A generalization of the local Hermitian and skew-Hermitian splitting iteration methods for the non-Hermitian saddle point problems

On generalized local Hermitian and skew-Hermitian splitting iterative method for block two-by-two linear systems

“…Both theoretical analysis and numerical experiments have shown that the GLHSS iterative method is feasible and effective. Beside, as the conclusion that u à Su ¼ 0 for all u 2 C n holds when S is a skew-symmetric matrix is not correct, the correctness of the existing Corollaries in real number space established in [23,28] is questioned.…”

“…For the real case with D ¼ 0, some simplified results are presented by Jiang and Zhu, denoted as Corollaries 2.1-2.7 in [23] and Corollaries 2.6-2.10 in [28]. However, these results are perhaps incorrect, because they are all based on Lemma 2.3.…”

“…It must be noted that Corollary 2.13 is just Theorem 2.2 in [23] and Theorem 2.5 in [28], but that the convergence condition in Theorem 2.5 in [28] is not sufficient, The condition ða þ cÞ 2 À ðc þ s 1 Þ 2 À b 2 > 0 must be satisfied and added.…”

“…Algorithm 1.1 [28] (GLHSS iterative method). Assume A 2 C nÂn is non-Hermitian and its Hermitian part H ¼ 1 2 ðA þ A Ã Þ is positive definite, B has full rank.…”

Semi-convergence analysis of the GPIU method for singular nonsymmetric saddle-point problems

“…Since the coefficient matrix is often large and sparse, iterative methods become more attractive than direct methods for the saddle point problem (1). Many efficient methods have been proposed, such as the Uzawa-type schemes ( [10,15,16,19,22]), the Krylov subspace methods ( [25,33,34]), the SOR-like methods ( [9,20,26,30,35]), the Hermitian and skew-Hermitian splitting (HSS) iteration methods ( [2,[4][5][6][7][8]29]) and some other iterative methods [15,28,42], we mention just a few.…”