From Potential Theory to Matrix Iterations in Six Steps
Tobin A. Driscoll, Kim-Chuan Toh, Lloyd N. Trefethen · SIAM Review · 1998
The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor $\rho \le 1$ can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in asystematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.