Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method

A Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems

“…where K is of simple form (e.g., diagonal) so that A = G + (S + K), Benzi [13] constructed a generalization of the Hermitian and skew-Hermitian splitting iteration method and Aghazadeh constructed a two-parameter generalized HSS iteration [2]. The TGHSS iteration method for solving the non-Hermitian positive definite linear system Ax = b can be described as the following algorithm.…”

“…It has been shown in [2] that the TGHSS iteration method allows one to have more control on each of the half-steps of the TGHSS than that of the GHSS iteration method. Therefore, by choosing suitable iterative parameters, the TGHSS iteration method will be more efficient than the GHSS iteration method.…”

“…Obviously, if P = I, then the PTGHSS iteration method reduces to the TGHSS iteration method [2]. If α = β and P = I, then the PTGHSS iteration method reduces to the GHSS iteration method [13].…”

“…In this paper, based on the generalization of the Hermitian and skew-Hermitian splitting (GHSS) iteration scheme proposed by Benzi [13] and the two-parameter GHSS (TGHSS) [2] iteration method for solving non-Hermitian positive definite linear system, we will first establish a preconditioned TGHSS (PTGHSS) iteration method. Then we propose a class of PTGHSS-based iteration methods, named by Picard-PTGHSS and nonlinear PTGHSS-like iteration methods for solving large sparse weakly nonlinear equations (1.1).…”

“…Unlike the HSS iteration as the inner iteration, the new iteration methods consist in the fact that the solution of systems with coefficient matrix αP + S + K by inner iterations is made easier, since this matrix is more "diagonally dominant" (loosely speaking) and typically better conditioned than αP + S [13]. Following [2], the quasi-optimal iterative parameters are given. Therefore, the new iteration methods are easy to be implemented.…”