ITERATIVE SOLUTION OF SKEW-SYMMETRIC LINEAR SYSTEMS

摘要

We offer a systematic study of Krylov subspace methods for solving skew-symmetric linear systems. For the method of conjugate gradients we derive a backward stable block decomposition of skew-symmetric tridiagonal matrices and set search directions that satisfy a special relationship, which we call skew-A-conjugacy. Imposing Galerkin conditions, the resulting scheme is equivalent to the CGNE algorithm, but the derivation does not rely on the normal equations. We also discuss minimum residual algorithms, review recent related work, and show how the iterations are derived. The important question of preconditioning is then addressed. The preconditioned iterations we develop are based on preserving the skew-symmetry, and we introduce an incomplete 2 x 2 block LDLT decomposition. A numerical example illustrates the convergence properties of the algorithms and the effectiveness of the preconditioning approach.