Skip to main content
SpringerLink
Log in

A Comparison of Methods for Solving Parametric Interval Linear Systems with General Dependencies

  • Conference paper
Numerical Methods and Applications (NMA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6046))

Included in the following conference series:

Abstract

This study compares two methods for solving interval linear systems whose coefficients are functions of interval parameters: the generalized Rump’s fixed-point iteration and Skalna’s Direct Method. Both methods have the same scope of application and require estimating the range of the same functions over a box. Evaluation of functional ranges using the simplest form of interval analysis produces wide intervals. This is due in a large part to the so-called interval dependency. To cope with the dependence problem, revised affine arithmetic with a new affine approximation of a product is used. Numerical examples are provided to show the advantages of Skalna’s Direct Method over generalized Rump’s fixed point iteration.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akhmerov, R.R.: Interval-affine Gaussian algorithm for constrained systems. Reliable Computing 11(5), 323–341 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. El-Owny, H.: Parametric Linear System of Equations, whose Elements are Nonlinear Functions. In: 12th GAMM - IMACS International Symposion on Scientific Computing, Computer Arithmetic and Validated Numerics, vol. 16 (2006)

    Google Scholar 

  3. Garloff, J., Popova, E.D., Smith, A.P.: Solving Linear Systems with Polynomial Parameter Dependency in the Reliable Analysis of Structural Frames. To appear in Proceedings of the 2nd International Conference on Uncertainty in Structural Dynamics, Sheffield, UK, June 15-17 (2009)

    Google Scholar 

  4. 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), Switzerland (2004)

    Google Scholar 

  5. Kolev, L.V.: Solving Linear Systems whose Elements are Non-linear Functions of Intervals. Numerical Algorithms 37, 213–224 (2004)

    Article  MathSciNet  Google Scholar 

  6. Kolev, L.V.: A new method for global solution of systems of non-linear equations. Reliable Computing 4, 125–146 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kolev, L.V.: Automatic computation of a linear interval enclosure. Reliable Computing 7, 17–18 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kulpa, Z., Pownuk, A., Skalna, I.: Analysis of linear mechanical structures with uncertainties by means of interval methods. Computer Assisted Mechanics and Engineering Sciences 5(4), 443–477 (1998), http://andrzej.pownuk.com/publications/IntervalEquations.pdf

    MathSciNet  MATH  Google Scholar 

  9. Messine, F.: Extentions of Affine Arithmetic: Application to Unconstrained Global Optimization. Journal of Universal Computer Science 8(11), 992–1015 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Miyajima, S., Miyata, T., Kashiwagi, M.: On the Best Multiplication of the Affine Arithmetic. Transactions of the Institute of Electronics, Information and Communication Engineers J86-A(2), 150–159 (2003)

    Google Scholar 

  11. Muhanna, R.L., Zhang, H., Mullen, R.L.: Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics. Reliable Computing 13(2), 173–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Neumaier, A.: Interval Methods for Systems of Equations, pp. xvi–255. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  13. Popova, E.D.: On the Solution of Parametrised Linear Systems. In: Kraemer, W., von Wolff Gudenberg, J. (eds.) Scientific Computing, Validated Numerics, Interval Methods, pp. 127–138. Kluwer Acad. Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

  14. Popova, E.: Generalizing the Parametric Fixed-Point Iteration. Proceedings in Applied Mathematics & Mechanics (PAMM) 4(1), 680–681 (2004)

    Article  Google Scholar 

  15. Rohn, J., Rex, G.: Enclosing solutions of linear equations. SIAM Journal Numerical Analysis 35(2), 524–529 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rump, S.M.: New Results on Verified Inclusions. In: Miranker, W.L., Toupin, R.A. (eds.) Accurate Scientific Computations. LNCS, vol. 235, pp. 31–69. Springer, Heidelberg (1986)

    Chapter  Google Scholar 

  17. Rump, S.M.: Verification methods for dense and sparse systems of equations. In: Herzberger, J. (ed.) Topics in Validated Computations, pp. 63–135. North-Holland, Amsterdam (1994)

    Google Scholar 

  18. Rump, S.M.: A note on epsilon-inflation. Reliable Computing 4, 371–375 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shary, S.P.: Solving tied interval linear systems. Sibirskii Zhurnal Vychislitiel’noi Matiematiki 7(4), 363–376 (2004)

    MATH  Google Scholar 

  20. Skalna, I.: A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters. Reliable Computing 12(2), 107–120 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Skalna, I.: Direct method for solving parametric interval linear systems with non-affine dependencies. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2009. LNCS, vol. 6068, pp. 485–494. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Iwona Skalna

Authors
  1. Iwona Skalna
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Computer and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev 25 A, 1113, Sofia, Bulgaria

    Ivan Dimov

  2. Faculty of Mathematics and Informatics, Department Numerical Methods and Algorithms, University of Sofia "St. Kliment Ohridski",, blvd. James Bourchier 5, 1164, Sofia, Bulgaria

    Stefka Dimova

  3. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Bonchev St.,Bl.8, 1113, Sofia, Bulgaria

    Natalia Kolkovska

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Skalna, I. (2011). A Comparison of Methods for Solving Parametric Interval Linear Systems with General Dependencies. In: Dimov, I., Dimova, S., Kolkovska, N. (eds) Numerical Methods and Applications. NMA 2010. Lecture Notes in Computer Science, vol 6046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18466-6_59

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-18466-6_59

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-18465-9

  • Online ISBN: 978-3-642-18466-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation