Equivalence of Krylov Subspace Methods for Skew-Symmetric Linear Systems
This theoretical result clarifies the relationship between methods for skew-symmetric systems, but is incremental as it unifies existing approaches rather than introducing new algorithms.
The paper proves that two recently proposed Krylov subspace methods for skew-symmetric linear systems are equivalent to the conjugate gradient method on the normal equations and Craig's method, respectively, in exact arithmetic. It also shows that projecting from the original subspace to a lower-dimensional one does not increase error or residual norms.
In recent years two Krylov subspace methods have been proposed for solving skew symmetric linear systems, one based on the minimum residual condition, the other on the Galerkin condition. We give new, algorithm-independent proofs that in exact arithmetic the iterates for these methods are identical to the iterates for the conjugate gradient method applied to the normal equations and the classic Craig's method, respectively, both of which select iterates from a Krylov subspace of lower dimension. More generally, we show that projecting an approximate solution from the original subspace to the lower-dimensional one cannot increase the norm of the error or residual.