  Complex valued systems with an indefinite matrix term arise in important
applications such as for certain time-harmonic partial differential equations
such as the Maxwell's equation and for the Helmholtz equation. Complex systems
with symmetric positive definite matrices can be solved readily by rewriting
the complex matrix system in two-by-two block matrix form with real matrices
which can be efficiently solved by iteration using the preconditioned square
block (PRESB) preconditioning method and preferably accelerated by the
Chebyshev method. The appearances of an indefinite matrix term causes however
some difficulties. To handle this we propose different forms of matrix
splitting methods, with or without any parameters involved. A matrix spectral
analyses is presented followed by extensive numerical comparisons of various
forms of the methods.
