I'm working on a problem that gives a matrix system of the form D + epsilon*S, where S is a skew-symmetric matrix. I'm interested in finding if any work has been done to develop a conjugate gradient method to exploit this structure (ideally for any epsilon, but possibly just for epsilon << 1), or if it's necessary to use biconjugate gradient or something similar.

I found the paper "ITERATIVE SOLUTION OF SKEW-SYMMETRIC LINEAR SYSTEMS" by CHEN GREIF AND JAMES M. VARAH which outlines what I have in mind (for epsilon = 0). After naively coding it up and seeing what happens when there is a non-zero diagonal I've found that their method will only converge for large epsilon.