In this paper, we are presenting a proposal for new modified
algorithms for RRGMRES and AGMRES. It is known that RRGMRES and
AGMRES are viable methods for solving linear discrete ill-posed
problems. In this paper we have focused on the residual norm and
have come-up with two improvements where successive updates and the
stabilization of decreases for the residual norm improve performance
respectively. Our numerical experiments confirm that our improved
algorithms are effective for linear discrete ill-posed problems.