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Optimal Preconditioning for the Interval Parametric Gauss–Seidel Method

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Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9553))

Abstract

We deal with an interval parametric system of linear equations, and focus on the problem how to find an optimal preconditioning matrix for the interval parametric Gauss–Seidel method. The optimality criteria considered are to minimize the width of the resulting enclosure, to minimize its upper end-point or to maximize its lower end-point. We show that such optimal preconditioners can be computed by solving suitable linear programming problems. We also show by examples that, in some cases, such optimal preconditioners are able to significantly decrease an overestimation of the results of common methods.

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References

  1. Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer, New York (2006)

    MATH  Google Scholar 

  2. Gau, C.-Y., Stadtherr, M.A.: New interval methodologies for reliable chemical process modeling. Comput. Chem. Eng. 26(6), 827–840 (2002)

    Article  Google Scholar 

  3. Hladík, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561–574 (2012)

    Article  MathSciNet  Google Scholar 

  4. Hladík, M.: New operator and method for solving real preconditioned interval linear equations. SIAM J. Numer. Anal. 52(1), 194–206 (2014)

    Article  MathSciNet  Google Scholar 

  5. Kearfott, R.B.: Preconditioners for the interval Gauss-Seidel method. SIAM J. Numer. Anal. 27(3), 804–822 (1990)

    Article  MathSciNet  Google Scholar 

  6. Kearfott, R.B.: Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems. Comput. 47(2), 169–191 (1991)

    Article  MathSciNet  Google Scholar 

  7. Kearfott, R.B.: A comparison of some methods for bounding connected and disconnected solution sets of interval linear systems. Comput. 82(1), 77–102 (2008)

    Article  MathSciNet  Google Scholar 

  8. Kearfott, R.B., Hu, C., Novoa, M.: A review of preconditioners for the interval Gauss-Seidel method. Interval Comput. 1991(1), 59–85 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Lin, Y., Stadtherr, M.A.: Advances in interval methods for deterministic global optimization in chemical engineering. J. Glob. Optim. 29(3), 281–296 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  12. Popova, E.: Quality of the solution sets of parameter-dependent interval linear systems. ZAMM, Z. Angew. Math. Mech. 82(10), 723–727 (2002)

    Article  MathSciNet  Google Scholar 

  13. Popova, E.D.: On the solution of parametrised linear systems. In: Krämer, W., von Gudenberg, J.W. (eds.) Scientific Computing, Validated Numerics, Interval Methods, pp. 127–138. Kluwer (2001)

    Google Scholar 

  14. Popova, E.D., Hladík, M.: Outer enclosures to the parametric AE solution set. Soft Comput. 17(8), 1403–1414 (2013)

    Article  Google Scholar 

  15. Popova, E.D., Krämer, W.: Visualizing parametric solution sets. BIT 48(1), 95–115 (2008)

    Article  MathSciNet  Google Scholar 

  16. Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999)

    Chapter  Google Scholar 

  17. Rump, S.M.: Verification methods: Rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author was supported by the Czech Science Foundation Grant P402-13-10660S.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Milan Hladík

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  1. Milan Hladík
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Correspondence to Milan Hladík .

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Editors and Affiliations

  1. Institute of Computer Science, University of Würzburg, Würzburg, Germany

    Marco Nehmeier

  2. Universität Würzburg, Würzburg, Germany

    Jürgen Wolff von Gudenberg

  3. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Warwick Tucker

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Hladík, M. (2016). Optimal Preconditioning for the Interval Parametric Gauss–Seidel Method. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds) Scientific Computing, Computer Arithmetic, and Validated Numerics. SCAN 2015. Lecture Notes in Computer Science(), vol 9553. Springer, Cham. https://doi.org/10.1007/978-3-319-31769-4_10

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  • DOI: https://doi.org/10.1007/978-3-319-31769-4_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-31768-7

  • Online ISBN: 978-3-319-31769-4

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