GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems

The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize $||p(A)b||$ over polynomials p of degree n. The difference is that p is normalized at $z = 0$ for GMRES and at z = ∞ for Amoldi. Analogous “ideal GMRES” and “ideal Amoldi” problems are obtained if one removes b from the discussion and minimizes $||p(A)||$ instead. Investigation of these true and ideal approximation problems gives insight into how fast GMRES converges and how the Amoldi iteration locates eigenvalues..