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Modified incomplete factorization strategies

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Preconditioned Conjugate Gradient Methods

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1457))

Abstract

We review here the conditioning properties of modified incomplete symmetric factorizations of Stieltjes matrices, appropriate for the PCG solution of symmetric positive definite linear systems. Emphasizing the algorithmic interpretation of the required assumptions, we analyse the theoretical support that can be given to practical factorization algorithms and the choice left open between “perturbed” and “unperturbed” policies. Recent results extending the scope of unperturbed modified factorizations are included and discussed.

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References

  1. O. Axelsson, On iterative solution of elliptic difference equations on a mesh-connected array of processors, Intern. Journal of High Speed Computing, 1 (1989), pp. 165–184.

    Article  MATH  Google Scholar 

  2. O. Axelsson and V. Barker, Finite Element Solution of Boundary Value Problems. Theory and Computation, Academic, New York, 1984.

    MATH  Google Scholar 

  3. O. Axelsson and V. Eijkhout, Rebust vectorizable preconditioners for three-dimensional elliptic difference equations with anisotropy, in Algorithms and Applications on Vector and Parallel Computers, H. te Riele, T. J. Dekker, and H. van der Vorst, eds., Elsevier (North Holland), Amsterdam, 1987, pp. 279–306.

    Google Scholar 

  4. O. Axelsson and I. Gustafsson, Preconditioning and two-level multigrid methods of arbitrary degree of approximation, Math. Comp., 40 (1983), pp. 214–242.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Beauwens, Upper eigenvalue bounds for pencils of matrices, Linear Algebra Appl., 62 (1984), pp. 87–104.

    Article  MathSciNet  MATH  Google Scholar 

  6. —, On Axelsson's perturbations, Linear Algebra Appl., 68 (1985), pp. 221–242.

    Article  MathSciNet  MATH  Google Scholar 

  7. —, Consistent ordering analysis, Scientific Report, Université Libre de Bruxelles, Brussels, 1986.

    Google Scholar 

  8. —, Lower eigenvalue bounds for pencils of matrices, Linear Algebra Appl., 85 (1987), pp. 101–119.

    Article  MathSciNet  MATH  Google Scholar 

  9. —, Approximate factorizations with S/P consistently ordered M-factors, BIT, 29 (1989), pp. 658–681.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Beauwens and R. Wilmet, Conditioning analysis of positive definite matrices by approximate factorizations, J. Comput. Appl. Math., 26 (1989), pp. 257–269.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. George and J. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, 1981.

    MATH  Google Scholar 

  12. I. Gustafsson, On first order factorization methods for the solution of problems with mixed boundary conditions and problems with discontinuous material coefficients, Research Report 77.13R, Dept. of Computer Sciences, Chalmers Univ. of Technology and Univ. of Goeteborg, Goeteborg, Sweden, 1977.

    Google Scholar 

  13. —, Modified incomplete Cholesky (MIC) methods, in Preconditioning Methods. Theory and Applications, D. Evans, ed., Gordon and Breach, New York-London-Paris, 1983, pp. 265–293.

    Google Scholar 

  14. F. Harary, Graph Theory, Addison-Wesley, Reading, 1969.

    MATH  Google Scholar 

  15. Y. Notay, Incomplete factorization of singular linear systems, BIT, 29 (1989), pp. 682–702.

    Article  MathSciNet  MATH  Google Scholar 

  16. —, Conditioning of Stieltjes matrices by S/P consistently ordered approximate factorizations, BIT, (1990). submitted.

    Google Scholar 

  17. —, Solving positive (semi)definite linear systems by preconditioned iterative methods, these Proceedings, (1990).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Young, Iterative methods for solving partial differential equations of elliptic type, Trans. Amer. Math. Soc., 76 (1954), pp. 91–111.

    Article  Google Scholar 

  20. —, Iterative solution of large linear systems, Academic, New York, 1971.

    MATH  Google Scholar 

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Owe Axelsson Lily Yu. Kolotilina

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© 1990 Springer-Verlag

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Beauwens, R. (1990). Modified incomplete factorization strategies. In: Axelsson, O., Kolotilina, L.Y. (eds) Preconditioned Conjugate Gradient Methods. Lecture Notes in Mathematics, vol 1457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090898

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  • DOI: https://doi.org/10.1007/BFb0090898

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53515-7

  • Online ISBN: 978-3-540-46746-5

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