Volume 12, pp. 205-215, 2001.

Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods

Bernd Fischer and Franz Peherstorfer

Abstract

Let ${\phi }_m$ be a polynomial satisfying some mild conditions. Given a set $R\subset \mathbf{C}$, a continuous function $f$ on $R$ and its best approximation $p_{n-1}^{\ast }$ from ${\mathrm{\Pi }}_{n-1}$ with respect to the maximum norm, we show that $p_{n-1}^{\ast }\circ {\phi }_m$ is a best approximation to $f\circ {\phi }_m$ on the inverse polynomial image $S$ of $R$, i.e. ${\phi }_m(S)=R$, where the extremal signature is given explicitly. A similar result is presented for constrained Chebyshev polynomial approximation. Finally, we apply the obtained results to the computation of the convergence rate of Krylov subspace methods when applied to a preconditioned linear system. We investigate pairs of preconditioners where the eigenvalues are contained in sets $S$ and $R$, respectively, which are related by ${\phi }_m(S)=R$.

Full Text (PDF) [397 KB], BibTeX

Key words

Chebyshev polynomial, optimal polynomial, extremal signature, Krylov subspace method, convergence rate.

AMS subject classifications

41A10, 30E10, 65F10.

ETNA articles which cite this article

Vol. 37 (2010), pp. 202-213 Valeria Simoncini: On a non-stagnation condition for GMRES and application to saddle point matrices

Vol. 41 (2014), pp. 159-166 Jörg Liesen and Petr Tichý: Max-min and min-max approximation problems for normal matrices revisited