Conjugate Gradient-Type Methods for Linear Systems with Complex Symmetric Coefficient Matrices

Conjugate gradient-type methods for the solution of large sparse linear systems $Ax = b$ with complex symmetric coefficient matrices $A = A^T $ are considered. Such linear systems arise in important applications, such as the numerical solution of the complex Helmholtz equation. Furthermore, most complex non-Hermitian linear systems which occur in practice are actually complex symmetric. Conjugate gradient-type iterations which are based on a variant of the nonsymmetric Lanczos algorithm for complex symmetric matrices are investigated. In particular, a new approach with iterates defined by a quasi-minimal residual property is proposed. The resulting algorithm presents several advantages over the standard biconjugate gradient method. Some remarks are also included on the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.