A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

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