# On GMRES for singular EP and GP systems

## TL;DR

This paper investigates the numerical behavior of GMRES for singular systems, identifying factors that cause accuracy deterioration and comparing it with RR-GMRES, supported by numerical experiments.

## Contribution

It reveals how inconsistency, initial residuals, and principal angles impact GMRES accuracy for singular systems and compares GMRES with RR-GMRES.

## Key findings

- GMRES accuracy deteriorates due to system inconsistency
- Initial residuals' proximity to nullspace affects solution quality
- Principal angles between ranges influence conditioning and accuracy

## Abstract

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.