Image numérique, GMRES et polynômes de Faber

We first show that ‖Fn(A)‖⩽2, where A is a linear continuous operator acting in a Hilbert space, and Fn 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. To cite this article: B. Beckermann, C. R. Acad. Sci. Paris, Ser. I 340 (2005).