TABLE OF CONTENTS In the theory of iterative methods and preconditioners, we often use different measures of the eigenspectrum or the singular value spectrum to indicate levels of expected fast convergence. The key measure is clustering.
By cluster or clustering we mean that there are a large number of eigen-values (or singular values) that are inside a small interval [384] or close to a fixed point [28]. If we define, for any ? 1 ? ? 2, a complex row vector set by
then a more precise statement can be made as follows.
Definition 1.5.10. Given a square matrix A n n , if 
for some relatively small n 1 (with respect to n), we say ?( A) is clustered at point ? with a cluster size ? 1 and cluster radius ? 2 .
Here ? 2 is the radius of a disc, centering at ?, containing all the eigenvalues and ? 1 is the radius of a smaller disc that contains most of the eigenvalues (i.e., all eigenvalues except the first n 1 ?1).
Remark 1.5.11. As far as convergence of conjugate gradients methods is concerned, point clusterings imply that at step n 1 the underlying approximation in span ( q 1, , q n 1) (in Section 3.6 and Section 3.5) is almost as accurate as in span ( q 1, , q n). In...
