607 GMRES/CR AND ARNOLDI/LANCZOS AS MATRIX APPROXIMATION PROBLEMS A. Greenbaum, L. Trefethen, May 1992 The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize kp(A)bk over polynomials p of degree n. The difference is that p is normalized at z = 0 for GMRES and at z = inf for Arnoldi. Analogous "ideal GMRES" and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes kp(A)k instead. Investigation of these true and ideal approximation problems gives insight into how fast GMRES converges and how the Arnoldi iteration locates eigenvalues. 608 MATRICES THAT GENERATE THE SAME KRYLOV RESIDUAL SPACES