Abstract

We consider conjugate gradient type methods for the solution of large linear systems Az = b with complex coefficient matrices of the type A = T + io1 where T is Hermitian and u a real scalar. Three different conjugate gradient type approaches with it-erates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunders’s SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure of A can be preserved by using polynomial precon-ditioning, 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.

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