Eiermann, M.; Varga, R. S.; Niethammer, W. 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 Jahresber. Dtsch. Math.-Ver. 89, 1-32 (1987). 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 Cited in 6 Documents MSC: 65F10 Iterative numerical methods for linear systems 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) Keywords:nonsymmetric matrix; semiiterative methods; convergence acceleration; minimal asymptotic convergence factor; conformal mappings; Faber polynomials × Cite Format Result Cite Review PDF