Research On Iterative Algorithms Of Constraint Matrix Equation AXB+CXD=F

The research of constrained matrix equation is closely related to the practical application in real life.Many complex problems in modern science and technology need to be solved by matrix equation theory and methods.Nowadays,many complex problems in engineering and scientific fields,such as modern financial theory,automatic control theory,parameter identification,information theory and vibration theory,can be solved by solving constraint matrix equations.The research on related matrix theories and methods has greatly promoted the development of science and technology,so the speed and high accuracy of solving matrix equations are the important contents of future research.In order to accelerate the solving speed of matrix equation and reduce the number of iterations,this master's thesis mainly improves the iterative algorithm of solving matrix,and applies Anderson Acceleration to the multi-step iterative algorithm of solving constraint matrix equation.The main research questions of this paper are as follows:Problem ?.Given matrices A,C?Rm×n,B,D?Rn×p,F?Rm×p.Find X ? Rn×n,such thatProblem ?.Given matrices X ?Rn×n.Find X ? SE,such thatProblem ?.Given matrices A,C ? Rm×n,B,D ? Rn×p,F?Rm×p,L?Rn×n,U?Rm×m.Find X ? SRn×n,such thatWhere ? is a given constant,and SE is the solution set of problem ?.?min(X)is the minimum eigenvalue of matrix X,The inequality X?Y is Xij?Yij for any two real matrices,where Xij and Yij represent the ij terms of X and Y respectively.Firstly,based on the idea of fixed point iterative algorithm,combined with Anderson acceleration,a multi-step iterative algorithm is proposed to solve the constrained matrix equation and its least squares problem.Secondly,the accelerated multi-step iterative algorithm for solving problems ? and ? is given,and the convergence of the accelerated multi-step iterative algorithm is proved.Finally,numerical experiments show that the accelerated multi-step iterative algorithm has better convergence effect in solving constrained matrix equations.Among them,problem ? can also be solved by the accelerated multi-step iterative algorithm through the corresponding matrix deformation.Numerical experiments show the effectiveness of the accelerated algorithm...