Summary
The Richardson iteration method is conceptually simple, as well as easy to program and parallelize. This makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues. We change the ordering of the relaxation parameters of a Richardson iteration method proposed by Eiermann, Niethammer and Varga for the solution of such problems. The new method obtained is shown to be stable and to have better convergence properties.
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References
[AG] Anderssen, R.S., Golub, G.H.: Richardson's non-stationary matrix iterative procedure. Rep. STAN-CS-72-304, Computer Science Dept., Stanford University 1972
[Br] Brigham, E.O.: The fast Fourier transform. New York: Prentice-Hall 1974
[Bu] Buneman, O.: Conversion of FFT's to fast Hartley transforms. SIAM J. Sci. Stat. Comput.7, 624–638 (1986)
[C] Curtiss, J.H.: Riemann sums and the fundamental polynomials of Lagrange interpolation. Duke Math. J.8, 525–532 (1941)
[El] Ellacott, S.W.: Computation of Faber series with application to numerical polynomial approximations in the complex plane. Math. Comput.40, 575–587 (1983)
[EN] Eiermann, M., Niethammer, W.: On the construction of semiiterative methods. SIAM J. Numer. Anal.20, 1153–1160 (1983)
[ENV] Eiermann, M., Niethammer, W., Varga, R.S.: A study of semiiterative methods for nonsymmetric systems of linear equations. Numer. Math.47, 505–533 (1985)
[ES] Elman, H.C., Streit, R.L.: Polynomial iteration for nonsymmetric indefinite linear systems. In: Hennart, J.P. (ed.) Numerical analysis, proceedings. Lect. Notes Math. No. 1230, Berlin Heidelberg New York: Springer 1986
[EVN] Eiermann, M., Varga, R.S., Niethammer, W.: Iterationsverfahren für nichtsymmetrische Gleichungssysteme und Approximationsmethoden im Komplexen. Jahresber. Dtsch. Math.-Ver.89, 1–32 (1987)
[FGHLW] Fischer, D., Golub, G., Hald, O., Leiva, C., Widlund, O.: On Fourier-Toeplitz methods for separable elliptic problems. Math. Comput.28, 349–368 (1974)
[Ga] Gaier, D.: Vorlesungen über Approximation im Komplexen. Basel: Birkhäuser 1980
[Gu] Gutknecht, M.H.: Numerical conformal mapping methods based on fuction conjugation. J. Comput. Appl. Math.14, 31–77 (1986)
[H] Henrici, P.: Applied and computational complex analysis, vol. 3. New York: Wiley 1986
[LF] Lebedev, V.I., Finogenov, S.A.: Utilization of ordered Chebyshev parameters in iterative methods. U.S.S.R. Comput. Math. Math. Phys.16, 70–83 (1976)
[Man] Manteuffel, T.A.: The Tschebychev iteration for nonsymmetric linear systems. Numer. Math.28, 307–327 (1977)
[Mar] Marchuk, G.I.: Methods of numerical mathematics, 2nd Ed. New York: Springer 1982
[OS] Opfer, G., Schober, G.: Richardson's iteration for nonsymmetric matrices. Linear Algebra Appl.58, 343–361 (1984)
[R] Reichel, L.: A fast method for solving certain integral equations of the first kind with application to conformal mapping. J. Comput. Appl. Math.14, 125–142 (1986)
[Sa] Samarskij, A.A.: Theorie der Differenzenverfahren. Leipzig: Akademische Verlagsgesell-schaft Geest & Portig 1984
[Sm] Smith, G.D.: Numerical solution of partial differential equations: Finite difference methods: Oxford University Press, New York 1978
[Sw] Swarztrauber, P.N.: A direct method for the discrete solution of separable elliptic equations. SIAM J. Numer. Anal.11, 1136–1150 (1974)
[T1] Trefethen, L.N. (ed.): Numerical conformal mapping. Special Issue of J. Comput. Appl. Math.14 (1986)
[T2] Trefethen, L.N.: SCEXT, program for computing the Schwarz-Christoffel mapping from the upper half plane to the exterior of a polygon. Dept. of Mathematics. M.I.T. 1987
[W] Walsh, J.L.: Interpolation and approximation by rational functions in the complex domain, 3rd Ed. Amer. Math. Soc., Providence, RI 1960
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Research supported by the National Science Foundation under Grant DMS-8704196
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Fischer, B., Reichel, L. A stable Richardson iteration method for complex linear systems. Numer. Math. 54, 225–242 (1989). https://doi.org/10.1007/BF01396976
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DOI: https://doi.org/10.1007/BF01396976