On Ill-Conditioned Linear Systems and Regularized GMRES Method

In this paper we consider solving ill-conditioned square linear systems with noisy right hand sides. To have meaningful solutions of such systems, we use Tikhonov regularization with the GMRES method in three different ways. Our computational experiments show that solving the noisy system by regularizing the least square problem inside the GMRES algorithm gives us meaningful solutions and is much faster than regularizing the original least square model itself or its least square problem within the GMRES method.