Iterationsverfahren für nichtsymmetrische Gleichungssysteme und Approximationsmethoden im Komplexen. (Iterative methods for nonsymmetric systems of equations and approximation methods in the complex domain).(German)Zbl 0632.65031

This survey article describes the theory of semiiterative methods for the convergence acceleration of linear systems of equations. When the spectrum of the iteration matrix is in a known compact set $$\Omega$$ then the optimal semiiterative methods (i.e. those with minimal asymptotic convergence factor) are closely related to optimal polynomial approximations of 1/(1-z) on $$\Omega$$. In particular, when $${\bar {\mathbb{C}}}\setminus \Omega$$ is simply connected, optimal semiiterative methods can be constructed with the help of conformal mappings and Faber polynomials.
Reviewer: A.Neumaier

MSC:

 65F10 Iterative numerical methods for linear systems 65E05 General theory of numerical methods in complex analysis (potential theory, etc.)