  In this paper, we study the restarted Krylov subspace method, which is
typically represented by the GMRES(m) method. Our work mainly focused on the
amount of change in the iterative solution of GMRES(m) at each restart. We
propose an extension of the GMRES(m) method based on the idea of projection.
The algorithm is named as LGMRES. In addition, LLBGMRE method is also obtained
by adding backtracking restart technology to LGMRES. Theoretical analysis and
numerical experiments show that LGMRES and LLBGMRES have better convergence
than traditional restart GMRES(m) method.
