Properties of Worst-Case GMRES

0421797 - ÚI 2014 RIV US eng J - Journal Article
Faber, V. - Liesen, J. - Tichý, Petr
Properties of Worst-Case GMRES.
SIAM Journal on Matrix Analysis and Applications. Roč. 34, č. 4 (2013), s. 1500-1519. ISSN 0895-4798. E-ISSN 1095-7162
R&D Projects: GA ČR GA13-06684S
Grant - others:GA AV ČR(CZ) M10041090
Institutional support: RVO:67985807
Keywords : GMRES method * worst-case convergence * ideal GMRES * matrix approximation problems * minmax
Subject RIV: BA - General Mathematics
Impact factor: 1.806, year: 2013

In the convergence analysis of the GMRES method for a given matrix A, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step k, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for A and k. We show that the worst case behavior of GMRES for the matrices A and A transposed is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we prove that such vectors satisfy a certain "cross equality". We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is strict at all iteration steps k greater than 3.
Permanent Link: http://hdl.handle.net/11104/0228050