Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices
Mark Embree, Josef Sifuentes, Kirk M. Soodhalter, Daniel B. Szyld, Fei Xue · SIAM Journal on Matrix Analysis and Applications · 2012
The progressive GMRES algorithm, introduced by Beckermann and Reichel in 2008, is a residual-minimizing short-recurrence Krylov subspace method for solving a linear system in which the coefficient matrix has a low-rank skew-Hermitian part. We analyze this algorithm, observing a critical instability that makes the method unsuitable for some problems. To work around this issue we introduce a different short-term recurrence method based on Krylov subspaces for such matrices, which can be used as either a solver or a preconditioner. Numerical experiments compare this method to alternative algorithms.