A Class Of Improved Algorithms For The Least Squares Problem Of Generalized Sylvester Equations

Matrix equation theory is a very important part of linear algebra and it has been widely used in algebra,combination,graph theory,control and other fields.The Sylvester matrix equation first appeared in the 1880s and has received much attention from many scholars in various fields.The earliest research on the least squares problem of Sylvester matrix equation was mainly applied in matrix perturbation analysis.Later,with the emergence of large-scale problems,the research on the least squares problem of generalized Sylvester matrix equations are more and more extensive.The main problem studied in this paper is the least squares problem of generalized Sylvester matrix equations under kernel norm and spectral norm and this kind of problem has been widely used in the fields of financial theory,image restoration.Li Jiaofen et al.used inexact alternating direction method of multipliers to discuss the constrained least squares problem of generalized Sylvester matrix equations under the kernel norm and spectral norm,the algorithm is(?) where S is a closed convex set.Based on Li Jiaofen et al,this paper simplifies the subproblem by introducing new variables and applies alternating direction method of multipliers to solve it.Each subproblem can be solved accurately,more important,it has explicit expressions and we prove the convergence of the algorithm in theory.Numerical experiments show that the improved algorithm is greatly improved in terms of time and number of iterations.