Iterative methods for solving Ax = b GMRES/FOM versus QMR/BiCG

Abstract

We study the convergence of GMRES/FOM and QMR/BiCG methods for solving nonsymmetric Az = b. We prove that given the results of a BiCG computation on Az = b, we can obtain a matrix B with the same eigenvalues as A and a vector c such that the residual norms generated by a FOM computation on Bz = c are identical to those generated by the BiCG computations. Using a unitary equivalence for each of these methods, we obtain test problems where we can easily vary certain spectral properties of the matrices. We use these test problems to study the effects of nonnormality on the convergence of GMRES and QMR, to study the effects of eigenvalue outliers on the convergence of QMR, and to compare the convergence of restarted GMRES and QMR across a family of normal and nonnormal problems. Our GMRES tests on nonnormal test matrices indicate that nonnormality can have unexpected effects upon the residual norm convergence, giving misleading indications of superior convergence when the error norms for GMRES are not significantly different from those for QMR. Our QMR tests indicate that the convergence of the QMR residual and error norms is infLuenced predominantly by small and large eigenvalue outliers and by the character, real, complex, or nearly real, of the outliers and the other eigenvalues. In our comparison tests QMR outperformed GMRES(10) and GMRES(20) on both the normal and nonnormal test matrices. If you have difficulty viewing the second part of the linked postscript file, open the file: http://www.cs.umd.edu/fs/ftp/pub/papers/papers/3587.figures.ps. This is the second part of the paper in a separate file. (Also cross-referenced as UMIACS-TR-96-2)