Numerical Equivalences among Krylov Subspace Algorithms for Skew-Symmetric Matrices

Abstract

We briefly review Krylov subspace methods based on the Galerkin and minimum residual conditions for solving $Ax=b$ with real $A$ and $b$, followed by two implementations: the conjugate gradient (CG) based methods CGNE and CGNR. We then show the numerical equivalence of Lanczos tridiagonalization and Golub--Kahan bidiagonalization for any real skew-symmetric matrix $A$. We give short derivations of two algorithms for solving $Ax=b$ with skew-symmetric $A$ and use the above equivalence to show that these are numerically equivalent to the Golub--Kahan bidiagonalization variants of CGNE and CGNR. These last two numerical equivalences add to the theoretical equivalences in the work by Eisenstat [Equivalence of Krylov Subspace Methods for Skew-Symmetric Linear Systems, Department of Computer Science, Yale University, preprint, arXiv:1512.00311, 2015] that unified and extended earlier work. We next present a method based on the Lanczos tridiagonalization process for minimizing $\|A^T (b-Ax_k)\|_2$ when $A^T\! =-A$ and show that for skew-symmetric systems it is numerically equivalent to LSMR developed by Fong and Saunders [SIAM J. Sci. Comput., 33 (2011), pp. 2950--2971]. Finally, we illustrate the typical convergence behaviors of these algorithms with a numerical example and use these and an analysis to give new insights into algorithm choices for general large sparse matrix solution of equations problems.