Theoretical and numerical comparisons of GMRES and WZ-GMRES
The authors study the numerical stability of the WZ-GMRES method proposed by \textit{H. F. Walker} and \textit{L. Zhou} [Numer. Linear Algebra Appl. 1, No. 6, 571--581 (1994; Zbl 0838.65030)] and compare the stability of the WZ-GMRES method with that of the GMRES method for solving systems of linear equations \(Ax = b\) with a non-symmetric matrix \(A\). The main result of the paper is that the mathematically equivalent GMRES and WZ-GMRES methods are not numerically equivalent when the methods are near convergence. Some relationships between the condition numbers of the triangular matrices \(R_{G}\) and \(R_{WZ}\) and the matrix \(A\) are derived. Here, \(R_{G}\) and \(R_{WZ}\) are the matrices of the systems of equations which one has to solve within the GMRES and the WZ-GMRES iteration, respectively. It is shown that generally the matrix \(R_{WZ}\) is worse conditioned than the matrix \(R_{G}\) and that the matrix \(R_{WZ}\) is ill-conditioned when the method is near convergence. This makes the WZ-GMRES method numerically unstable. Therefore, it is more preferable to use the original GMRES implementation. Additionally, an upper bound of the residual norm is established. Numerical experiments confirm the theoretical results.
scientific article; zbMATH DE number 2142392
scientific article