Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning

A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

“…In this paper, we present the nested splitting conjugate gradient (NSCG) iterative method for solving the generalized Sylvester equation AXB + CXD = E with large sparse coefficient matrices, which is an extension of [16][17][18]. Based on the NSCG method, we propose the preconditioned nested splitting conjugate gradient (PNSCG) iterative method, which extends the scope of the NSCG iteration method in applications.…”

“…In this section, we will give an implementation of the NSCG method for the generalized Sylvester equation (1.1) and present sufficient conditions for the convergent splitting when the coefficient matrices are nonsymmetric, which extend those results in [16][17][18].…”

“…Our purpose in the present paper is twofold. Firstly, we present the nested splitting conjugate gradient (NSCG) iterative method for solving the generalized Sylvester equation (1.1) by using the symmetric and skew-symmetric splitting of the matrices A, B, C and D. This method was first proposed in [16] for solving system of linear equations Ax = b, then was extended to solve the matrix equation AXB = C and AX + XB = C , respectively, see [17,18]. For more detail on this method, we refer to [28,29] and references therein.…”

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

“…Such problems arise in the solution of large eigenvalue problems and in the boundary value problems . They play an important role in linear control and filtering theory for continuous or discrete‐time large‐scale dynamical systems, image restoration and processing, and other applications such as model reduction, numerical solution of matrix differential Riccati equations, and many more; see .…”

“…In , Bai gave the HSS iteration method for solving large sparse continuous Sylvester equations with non‐Hermitian and positive definite/semi‐definite matrices. In , the NSCG method was presented for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non‐Hermitian) positive semi‐definite, and at least one of them is positive definite. In , Bouhamidi and Jbilou proposed a Krylov subspace method for solving large and sparse generalized Sylvester matrix equation.…”