An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems

In this work, by introducing a scalar matrix $\alpha I$, we transform the complex symmetric indefinite linear systems $(W+iT)x=b$ into a block two-by-two complex equations equivalently, and propose an efficient relaxed shift-splitting (ERSS) preconditioner. By adopting the relaxation technique, the ERSS preconditioner is not only a computational advantage but also closer to the original two-by-two of complex coefficient matrix. The eigenvalue distributions of the preconditioned matrix are analysed. An efficient and practical formula for computing the parameter value $\alpha$ is also derived by computing the Frobenius norm of symmetric indefinite matrix $T$. Numerical examples on a few model problems are illustrated to verify the performances of the ERSS preconditioner.