GMRES and the Minimal Polynomial
 ABSTRACT
 We present bounds on the convergence rate of the Generalized Minimal
Residual (GMRES)
method for solving nonsingular systems of linear equations Ax = b
in exact arithmetic.
The results are valid in finite and infinite dimensional spaces.
The bounds are derived by constructing residual polynomials
that are related to the minimal polynomial of A and can be
interpreted as follows:
If the eigenvalues of A consist of a single cluster plus outliers
then the convergence factor is bounded by the cluster radius,
while the asymptotic error constant is proportional
to the deviation of A from normality as well as the indices
of the outliers and their distance from the cluster.
If the eigenvalues of A consist of several close clusters, then
GMRES treats the clusters as a single big cluster, and the
convergence factor is the radius of this big cluster. Our bounds also lead to
a simpler proof of existing rsuperlinear convergence
results in Hilbert space. This research was partially supported by National Science Foundation grants
DMS9122745 & D9423705 (Campbell), CCR9102853 (Ipsen), DMS9321938 (Kelley), and
DMS9020915 & DMS9403224 (Meyer).
 JOURNAL
 BIT
 Vol. 36, 1996, pp. 3243
 COAUTHORS
 S. L. Campbell
 I. C. F. Ipsen
 C. T. Kelley

C. D. Meyer
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