Skip to main content
SpringerLink
Log in

A Parallelized Version of the Covering Algorithm for Solving Parameter-Dependent Systems of Nonlinear Equations

  • Published:
Reliable Computing

Abstract

The so-called covering algorithm for enclosing the solution set of parameter—dependent systems of nonlinear equations has been recently proposed by Neumaier (The Enclosure of Solutions of Parameter Dependent Systems of Equations, in: Moore, R. E. (ed.), Reliability in Computing: The Role of Interval Methods in Scientific Computations, Academic Press, 1988). However, in the covering algorithm, only one box is processed in each iteration. This paper presents a parallelized version of the covering algorithm, in which all boxes present are processed simultaneously in each iteration. It is shown through several examples that this strategy results in speed-up of the algorithm by several orders of magnitude, particularly so in demanding problems. The proposed parallelized version can be run even on ordinary computers, i.e., it does not require a parallel computer.

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. Forte Fortran 95 User Manual, Sun Microsystems, Palo Alto, CA, 2001.

  2. Kearfott, R. B.: Abstract Generalized Bisection and a Cost Bound, