The paper focuses on the local Hermitian and skew-Hermitian splitting (LHSS) iteration method and the modified LHSS (MLHSS) iteration method for solving generalized nonsymmetric saddle point problems with nonzero \((2,2)\) blocks: \[ \left (\begin{matrix} A & B \\ B^T & -C \end{matrix}\right ) \left (\begin{matrix} x \\ y \end{matrix}\right)=\left (\begin{matrix} f \\ g \end{matrix}\right) \tag{1} \] where \(A\in\mathbf R^{m\times m}\) is a positive definite matrix and \(A\neq A^T\), \(C\in\mathbf R^{m\times n}\) is a symmetric positive semi-definite matrix, \(B\in\mathbf R^{m\times n},\) \(m\geq n,\) is a full column rank matrix, \(f\in\mathbf R^{m},\) \(g\in\mathbf R^n\), are two given vectors.
The first section is an introduction in nature.
The second section presents three convergence theorems of the LHSS and MLHSS iteration methods. Thus, the first one gives a sufficient and necessary condition for guaranteeing the convergence of the MLHSS method. Based on the first theorem, the second one gives a description of the convergence of the LHSS method, while the third one derives the convergence condition of the MLHSS method.
The third section gives other formal MLHSS methods, presenting three different algorithms.
The feasibility and effectiveness of the presented iteration algorithms are illustrated by some numerical examples given in the fourth section.
The last section presents the main conclusions.