A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

“…In this section, we present applications of our proposed algorithms to the certain linear matrix equations. To show the effectiveness and capability of our algorithms, we compare our proposed algorithms to the mentioned existing algorithms as well as the direct methods (3) and (11). For convenience, we abbreviate TauOpt to represent our algorithms.…”

“…In practical applications, we solve the linear matrix equations of large dimensions by effective iterative methods. There are several ideas to formulate an iterative procedure, namely, one can use matrix sign function [5], block recursion [6,7], Krylov subspace [8,9], Hermitian and skew-Hermitian splitting [10,11], and other related research works; see, e.g., [12][13][14][15]. In the recent decade, the ideas of gradients, hierarchical identification and minimization of associated norm-error functions have encouraged and brought about many researches; see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Least‐squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

“…In Dehghan and Shirilord, 30,31 double‐step scale splitting (DSS) approaches were used to solve complex Sylvester matrix equation. Dehghan and Shirilord extended the modified Hermitian and skew‐Hermitian splitting (MHSS) method for solving large sparse Sylvester equation with non‐Hermitian and complex symmetric positive definite/semidefinite matrices 32 . Dehghan and Hajarian extended the CG method for finding the least Frobenius norm generalized centrosymmetric and central anti‐symmetric solutions of general coupled matrix equations 33 .…”

Iterative algorithm for the reflexive solutions of the generalized Sylvester matrix equation

“…Moreover, Hajarian [15] established the matrix form of the biconjugate residual (BCR) algorithm for computing the generalized reflexive (anti-reflexive) solutions of the generalized Sylvester matrix equation P [16] established some conditions for the existence and the representations for the Hermitian reflexive, anti-reflexive, and non-negative definite reflexive solutions to the matrix equation AX = B with respect to a generalized reflection P by using the Moore-Penrose inverse. Dehghan and Shirilord [17] presented a generalized MHSS approach for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semi-definite matrices based on the MHSS method. Dehghan and Hajarian [18] proposed two algorithms for solving the generalized coupled Sylvester matrix equations over reflexive and anti-reflexive matrices.…”