Two efficient iteration methods for complex symmetric indefinite linear systems

In this paper, we consider addressing the complex symmetric indefinite linear systems with iteration methods. Based on a technical reformulation of original indefinite systems, we propose two iteration methods for solving the indefinite systems by multiplying complex numbers through both sides of the systems. Notably, the first method is unconditionally convergent despite the challenges posed by the indefinite term in complex symmetric linear systems. The second method extends the first by introducing an additional parameter, resulting in improved efficiency and convergence under appropriate conditions. Furthermore, we establish convergence theorems for both methods and derive explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices. Numerical experiments on a set of model problems demonstrate the efficiency of the proposed methods in comparison to existing approaches.