A stabilized GMRES method for singular and severely illconditioned systems of linear equations
Abstract
Consider using the rightpreconditioned GMRES (ABGMRES) for obtaining the minimumnorm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and illconditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very illconditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying ABGMRES to underdetermined systems, but also for applying GMRES to severely illconditioned rangesymmetric systems of linear equations.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.10853
 Bibcode:
 2020arXiv200710853L
 Keywords:

 Mathematics  Numerical Analysis;
 65F10;
 G.1.3
 EPrint:
 27 pages, 13 figures, 8 tables, Fig. 9 added, Modified comment after theorem 7, other minor changes