GMRES/CR AND ARNOLDI-LANCZOS AS MATRIX APPROXIMATION-PROBLEMS

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