Soit le polynôme de Faber de degré n associé à l'image numérique d'un opérateur linéaire continu A sur un espace de Hilbert. Nous montrons dans un premier temps que . Nous en déduisons ensuite, en terme d'image numérique, de nouvelles estimations d'erreur pour la méthode GMRES, méthode itérative adaptée à la résolution des systèmes linéaires non-hermitiens.
We first show that , where A is a linear continuous operator acting in a Hilbert space, and is the Faber polynomial of degree n corresponding to the numerical range of A. Then we deduce several new error bounds based on the numerical range for GMRES, an iterative method for solving non-Hermitian systems of linear equations.
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@article{CRMATH_2005__340_11_855_0,
author = {Bernhard Beckermann},
title = {Image num\'erique, {GMRES} et polyn\^omes de {Faber}},
journal = {Comptes Rendus. Math\'ematique},
pages = {855--860},
publisher = {Elsevier},
volume = {340},
number = {11},
year = {2005},
doi = {10.1016/j.crma.2005.04.027},
language = {fr},
}
Bernhard Beckermann. Image numérique, GMRES et polynômes de Faber. Comptes Rendus. Mathématique, Volume 340 (2005) no. 11, pp. 855-860. doi : 10.1016/j.crma.2005.04.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.027/
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