From Potential Theory to Matrix Iterations in Six Steps

Authors:
Tobin A. Driscoll;Kim-Chuan Toh;Lloyd N. Trefethen
Affiliations:
-;-;-
Venue:
SIAM Review
Year:
1998

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Abstract

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 a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.