Equivalence of Krylov Subspace Methods for Skew-Symmetric Linear Systems

TL;DR

This paper proves the equivalence of two Krylov subspace methods for skew-symmetric linear systems to well-known methods, showing that their iterates are identical to those of conjugate gradient and Craig's method in exact arithmetic.

Contribution

It provides new, algorithm-independent proofs of the equivalence between these Krylov methods and classical algorithms, clarifying their theoretical relationship.

Findings

Iterates for the methods are identical to conjugate gradient and Craig's method in exact arithmetic.

Projecting solutions to lower-dimensional subspaces does not increase error or residual norms.

The proofs are independent of specific algorithm implementations.

Abstract

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.