under the Symplectic geometry frame.The third and fourth chapter we present some academic result and familiar standard form of Hamiltonian matrix, at the same time introduce cursorily the SR algorithm. The following emphasis is the Lanczos process and Lanczos algorithm computing all or partial eigenvalues of Hamiltonian matrix, its stop conditions and error estimate also are analyzed and explained. Finally we introduce the square strategy into the SR algorithm that reduce operations, and then analyze and contrast the implicit and explicit SR algorithm. The most important is we explore their numerical property.