SOR for AX−XB=C

On Hermitian and Skew-Hermitian Splitting Ietration Methods for the Continuous Sylvester Equations

“…For example, the matrices in (3.1) are shifted Hermitian positive definite or positive semi-definite matrices and, thus, are Hermitian positive definite; and the matrices in (3.2) are shifted skew-Hermitian matrices and, thus, are positive definite but not Hermitian. To further improve the computational efficiency of the HSS iteration method, we can solve the two sub-problems (3.1) and (3.2) inexactly by utilizing certain effective iteration methods, e.g., the (block) Gauss-Seidel, the (block) SOR, the ADI, the conjugate gradient or the Krylov subspace methods; see [11,12,24,33,38,40]. This naturally results in the following inexact HSS iteration method for solving the continuous Sylvester equation (1.1).…”

“…When the matrices A and B are large and sparse, iterative methods such as the Smith's method [37], the alternating direction implicit (ADI) method [11,24,33,40], the block successive overrelaxation (BSOR) method [38], the preconditioned conjugate gradient method [12], the matrix sign function method [27], and the matrix splitting methods [17] are often the methods of choice for efficiently and accurately solving the continuous Sylvester equation (1.1).…”

A note on the iterative solutions of general coupled matrix equation

“…Starke and Niethammer reported an iterative method for solutions of CT Sylvester equations by using the SOR (successive overrelaxation) technique [28], Mukaidani, Xu, and Mizukami discussed an iterative algorithm for generalized algebraic Lyapunov equations [24], and El Guennouni et al [13] used Krylov methods to solve Sylvester equations, and Bai constructed a class of unconditional convergent Hermitian and skew-Hermitian splitting (HSS) iteration methods for solving the CT Sylvester equations [1]. See also [3,4,7,8,14,15,22,23,25,27].…”

New SOR-like methods for solving the Sylvester equation

“…[3, pp. 245-246], [8,9]). Two direct methods, the Bartels-Stewart algorithm ( [1]) and the Hessenberg-Schur algorithm ( [4]), are widely known and used (the former is implemented within Matlab and Octave).…”

“…Another way is to use Krylov-subspace methods, which are available in the literature (see [5]). In [9] Starke and Niethammer proposed another routine: the Successive Over-Relaxation method for solving linear systems applied to the Sylvester equation. This however results in a block-iterative method not iterative one, as it requires solving a number of linear systems per iteration.…”