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.


  1. GMRES method
  2. singular linear systems
  3. least squares problems
  4. group inverse
  5. EP matrix
  6. GP matrix

MSC codes

  1. 65F10
  2. 15A12
  3. 15A09

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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1033 - 1048
ISSN (online): 1095-7162


Submitted: 8 May 2017
Accepted: 29 March 2018
Published online: 5 June 2018


  1. GMRES method
  2. singular linear systems
  3. least squares problems
  4. group inverse
  5. EP matrix
  6. GP matrix

MSC codes

  1. 65F10
  2. 15A12
  3. 15A09



Miroslav Rozložník

Funding Information

Czech Science Foundation : GA17-12925S
Japan Society for the Promotion of Science https://doi.org/10.13039/501100001691 : 16K17639

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