Abstract
An extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues, is investigated. Sufficient (and necessary) conditions are given such that convergence is assured. An analytic determination of good (and sometimes optimum) values of the involved real parameter is presented in terms of certain bounds on the eigenvalues of the iteration matrix. The usefulness of the developed theory is shown through a simple application to the conventional Jacobi method.
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References
J. de Pillis and M. Neumann,Iterative methods with k-part splittings, IMA J. Numer. Anal. 1, (1981), 65–79.
D. J. Evans and N. M. Missirlis.The preconditioned simultaneous displacement method (PSD method), MACS 22, (1980), 256–263.
E. Isaacson and H. B. Keller,Analysis of Numerical Methods, John Wiley and Sons, New York (1966).
A. Hadjidimos,The optimal solution of the extrapolation problem of a first order scheme, Intern. J. Comput. Math. 13, (1983), 153–168.
G. Kjellberg,On the convergence of successive over-relaxation applied to a class of linear systems of equations with complex eigenvalues, Ericsson Technics, Stockholm 2, (1958), 245–258.
B. Kredell,On complex successive overrelaxation, BIT 2, (1962), 143–152.
T. A. Manteuffel,The Chebyshev iteration for non-symmetric linear systems, Numer. Math. 28 (1977), 307–327.
N. M. Missirlis and D. J. Evans,On the convergence of some generalised preconditioned iterative methods, SIAM J. Numer. Anal. 18 (1981), 581–596.
N. M. Missirlis and D. J. Evans,The extrapolated successive overrelaxation (ESOR) method for consistently ordered matrices, Intern. J. Math. and Math. Sciences (to appear).
W. Niethammer,On different splittings and the associated iteration methods, SIAM J. Numer. Anal. 16, (1979), 186–200.
W. Niethammer,Konvergenzbeschleunigung bei einstufigen Iterationsverfahren durch Summierungs-methoden, Iterationsverfahren, Numer. Math., Approximations-theorie, Birkhauser, (1970), 235–243.
W. Niethammer and R. S. Varga,The analysis of k-step iterative methods for linear systems from summability theory, Numer. Math. 41, (1983), 177–206.
R. S. Varga,Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, New Jersey, (1962).
D. M. Young and H. E. Eidson,On the determination of the optimum relaxation factor for the SOR method when the eigenvalues of the Jacobi method are complex, Report CNA-1, Center for Numerical Analysis, Univ. Texas, Austin, Texas, (1970).
D. M. Young,Iterative Solution of Large Linear Systems, Academic Press, New York & London (1971).
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Missirlis, N.M. The extrapolated first order method for solving systems with complex eigenvalues. BIT 24, 357–365 (1984). https://doi.org/10.1007/BF02136034
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DOI: https://doi.org/10.1007/BF02136034