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Preprint 19-2005

Worst-case and ideal GMRES for a Jordan block

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Author(s) : Petr Tichý , Jörg Liesen

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 19-2005

MSC 2000

65F10 Iterative methods for linear systems

Abstract :
We investigate the convergence of GMRES for an $n$ by $n$ Jordan block. For each $k$ that divides $n$ we derive the exact form of the $k$th ideal GMRES polynomial. We show that for a Jordan block, the worst-case and ideal GMRES approximations are the same in these steps. Moreover, we derive lower and upper bounds on the norm of the $k$th ideal GMRES matrix polynomial. For the Jordan block with eigenvalue one, we present an explicit formula for its singular value decomposition and use it to improve the bound on the ideal GMRES residual norm in the considered steps $k$.

Keywords : Krylov subspace methods, convergence behavior, ideal GMRES, worst-case GMRES, Jordan block, SVD

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