Abstract 
Consider using the rightpreconditioned generalized minimal residual (ABGMRES) method, which is an efficient method for solving underdetermined least squares problems. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and illconditioned problems, the iterates of the ABGMRES method 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. Thus, the process becomes numerically stable and the system becomes consistent, rendering better convergence and a more accurate solution. Numerical experiments show that the proposed method is robust and efficient for solving inconsistent and illconditioned underdetermined least squares problems. The method can be considered as a way of making the GMRES stable for highly illconditioned inconsistent problems.
