Skip to main content
SpringerLink
Log in

Parallel Computing of Linear Systems with Linearly Dependent Intervals in MATLAB

  • Conference paper
  • First Online:
Book cover Parallel Processing and Applied Mathematics (PPAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10778))

Abstract

We implemented several known algorithms for finding an interval enclosure of the solution set of a linear system with linearly dependent interval parameters. To do that we have chosen MATLAB environment with use of INTLAB and VERSOFT libraries. Because our implementation is tested on Toeplitz and symmetric matrices, among others, there is a problem with a sparsity. We introduce straightforward format for representing such matrices, which seems to be almost as effective as the standard matrix representation but with less memory demands. Moreover, we take an advantage of Parallel Computing Toolbox to enhance the performance of implemented methods and to get more insights on how the methods stands in a scope of a tightness-performance ratio. The contribution is a time-tightness performance comparison of such methods, memory efficient representation and an exploration of explicit parallelization impact.

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

References

  1. Bauer, F.L.: Genauigkeitsfragen bei der Lösung linearer Gleichungssysteme. ZAMM 46(7), 409–421 (1966)

    Article  MATH  Google Scholar 

  2. Hladík, M.: Optimal preconditioning for the interval parametric Gauss–Seidel method. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds.) SCAN 2015. LNCS, vol. 9553, pp. 116–125. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31769-4_10

    Chapter  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  MATH  Google Scholar 

  4. Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis. Springer, London (2001). https://doi.org/10.1007/978-1-4471-0249-6

    Book  MATH  Google Scholar 

  5. Kolev, L.V.: Improvement of a direct method for outer solution of linear parametric systems. Reliable Comput. 12(3), 193–202 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mayer, G.: Three short descriptions of the symmetric and of the skew-symmetric solution set. Linear Algebra Appl. 475, 73–79 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  8. Popova, E.D.: Computer-assisted proofs in solving linear parametric problems. In: 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN 2006, Conference Post-Proceedings, Duisburg, Germany, 26–29 September, p. 35. IEEE Computer Society Press (2006)

    Google Scholar 

  9. Popova, E.D.: Solvability of parametric interval linear systems of equations and inequalities. SIAM J. Matrix Anal. Appl. 36(2), 615–633 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rohn, J.: Cheap and tight bounds: the recent result by E. Hansen can be made more efficient. Interval Comput. 4(13–21), 2 (1993)

    MATH  Google Scholar 

  11. Rohn, J.: VERSOFT: Verification software in MATLAB/INTLAB, version 10 (2009). http://www.nsc.ru/interval/Programing/versoft/

  12. Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999). http://www.ti3.tu-harburg.de/rump/

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Skalna, I.: A method for outer interval solution of systems of linear equations depending linearly on interval parameters. Reliable Comput. 12(2), 107–120 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Skalna, I.: Enclosure for the solution set of parametric linear systems with non-affine dependencies. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds.) PPAM 2011. LNCS, vol. 7204, pp. 513–522. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31500-8_53

    Chapter  Google Scholar 

  16. Skeel, R.D.: Scaling for numerical stability in Gaussian elimination. J. ACM 26(3), 494–526 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors were supported by the Czech Science Foundation Grant P402-13-10660S.

Author information

Authors and Affiliations

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

    Ondřej Král & Milan Hladík

Authors
  1. Ondřej Král
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milan Hladík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ondřej Král .

Editor information

Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Knoxville, Tennessee, USA

    Jack Dongarra

  3. University of Southern California, Marina Del Rey, California, USA

    Ewa Deelman

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Král, O., Hladík, M. (2018). Parallel Computing of Linear Systems with Linearly Dependent Intervals in MATLAB. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science(), vol 10778. Springer, Cham. https://doi.org/10.1007/978-3-319-78054-2_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-78054-2_37

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78053-5

  • Online ISBN: 978-3-319-78054-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation