# A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

## TL;DR

This paper introduces a modified generalized shift-splitting preconditioner and iteration method for nonsymmetric saddle point problems, proving their convergence and demonstrating superior numerical performance over existing methods.

## Contribution

The paper develops a new MGSSP preconditioner and iteration method, extending previous work, with proven convergence and improved efficiency for nonsymmetric saddle point problems.

## Key findings

- MGSSP iteration is unconditionally convergent and semi-convergent.
- Numerical results show MGSSP outperforms existing methods.
- MGSSP preconditioner is more effective than other preconditioners.

## Abstract

For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.