Preconditioning Complex Symmetric Linear Systems

A new preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to conjugate orthogonal conjugate gradient (COCG) or conjugate orthogonal conjugate residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. To reduce the computational cost the preconditioner is approximated with an inexact variant based on incomplete Cholesky decomposition or on orthogonal polynomials. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the inexact polynomial version completes the description of the preconditioner.