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SOR for \(AX-XB=C\). (English) Zbl 0736.65031

The authors consider a new approach to the block SOR method applied to linear systems of equations which can be written as a matrix equation \(AX-XB=C\). Such systems arise from finite differencing of separable elliptic boundary value problems on rectangular domains. The translation to the matrix equation enables us to handle an iterative method for the solution of such equations including Lyapunov’s matrix equation as a special case.
Further it gives us a more compact form of the problem of choosing optimal parameters for the block SOR method. This is particularly helpful for non-selfadjoint problems of elliptic type.
Applying the technique, the optimal parameters for the model problem of a convection-diffusion equation are determined under more general assumptions than those of R. C. Y. Chin and {T. A. Manteuffel} [SIAM J. Numer. Anal. 25, No. 3, 564-585 (1988; Zbl 0655.65060)]. Numerical data are given for the optimal parameters and the spectral radii of the optimized iteration matrices.

MSC:

65F30 Other matrix algorithms (MSC2010)
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
15A24 Matrix equations and identities
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Citations:

Zbl 0655.65060
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References:

[1] Barnett, S., Matrices in Control Theory (1971), Van Nostrand Reinhold: Van Nostrand Reinhold London · Zbl 0245.93002
[2] Barnett, S.; Storey, C., Some application of the Lyapunov matrix equation, J. Inst. Math. Appl., 4, 33-42 (1968) · Zbl 0155.12902
[3] Chin, R. C.Y.; Manteuffel, T. A., An analysis of block successive overrelaxation for a class of matrices with complex spectra, SIAM J. Numer. Anal., 25, 564-585 (1988) · Zbl 0655.65060
[4] Elman, H. C.; Golub, G. H., Iterative methods for cyclically reduced non-self-adjoint linear systems, Math. Comp., 54, 671-700 (1990) · Zbl 0699.65021
[5] Elman, H. C.; Golub, G. H., Iterative methods for cyclically reduced non-self-adjoint linear systems II, Math. Comp., 56, 215-242 (1991) · Zbl 0716.65096
[6] Gantmacher, F. R., Matrizentheorie (1986), Springer-Verlag: Springer-Verlag New York
[7] Golub, G. H.; Nash, S.; van Loan, C., A Hessenberg-Schur method for the problem \(AX + XB =C\), IEEE Trans. Automat. Control., AC-24, 909-913 (1979) · Zbl 0421.65022
[8] Kjellberg, G., On the convergence of successive overrelaxation applied to a class of linear systems of equations with complex eigenvalues, Ericsson Technics, 2, 245-258 (1958)
[9] Niethammer, W., Iterationsverfahren bei der konformen Abbildung, Computing, 2, 146-153 (1966) · Zbl 0161.12005
[10] Ostrowski, A.; Schneider, H., Some theorems on the inertia of general matrices, J. Math. Anal. Appl., 4, 72-84 (1962) · Zbl 0112.01401
[11] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes—The Art of Scientific Computing (1986), Cambridge U.P: Cambridge U.P New York · Zbl 0587.65003
[12] Smith, R. A., Matrix equation \(XA + BX =C\), SIAM J. Appl. Math., 16, 198-201 (1968) · Zbl 0157.22603
[14] Starke, G., Rationale Minimierungsprobleme in der komplexen Ebene im Zusammenhang mit der Bestimmung optimaler ADI-Parameter, (Ph.D. Thesis (1989), Univ. Karlsruhe) · Zbl 0696.65030
[15] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1983), Springer: Springer New York · Zbl 1004.65001
[16] Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall New York · Zbl 0133.08602
[17] Wachspress, E. L., Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1, 87-90 (1988) · Zbl 0631.65037
[18] Young, D. M., Iterative Solution of Large Linear Systems (1971), Academic: Academic New York · Zbl 0204.48102
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