Algorithms For Several Nonlinear Matrix Equations

Nonlinear matrix equations arise in areas of control theory, ladder networks, dy-namic programming, queueing theory, stochastic filtering and statistics. It has animportant theoretical meaning and high practical value of researching on numericalmethod to solve these nonlinear matrix equations.In this thesis, we use fixed point iteration method (FPI), Newton method (NM),cyclic reduction method (CR), and structure-preserving doubling algorithm (SDA) tosolve some kinds of nonlinear matrix equation.In Chapter Two, based on the fixed point theorems, we use fixed point iterationmethod to solve the maximal positive definite solution of symmetric nonlinear matrixequation, and get the convergence and convergence order theorems under certain con-ditions. We apply the weighting to fixed point iteration method, and get a sort of newfixed point iteration.In Chapter Three, we use Newton method to solve symmetric nonlinear matrixequation, and get the convergence and convergence order theorems under certain con-ditions. In addition, we first extend this method to solve the general nonlinear matrixequation A theorem for the existence of the quasi-maximal solu-tion is derived. Newton method is constructed to compute the quasi-maximal solution,and the convergence and convergence rate theorems are given.In Chapter Four, by using of even-odd permutation, we use cyclic reductionmethod to compute the maximal Hermitian positive definite solution and minimumpositive definite solution of a sort of nonlinear matrix equation. It has good numericalstability, and quadratical convergence rate.In Chapter Five, based on properties of the doubling transformation, we use thestructure-preserving doubling algorithm to compute the maximal positive definite solu-tion of symmetric nonlinear matrix equations, the symmetric positive semidefinite solu-tion of discrete-time algebraic Riccati equation, and the extremal solution of quadraticmatrix equation. This algorithm has good numerical stability, low cost computationalcost per step, and quadratical convergence rate.