Research On Iterative Algorithms For Several Kinds Of The Nonlinear Matrix Equations

Matrix theory and method are one of the important research topics in computational mathematics.It is widely used in modern engineering technology fields such as physics,engineering,finance and biology.In recent years,many scholars have studied this problem and made a series of important research findings.In this paper,several kinds of matrix iterative algorithms are systematically studied.These algorithms have broad application prospects in the field of numerical algorithm.In this paper,the theoretical and numerical methods of the following kinds of nonlinear matrix equation problems are systematically explored.Chapter 2:The Stein equation problem(?) where A1,A2,…Am are n ×n nonsingular matrices,and Q is a n × n positive definite matrix,p≥1is a positive integer.Based on the fixed-point principle and Anderson acceleration algorithm,a fixed-point acceleration algorithm is designed to solve this problem.The convergence theorem and error estimation formula of the algorithm are given by using the basic characteristics of Thompson metric.Finally,numerical experiments show that the algorithm is feasible and effective.Chapter 3:Study the problem of nonlinear matrix equations(?) where A1,A2,…Am are n × n nonsingular matrices,and Q is a n × n positive definite matrix,p≥1is a positive integer.Based on the fixed-point iterative algorithm and the minimum polynomial extrapolation method,a fixed point accelerated algorithm is designed to solve this problem.The convergence theorem and error estimation formula of the algorithm are given by using the basic characteristics of Thompson metric.Finally,numerical experiments show that the algorithm is feasible and effective.Chapter 4:Study the problem of generalized symmetric matrix equations AX=B,s.t.X ∈ GSRn×n(M,N),And matrix approximation problem(?)Paige algorithm is designed to solve these two problems,and the convergence of the algorithm is proved.Finally,numerical experiments show that the algorithm is feasible and effective.