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Improvements of Monotonicity Approach to Solve Interval Parametric Linear Systems

  • Iwona SkalnaEmail author
  • Marcin Pietroń
  • Milan Hladík
Conference paper
  • 113 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12044)

Abstract

Recently, we have proposed several improvements of the standard monotonicity approach to solving parametric interval linear systems. The obtained results turned out to be very promising; i.e., we have achieved narrower bounds, while generally preserving the computational time. In this paper we propose another improvements, which aim to further decrease the widths of the interval bounds.

Keywords

Parametric linear systems Monotonicity approach Revised affine forms Matrix equation 

Notes

Acknowledgments

M. Hladík was supported by the Czech Science Foundation Grant P403-18-04735S.

References

  1. 1.
    Comba, J., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: Proceedings of SIBGRAPI 1993 VI Simpósio Brasileiro de Computação Gráfica e Processamento de Imagens (Recife, BR), pp. 9–18 (1993)Google Scholar
  2. 2.
    Dehghani-Madiseh, M., Dehghan, M.: Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation \(A(p)X = B(p)\). Numer. Algorithms 73(1), 245–279 (2016).  https://doi.org/10.1007/s11075-015-0094-3MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hladík, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561–574 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hladík, M.: Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl. 30(2), 509–521 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Horáček, J., Hladík, M., Černý, M.: Interval linear algebra and computational complexity. In: Bebiano, N. (ed.) MAT-TRIAD 2015. SPMS, vol. 192, pp. 37–66. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-49984-0_3CrossRefzbMATHGoogle Scholar
  6. 6.
    Kolev, L.: Parameterized solution of linear interval parametric systems. Appl. Math. Comput. 246, 229–246 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Král, O., Hladík, M.: Parallel computing of linear systems with linearly dependent intervals in MATLAB. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds.) PPAM 2017. LNCS, vol. 10778, pp. 391–401. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78054-2_37CrossRefGoogle Scholar
  8. 8.
    Mayer, G.: An Oettli-Prager-like theorem for the symmetric solution set and for related solution sets. SIAM J. Matrix Anal. Appl. 33(3), 979–999 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Messine, F.: New affine forms in interval branch and bound algorithms. Technical report, R2I 99–02, Université de Pau et des Pays de l’Adour (UPPA), France (1999)Google Scholar
  10. 10.
    Neumaier, A., Pownuk, A.: Linear systems with large uncertainties, with applications to truss structures. Reliab. Comput. 13, 149–172 (2007).  https://doi.org/10.1007/s11155-006-9026-1MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Popova, E.: Computer-assisted proofs in solving linear parametric problems. In: 12th GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN 2006, Duisburg, Germany, p. 35 (2006)Google Scholar
  12. 12.
    Popova, E.D.: Enclosing the solution set of parametric interval matrix equation \(A(p)X = B(p)\). Numer. Algorithms 78(2), 423–447 (2018).  https://doi.org/10.1007/s11075-017-0382-1MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rohn, J.: A method for handling dependent data in interval linear systems. Technical report, 911, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2004). https://asepactivenode.lib.cas.cz/arl-cav/en/contapp/?repo=crepo1&key=20925094170
  14. 14.
    Skalna, I.: On checking the monotonicity of parametric interval solution of linear structural systems. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2007. LNCS, vol. 4967, pp. 1400–1409. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-68111-3_148CrossRefGoogle Scholar
  15. 15.
    Skalna, I.: Parametric Interval Algebraic Systems. Studies in Computational Intelligence. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-75187-0CrossRefzbMATHGoogle Scholar
  16. 16.
    Skalna, I., Hladík, M.: A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms. Numer. Algorithms 76(4), 1131–1152 (2017).  https://doi.org/10.1007/s11075-017-0300-6MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Skalna, I., Duda, J.: A study on vectorisation and paralellisation of the monotonicity approach. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds.) PPAM 2015. LNCS, vol. 9574, pp. 455–463. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-32152-3_42CrossRefGoogle Scholar
  18. 18.
    Skalna, I., Hladík, M.: A new method for computing a \(p\)-solution to parametric interval linear systems with affine-linear and nonlinear dependencies. BIT Numer. Math. 57(4), 1109–1136 (2017).  https://doi.org/10.1007/s10543-017-0679-4MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Skalna, I., Hladík, M.: Enhancing monotonicity checking in parametric interval linear systems. In: Martel, M., Damouche, N., Sandretto, J.A.D. (eds.) Trusted Numerical Computations, TNC 2018. Kalpa Publications in Computing, vol. 8, pp. 70–83. EasyChair (2018)Google Scholar
  20. 20.
    Vu, X.H., Sam-Haroud, D., Faltings, B.: A generic scheme for combining multiple inclusion representations in numerical constraint propagation. Technical report no. IC/2004/39, Swiss Federal Institute of Technology in Lausanne (EPFL), Lausanne, Switzerland, April 2004. http://liawww.epfl.ch/Publications/Archive/vuxuanha2004a.pdf

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.AGH University of Science and TechnologyKrakowPoland
  2. 2.Charles UniversityPragueCzech Republic

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