Skip to main content
SpringerLink
Log in

Improvements of Monotonicity Approach to Solve Interval Parametric Linear Systems

  • Conference paper
  • First Online:
Parallel Processing and Applied Mathematics (PPAM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12044))

  • 588 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    All the methods used in the computation are based on revised affine forms.

  2. 2.

    We first substitute interval parameters by respective revised affine forms and then we performs computation of them.

  3. 3.

    For details on Interval-affine Gauss-Seidel iteration see [18].

  4. 4.

    Some attempts to use parallelization for increasing the efficiency of a few methods for solving interval parametric linear systems are described in [7].

References

  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. 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/s11075-015-0094-3

    Article  MathSciNet  MATH  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.: Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl. 30(2), 509–521 (2008)

    Article  MathSciNet  Google Scholar 

  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_3

    Chapter  MATH  Google Scholar 

  6. Kolev, L.: Parameterized solution of linear interval parametric systems. Appl. Math. Comput. 246, 229–246 (2014)

    MathSciNet  MATH  Google Scholar 

  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_37

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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/s11155-006-9026-1

    Article  MathSciNet  MATH  Google Scholar 

  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. 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/s11075-017-0382-1

    Article  MathSciNet  MATH  Google Scholar 

  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. 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_148

    Chapter  Google Scholar 

  15. Skalna, I.: Parametric Interval Algebraic Systems. Studies in Computational Intelligence. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-75187-0

    Book  MATH  Google Scholar 

  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/s11075-017-0300-6

    Article  MathSciNet  MATH  Google Scholar 

  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_42

    Chapter  Google Scholar 

  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/s10543-017-0679-4

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. AGH University of Science and Technology, Krakow, Poland

    Iwona Skalna & Marcin Pietroń

  2. Charles University, Prague, Czech Republic

    Milan Hladík

Authors
  1. Iwona Skalna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcin Pietroń
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milan Hladík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iwona Skalna .

Editor information

Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Southern California, Marina del Rey, CA, USA

    Ewa Deelman

  3. University of Tennessee, Knoxville, TN, USA

    Jack Dongarra

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Skalna, I., Pietroń, M., Hladík, M. (2020). Improvements of Monotonicity Approach to Solve Interval Parametric Linear Systems. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019. Lecture Notes in Computer Science(), vol 12044. Springer, Cham. https://doi.org/10.1007/978-3-030-43222-5_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-43222-5_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43221-8

  • Online ISBN: 978-3-030-43222-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation