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\(H\)-splittings and two-stage iterative methods. (English) Zbl 0764.65018

The convergence of stationary and non-stationary two-stage iterative methods is discussed. Alternative proofs of the convergence results obtained by P. J. Lanzkron, D. J. Rose, and the second author [ibid. 58, No. 7, 685-702 (1991; Zbl 0718.65022)] are given. After the introduction of \(H\)-splittings and \(H\)-compatible splittings of \(H\)- matrices similar convergence results for splittings of \(H\)-matrices are presented.
Reviewer: M.Jung (Chemnitz)

MSC:

65F10 Iterative numerical methods for linear systems

Citations:

Zbl 0718.65022
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References:

[1] Alefeld, G. (1982): On the convergence of the symmetric SOR method for matrices with red-black ordering. Numer. Math.39, 113-117 · Zbl 0485.65024
[2] Alefeld, G., Varga, R.S. (1976): Zur Konvergenz des symmetrischen Relationsverfahrens. Numer. Math.25, 291-295 · Zbl 0319.65030
[3] Axelsson, O. (1977): Solution of linear systems of equations: Iterative methods. In: V.A. Baker, ed., Sparse matrix techniques ? Copenhagen 1976. Lecture Notes in Mathematics 572. Springer, Berlin Heidelberg New York, pp. 1-15
[4] Beawens, R. (1979): Factorization iterative methods,M-operators andH-operators. Numer. Math.31, 335-357; Errata in Numer. Math.49, 457 (1986) · Zbl 0431.65012
[5] Berman, A., Plemmons, J. (1979): Nonnegative matrices in the mathematical sciences. Academic Press, New York · Zbl 0484.15016
[6] Collatz, L. (1952): Aufgaben monotoner Art. Arch. Math.3, 366-376 · Zbl 0048.09802
[7] Dembo, R.S., Eisenstat, S.C., Steihaug, T. (1982): Inexact Newton methods. SIAM J. Numer. Anal.19, 400-408 · Zbl 0478.65030
[8] Fan, K. (1958): Topological proofs of certain theorems on matrices with non-negative elements. Monatshefte für Mathematik62, 219-237 · Zbl 0081.25104
[9] Frommer, A., Mayer, G. (1989): Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl.119, 141-152 · Zbl 0676.65022
[10] Golub, G.H., Overton, M.L. (1982): Convergence of a two-stage Richardson iterative procedure for solving systems of linear equations. In: G.A. Watson, ed., Numerical analysis (Proceedings of the Ninth Biennial Conference, Dundee, Scotland, 1981). Lecture Notes in Mathematics 912, Springer, Berlin Heidelberg New York, pp. 128-139
[11] Golub, G.H., Overton, M.L. (1988): The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems. Numer. Math.53, 571-593 · Zbl 0661.65033
[12] Johnson, C.R., Bru, R. (1990): The spectral radius of a product of nonnegative matrices. Linear Algebra Appl.141, 227-240 · Zbl 0712.15013
[13] Krishna, L.B. (1983): Some results on unsymmetric successive overrelaxation method. Numer. Math.42, 155-160 · Zbl 0497.65015
[14] Lanzkron, P.J., Rose, D.J., Szyld, D.B. (1991): Convergence of nested classical iterative methods for linear systems. Numer. Math.58, 685-702 · Zbl 0718.65022
[15] Marek, I., Szyld, D.B. (1990): Comparison theorems for weak splittings of bounded operators. Numer. Math.58, 387-397 · Zbl 0694.65023
[16] Marek, I., Szyld, D.B. (1990): Splittings ofM-operators: Irreducibility and the index of the iteration operator. Numer. Funct. Anal. Optimization,11, 529-553 · Zbl 0714.65059
[17] Mayer, G. (1987): Comparison theorems for iterative methods based on strong splittings. SIAM J. Numer. Anal.24, 215-227 · Zbl 0614.65030
[18] Moré, J.J. (1971): Global convergence of Newton-Gauss-Seidel methods. SIAM J. Numer. Anal.8, 325-336 · Zbl 0218.65018
