Matrix Iterations: The Six Gaps Between Potential Theory and Convergence

Abstract

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, ...) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor rho less than 1 can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in reducing the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.