  While there is no lack of efficient Krylov subspace solvers for Hermitian
systems, there are few for complex symmetric, skew symmetric, or skew Hermitian
systems, which are increasingly important in modern applications including
quantum dynamics, electromagnetics, and power systems. For a large consistent
complex symmetric system, one may apply a non-Hermitian Krylov subspace method
disregarding the symmetry of $A$, or a Hermitian Krylov solver on the
equivalent normal equation or an augmented system twice the original dimension.
These have the disadvantages of increasing either memory, conditioning, or
computational costs. An exception is a special version of QMR by Freund (1992),
but that may be affected by non-benign breakdowns unless look-ahead is
implemented; furthermore, it is designed for only consistent and nonsingular
problems. For skew symmetric systems, Greif and Varah (2009) adapted CG for
nonsingular skew symmetric linear systems that are necessarily and
restrictively of even order.
  We extend the symmetric and Hermitian algorithms MINRES and MINRES-QLP by
Choi, Paige and Saunders (2011) to complex symmetric, skew symmetric, and skew
Hermitian systems. In particular, MINRES-QLP uses a rank-revealing QLP
decomposition of the tridiagonal matrix from a three-term recurrent
complex-symmetric Lanczos process. Whether the systems are real or complex,
singular or invertible, compatible or inconsistent, MINRES-QLP computes the
unique minimum-length, i.e., pseudoinverse, solutions. It is a significant
extension of MINRES by Paige and Saunders (1975) with enhanced stability and
capability.
