An Inexact Newton Generalized Conjugate Gradient Iteration Method For Solving Quadratic Matrix Equation

This paper investigates numerical algorithms for solving quadratic matrix equation AX2+BX+C=0.Quadratic matrix equation has extensive and profound applications in material science,physics,engineering,control theory and computer science,and et al..It is important to study efficient numerical methods for solving this problem in numerical algebra.Especially in recent years,with the rapid development of computer technology,numerical methods for nonlinear equations have been gradually developed into a very popular topic in the field of computational mathematics and engineering control areas.In 2001,Higham et al.proposed an exact Newton method with an exact line search for solving quadratic matrix equation,in which a linear Sylvester matrix equation is required computing exactly.This leads to large amount of computation.In the paper,we first discuss several important properties of the OROD method for solving the Sylvester equation AXB+CX=D,secondly,we introduce an inexact Newton-generalized conjugate gradient method for solving the quadratic matrix equation AX2+BX+C=0 The paper is organized as follows:In Chapter 1,we simply introduce the background and related preparatory knowledge.In Chapter 2,we discuss the OROD method for solving the Sylvester equation AXB+CX=D and study its properties.We show that the error of the iterative sequence generated by this algorithm is decreasing.Moreover,we give its exact minimization characteristic.In Chapter 3,we present an inexact Newton-generalized conjugate gradient method for solving the quadratic matrix equation AX2+BX+C=0.We adopt the OROD method to solve the subproblem Sylvester equation approximately.We also study the perturbation analysis of the solution and the backward error analysis on the method.We do some numerical experiments to show its efficiency.