On conjugate gradient type methods and polynomial preconditioners for a class of complex non-hermitian matrices

We consider conjugate gradient type methods for the solution of large linear systemsA x=b with complex coefficient matrices of the typeA=T+i¿I whereT is Hermitian and ¿ a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure ofA can be preserved by using polynomial preconditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.