On GMRES for singular EP and GP systems

0490058 - MÚ 2019 RIV US eng J - Journal Article
Morikuni, K. - Rozložník, Miroslav
On GMRES for singular EP and GP systems.
SIAM Journal on Matrix Analysis and Applications. Roč. 39, č. 2 (2018), s. 1033-1048. ISSN 0895-4798. E-ISSN 1095-7162
Institutional support: RVO:67985840
Keywords : GMRES method * singular linear systems * least squares problems * group inverse
OECD category: Pure mathematics
Impact factor: 1.912, year: 2018
https://epubs.siam.org/doi/10.1137/17M1128216

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, and (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 method. Numerical experiments show typical behaviors of GMRES for small problems with EP and GP matrices.
Permanent Link: http://hdl.handle.net/11104/0284355