On GMRES for singular EP and GP systems
For researchers solving singular linear systems, this work provides a theoretical and numerical analysis of GMRES accuracy limitations, though it is incremental as it extends known issues to specific matrix classes.
The paper identifies three factors causing accuracy deterioration in GMRES for singular EP and GP systems: inconsistency, initial residual distance to nullspace, and principal angles between range spaces. Numerical experiments demonstrate these effects on small problems.
In this contribution, we study the numerical behavior of the Generalized Minimal Residual (GMRES) method for solving singular linear systems. It is known that GMRES determines a least squares solution without breakdown if the coefficient matrix is range-symmetric (EP), or if its range and nullspace are disjoint (GP) and the system is consistent. We show that the accuracy of GMRES iterates may deteriorate in practice due to three distinct factors: (i) the inconsistency of the linear system; (ii) the distance of the initial residual to the nullspace of the coefficient matrix; (iii) the extremal principal angles between the ranges of the coefficient matrix and its transpose. These factors lead to poor conditioning of the extended Hessenberg matrix in the Arnoldi decomposition and affect the accuracy of the computed least squares solution. We also compare GMRES with the range restricted GMRES (RR-GMRES) method. Numerical experiments show typical behaviors of GMRES for small problems with EP and GP matrices.