Minimal Residual Methods for Complex Symmetric, Skew Symmetric, and Skew Hermitian Systems

While there is no lack of efficient Krylov subspace solvers for Hermitian systems, few exist 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 memory, conditioning, or computational costs. An exception is a special version of QMR by Freund (1992), but that may be affected by nonbenign breakdowns unless look-ahead is implemented; furthermore, it is designed for only consistent and nonsingular problems. 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 minimumlength (i.e., pseudoinverse) solutions. It is a significant extension of MINRES by Paige and Saunders (1975) with enhanced stability and capability.