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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.

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How large is your system? Where does it come from? I don't understand your second paragraph; if $ \epsilon = 0 $ don't you just have a diagonal matrix? Since $ D + \epsilon S $ is a small perturbation of a diagonal matrix, I would expect a very simple scheme like Jacobi or Gauss-Seidel to converge fairly quickly. To be a little more sophisticated you could try diagonally preconditioned GMRES. Finally, you might get more answers at scicomp.stackexchange.com –  Andrew T. Barker Sep 21 '12 at 19:09
    
I recommend asking this question on scicomp.stackexchange.com. –  David Ketcheson Oct 10 '12 at 18:19

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