Skip to main content
SpringerLink
Log in

On Maximal Inner Estimation of the Solution Sets of Linear Systems with Interval Parameters

  • Published:
Reliable Computing

Abstract

The purpose of this paper is to inquire the connection between maximal inner interval estimates of the solution sets to interval linear system and solutions of the dualization equation in Kaucher interval arithmetic. The results of our work are as follows: 1) a criterion of inner interval estimate of the solution set, 2) a criterion for a solution of dualization equation to be a maximal inner interval estimate of the solution set, 3) a criterion for multiplication by an interval matrix to be upper strictly isotone.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gardeñes, E. and Trepat, A.: Fundamentals of SIGLA, an Interval Computing System over the Completed Set of Intervals, Computing 24 (1980), pp. 161-179.

    Article  MATH  MathSciNet  Google Scholar 

  2. Kaucher, E.: Interval Analysis in the Extended Interval Space ℝ, Computing Suppl. 2(1980), pp. 33-49.

    MATH  MathSciNet  Google Scholar 

  3. Kleene, S. C.: Mathematical Logic, John Wiley, New York-London-Sydney, 1967.

    MATH  Google Scholar 

  4. Kupriyanova, L.: Inner Estimation of theUnited Solution Set of Interval LinearAlgebraic System, Reliable Computing 1 (1) (1995), pp. 15-31.

    Article  MATH  MathSciNet  Google Scholar 

  5. Lakeyev, A. V.: Linear Algebraic Equations in Kaucher Arithmetic, in: Int. J. of Reliable Comput. Supplement 1995, Extended Abstracts of APIC'95: International Workshop on Applications of Interval Computations, El Paso, Texas, February 23-25, 1995, UTEP, El Paso, pp. 130-133.

  6. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  7. Neumaier, A.: Tolerance Analysis with Interval Arithmetic, Freiburger Intervall-Berichte 86 (9), pp. 5-19.

  8. Shary, S. P.: Algebraic Approach in the 'Outer Problem' for Interval Linear Systems, Computational Technologies 3 (2) (1998), pp. 67-114 (in Russian).

    MATH  MathSciNet  Google Scholar 

  9. Shary, S. P.: Algebraic Approach to the Analysis of Linear Static Systems under Interval Uncertainty, Izvestiya Akademii Nauk, Control Theory and Systems (3) (1997), pp. 51-61 (in Russian).

    MathSciNet  Google Scholar 

  10. Shary, S. P.:Algebraic Approach to the Interval Linear Static Identification, Tolerance and Control Problems, or One More Application of Kaucher Arithmetic, Reliable Computing 2 (1) (1996), pp. 3-33.

    Article  MATH  MathSciNet  Google Scholar 

  11. Shary, S. P.: Algebraic Solutions to Interval Linear Equations and Their Applications, in: Alefeld, G. and Herzberger, J. (eds), NumericalMethods and Error Bounds,Mathematical Research 89, Akademie Verlag, Berlin, 1996, pp. 224-233.

    Google Scholar 

  12. Shary, S. P.: A New Approach to the Analysis of Static Systems under Interval Data Uncertainty, Computational Technologies 2 (1) (1997), pp. 84-102 (in Russian).

    MathSciNet  Google Scholar 

  13. Shary, S. P.: A New Approach to the Analysis of Static Systems under Interval Uncertainty, in: Alefeld, G., Frommer, A., and Lang, B. (eds), Scientific Computing and Validated Numerics, Mathematical Research 90, Akademie Verlag, 1996, pp. 118-132.

  14. Shary, S. P.: Outer Estimation of Generalized Solution Sets to Interval Linear Systems, in: Csendes, T. (ed.),Developments in ReliableComputing,KluwerAcademic Publishers, Dordrecht, 1999, pp. 323-335.

    Google Scholar 

  15. Shary, S. P.: Solving the Linear Interval Tolerance Problem, Mathematics and Computers in Simulation 39 (1995), pp. 53-85.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Technologies, Novosibirsk, Russia

    Irene A. Sharaya

Authors
  1. Irene A. Sharaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sharaya, I.A. On Maximal Inner Estimation of the Solution Sets of Linear Systems with Interval Parameters. Reliable Computing 7, 409–424 (2001). https://doi.org/10.1023/A:1011428127620

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011428127620

Keywords

Search

Navigation