Optimal Parameters for Linear Second-Degree Stationary Iterative Methods

In this paper we show that the optimal parameters for linear second-degree stationary iterative methods applied to nonsymmetric linear systems can be found by solving the same minimax problem used to find optimal parameters for the Chebyshev iteration. In fact, the Chebyshev iteration is asymptotica...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1982-08, Vol.19 (4), p.833-839
Main Author: Manteuffel, T. A.
Format: Article
Language:English
Subjects:
Circles
Eigenvalues
Ellipses
Iterative methods
Linear systems
Methods
Minimax problems
Polynomials
Spectral theory
Steepest descent method
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Summary:In this paper we show that the optimal parameters for linear second-degree stationary iterative methods applied to nonsymmetric linear systems can be found by solving the same minimax problem used to find optimal parameters for the Chebyshev iteration. In fact, the Chebyshev iteration is asymptotically equivalent to a linear second-degree stationary method. The method of finding optimal parameters for the Chebyshev iteration given in Manteuffel [Numer. Math., 28 (1977), pp. 307-327] can be used to find optimal parameters for the stationary method as well.
ISSN:0036-1429
1095-7170
DOI:10.1137/0719058