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H-Splittings and two-stage iterative methods

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Summary

Convergence of two-stage iterative methods for the solution of linear systems is studied. Convergence of the non-stationary method is shown if the number of inner iterations becomes sufficiently large. TheR 1-factor of the two-stage method is related to the spectral radius of the iteration matrix of the outer splitting. Convergence is further studied for splittings ofH-matrices. These matrices are not necessarily monotone. Conditions on the splittings are given so that the two-stage method is convergent for any number of inner iterations.

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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

    Google Scholar 

  2. Alefeld, G., Varga, R.S. (1976): Zur Konvergenz des symmetrischen Relationsverfahrens. Numer. Math.25, 291–295

    Google Scholar 

  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

    Google Scholar 

  4. Beawens, R. (1979): Factorization iterative methods,M-operators andH-operators. Numer. Math.31, 335–357; Errata in Numer. Math.49, 457 (1986)

    Google Scholar 

  5. Berman, A., Plemmons, J. (1979): Nonnegative matrices in the mathematical sciences. Academic Press, New York

    Google Scholar 

  6. Collatz, L. (1952): Aufgaben monotoner Art. Arch. Math.3, 366–376

    Google Scholar 

  7. Dembo, R.S., Eisenstat, S.C., Steihaug, T. (1982): Inexact Newton methods. SIAM J. Numer. Anal.19, 400–408

    Google Scholar 

  8. Fan, K. (1958): Topological proofs of certain theorems on matrices with non-negative elements. Monatshefte für Mathematik62, 219–237

    Google Scholar 

  9. Frommer, A., Mayer, G. (1989): Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl.119, 141–152

    Google Scholar 

  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

    Google Scholar 

  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

    Google Scholar 

  12. Johnson, C.R., Bru, R. (1990): The spectral radius of a product of nonnegative matrices. Linear Algebra Appl.141, 227–240

    Google Scholar 

  13. Krishna, L.B. (1983): Some results on unsymmetric successive overrelaxation method. Numer. Math.42, 155–160

    Google Scholar 

  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

    Google Scholar 

  15. Marek, I., Szyld, D.B. (1990): Comparison theorems for weak splittings of bounded operators. Numer. Math.58, 387–397

    Google Scholar 

  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

    Google Scholar 

  17. Mayer, G. (1987): Comparison theorems for iterative methods based on strong splittings. SIAM J. Numer. Anal.24, 215–227

    Google Scholar 

  18. Moré, J.J. (1971): Global convergence of Newton-Gauss-Seidel methods. SIAM J. Numer. Anal.8, 325–336

    Google Scholar 

  19. Neumaier, A. (1984): New techniques for the analysis of linear interval equations. Linear Algebra Appl.58, 273–325

    Google Scholar 

  20. Neumaier, A. (1986): On the comparison ofH-matrices withM-matrices. Linear Algebra Appl.83, 135–141

    Google Scholar 

  21. Neumaier, A. (1990): Interval methods for systems of equations. Cambridge University Press, Cambridge New York

    Google Scholar 

  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

    Google Scholar 

  23. Neumann, M. (1984): On bounds for the convergence of the SSOR method forH-matrices. Linear Multilinear Algebra15, 13–21

    Google Scholar 

  24. Nichols, N.K. (1973): On the convergence of two-stage iterative processes for solving linear equations. SIAM J. Numer. Anal.10, 460–469

    Google Scholar 

  25. Ortega, J.M. (1972): Numerical analysis. A second course. Academic Press, New York (reprinted by SIAM, Philadelphia, 1990)

    Google Scholar 

  26. Ortega, J.M., Rheinboldt, W.C. (1970): Iterative solution of nonlinear equations in several variables. Academic Press, New York London

    Google Scholar 

  27. Ostrowski, A.M. (1937): Über die Determinanten mit überwiegender Hauptdiagonale. Comentarii Mathematici Helvetici10, 69–96

    Google Scholar 

  28. Ostrowski, A.M. (1956): Determination mit überwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationprozessen. Comentarii Mathematici Helvetici30, 175–210

    Google Scholar 

  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

    Google Scholar 

  30. Robert, F., Charnay, M., Musy, F. (1975): Itérations chaotiques série-parallèle pour des équations non-linéaires de point fixe. Aplikace Matematiky20, 1–38

    Google Scholar 

  31. Robert, F., (1969): Blocs-H-matrices et convergence des méthodes itératives classiques par blocs. Linear Algebra Appl.2, 223–265

    Google Scholar 

  32. Schneider, H. (1984): Theorems onM-splittings of a singularM-matrix which depend on graph structure. Linear Algebra Appl.58, 407–424

    Google Scholar 

  33. Sherman, A.H. (1978): On Newton-iterative methods for the solution of systems of nonlinear equations. SIAM J. Numer. Anal.15, 755–771

    Google Scholar 

  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

    Google Scholar 

  35. Szyld, D.B., Widlung, O.B. (1992): Variational analysis of some conjugate gradient methods. East-West J. Numer. Anal.1, 1–25

    Google Scholar 

  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

    Google Scholar 

  37. Varga, R.S. (1962): Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, New Jersey

    Google Scholar 

  38. Varga, R.S. (1976): On recurring theorems on diagonal dominance. Linear Algebra Appl.13, 1–9

    Google Scholar 

  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

    Google Scholar 

  40. Wachspress, E.L. (1966): Iterative Solution of Elliptic Systems. Prentice-Hall, Englewood Cliffs, New Jersey

    Google Scholar 

  41. Young, D.M. (1971): Iterative solution of large linear systems. Academic Press, New York

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Bergische Universität GH Wuppertal, Gauss-Strasse 20, W-5600, Wuppertal, Germany

    Andreas Frommer

  2. Department of Mathematics, Temple University, 19122-2585, Philadelphia, PA, USA

    Daniel B. Szyld

Authors
  1. Andreas Frommer
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  2. Daniel B. Szyld
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Additional information

This work was supported in part by a Temple University Summer Research Fellowship.

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Frommer, A., Szyld, D.B. H-Splittings and two-stage iterative methods. Numer. Math. 63, 345–356 (1992). https://doi.org/10.1007/BF01385865

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  • DOI: https://doi.org/10.1007/BF01385865

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