Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

Abstract

Abstract. For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skewHermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

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

“…We will also use S(W ) := 1 2 (W − W * ) to denote the skew-Hermitian part of the matrix W . Obviously, it holds that W = H(W ) + S(W ); see [2][3][4][5][6].…”

A New GSOR Method for Generalised Saddle Point Problems

“…In recent years, many iterative methods have been introduced to solve the problem (1.1), including Uzawa-type schemes [14,20,23,25,33,34,51,53], iterative projection methods [3], block and approximate Schur complement preconditioners [17,19,22,35,40,42,43], iterative null space methods [1,26,48], splitting methods [4,7,[9][10][11][12][13]18,29,30,36,38,39,41,46,50], indefinite preconditioning [31,37], and preconditioning methods based on approximate factorisation of the coefficient matrix [6,8,28,44]. A classical approach to solve (1.1) is the successive overrelaxation (SOR) iteration method [49], which can involve relatively low computation per iterative step.…”

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

“…being the symmetric and the skew-symmetric parts and A T being the transpose of the matrix A; see [1,14]. By modifying and preconditioning the HSS iteration method [10,11,13], recently Bai, Benzi, Chen and Wang [9] proposed and discussed a class of preconditioned modified HSS (PMHSS) iteration methods for solving the block two-by-two linear system (1.1); see also [7,8].…”