[19] Neumaier, A. (1984): New techniques for the analysis of linear interval equations. Linear Algebra Appl.58, 273-325 · Zbl 0558.65019
[20] Neumaier, A. (1986): On the comparison ofH-matrices withM-matrices. Linear Algebra Appl.83, 135-141 · Zbl 0605.15005
[21] Neumaier, A. (1990): Interval methods for systems of equations. Cambridge University Press, Cambridge New York · Zbl 0715.65030
[22] Neumaier, A., Varga, R.S. (1984): Exact convergence and divergence domains for the symmetric successive overrelaxation iterative (SSOR) method applied toH-matrices. Linear Algebra Appl.58, 261-272 · Zbl 0569.65021
[23] Neumann, M. (1984): On bounds for the convergence of the SSOR method forH-matrices. Linear Multilinear Algebra15, 13-21 · Zbl 0532.65022
[24] Nichols, N.K. (1973): On the convergence of two-stage iterative processes for solving linear equations. SIAM J. Numer. Anal.10, 460-469 · Zbl 0259.65040
[25] Ortega, J.M. (1972): Numerical analysis. A second course. Academic Press, New York (reprinted by SIAM, Philadelphia, 1990) · Zbl 0248.65001
[26] Ortega, J.M., Rheinboldt, W.C. (1970): Iterative solution of nonlinear equations in several variables. Academic Press, New York London · Zbl 0241.65046
[27] Ostrowski, A.M. (1937): Über die Determinanten mit überwiegender Hauptdiagonale. Comentarii Mathematici Helvetici10, 69-96 · JFM 63.0035.01
[28] Ostrowski, A.M. (1956): Determination mit überwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationprozessen. Comentarii Mathematici Helvetici30, 175-210 · Zbl 0072.13803
[29] Rheinboldt, W.C., Vandergraft, J.S. (1973): A simple approach to the Perron-Frobenius theory for positive operators on general partially-ordered finite-dimensional linear spaces. Math. Comput.27, 139-145 · Zbl 0255.15017
[30] Robert, F., Charnay, M., Musy, F. (1975): Itérations chaotiques série-parallèle pour des équations non-linéaires de point fixe. Aplikace Matematiky20, 1-38
[31] Robert, F., (1969): Blocs-H-matrices et convergence des méthodes itératives classiques par blocs. Linear Algebra Appl.2, 223-265 · Zbl 0182.21302
[32] Schneider, H. (1984): Theorems onM-splittings of a singularM-matrix which depend on graph structure. Linear Algebra Appl.58, 407-424 · Zbl 0561.65020
[33] Sherman, A.H. (1978): On Newton-iterative methods for the solution of systems of nonlinear equations. SIAM J. Numer. Anal.15, 755-771 · Zbl 0396.65019
[34] Szyld, D.B., Jones, M.T. (1992): Two-stage and multisplitting methods for the parallel solution of linear systems. SIAM J. Matrix Anal. Appl.13, 671-679 · Zbl 0754.65037
[35] Szyld, D.B., Widlung, O.B. (1992): Variational analysis of some conjugate gradient methods. East-West J. Numer. Anal.1, 1-25
[36] Varga, R.S. (1960): Factorization and normalized iterative methods. In: R.E. Langer, ed., Boundary problems in differential equations. The University of Wisconsin Press, Wisconsin, pp. 121-142
[37] Varga, R.S. (1962): Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, New Jersey · Zbl 0133.08602
[38] Varga, R.S. (1976): On recurring theorems on diagonal dominance. Linear Algebra Appl.13, 1-9 · Zbl 0336.15007
[39] Varga, R.S., Saff, E.B., Mehrmann, V. (1980): Incomplete factorizations of matrices and connections withH-matrices. SIAM J. Numer. Anal.17, 787-793 · Zbl 0477.65020
[40] Wachspress, E.L. (1966): Iterative Solution of Elliptic Systems. Prentice-Hall, Englewood Cliffs, New Jersey · Zbl 0161.12203
[41] Young, D.M. (1971): Iterative solution of large linear systems. Academic Press, New York · Zbl 0231.65034
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