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Zbl 0632.65031
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)
[J] Jahresber. Dtsch. Math.-Ver. 89, 1-32 (1987). ISSN 0012-0456; ISSN 1869-7135/e

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 ${\bar {\bbfC}}\setminus \Omega$ is simply connected, optimal semiiterative methods can be constructed with the help of conformal mappings and Faber polynomials.
[A.Neumaier]

MSC 2000:: *65F10 Iterative methods for linear systems
65E05 Numerical methods in complex analysis

Keywords: nonsymmetric matrix; semiiterative methods; convergence acceleration; minimal asymptotic convergence factor; conformal mappings; Faber polynomials

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