Methods equivalent to FOM or GMRES

In this chapter we review iterative methods which are equivalent to FOM or GMRES in the sense that, mathematically, they must deliver the same residual norms. However, as we will see, this is not always the case in finite precision arithmetic. The algorithms mathematically equivalent to GMRES either construct residual vectors \(r_k\) orthogonal to \(A\mathcal{K}_k(A,r_0)\) or explicitly minimize the residual norms.