chapter  3
26 Pages

Combinatorial Preconditioners

BySivan Toledo, Haim Avron

The Conjugate Gradient (CG) method is an iterative algorithm for solving linear systems of equations Ax = b, where A is symmetric and positive definite. The convergence of the method depends on the spectrum of A; when its eigenvalues are clustered, the method converges rapidly. In partic-

ular, CG converges to within a fixed tolerance in O( √ κ) iterations, where

κ = κ(A) = λmax(A)/λmin(A) is the spectral condition number of A. This upper bound is tight when eigenvalues are poorly distributed (i.e., they are not clustered). For example, if there are only two distinct eigenvalues, CG converges in two iterations.