Theoretical and numerical comparisons of GMRES and WZ-GMRES

WZ-GMRES, ‘a simpler GMRES’ proposed by Walker and Zhou, is mathematically equivalent to the generalized minimal residual method (GMRES) for solving large unsymmetric linear systems of equations. In this paper, relationships are established between two bases of an m-dimensional Krylov subspace Km(A, r0), and the condition number of the transition matrix between two bases is studied. Some relationships are derived between the condition numbers of the small matrices RG and RWZ resulting from GMRES and WZ-GMRES, respectively. A detailed analysis shows that generally RWZ is worse conditioned than RG, and in particular, RWZ is definitely ill conditioned when the method is near convergence. Furthermore, numerical behavior of WZ-GMRES is analyzed. It turns out that WZ-GMRES is not numerically equivalent to GMRES when the method is near convergence, and WZ-GMRES is numerically less stable than GMRES and can be numerically unstable. Numerical examples confirm the theoretical results.