Author(s) :
Jörg Liesen
,
Petr Tichý
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 28-2003
MSC 2000
- 41A10 Approximation by polynomials
-
41A44 Best constants
-
41A50 Best approximation, Chebyshev systems
-
65F10 Iterative methods for linear systems
Abstract :
The worst-case residual norms of the GMRES method
for linear algebraic systems can, in case of
a normal matrix, be characterized by a min-max approximation
problem on the matrix eigenvalues. In our paper
"The worst-case GMRES for normal matrices" we derive
a lower bound on this min-max value (worst-case residual norm) for each step of the GMRES
iteration. We conjecture that the lower bound and the min-max
value agree up to a factor of $4/\pi$, i.e. that the lower bound
multiplied by $4/\pi$ represents an upper bound. In this paper we
prove for several different iteration steps that our conjecture
is true for a special set of eigenvalues, namely the roots of unity.
This case is of interest, since numerical experiments indicate that
the ratio of the min-max value and our lower bound is
maximal when the eigenvalues are the roots of unity.
Keywords :
min-max problem, polynomial approximation on discrete set, best approximation, best constants, GMRES, evaluation of convergence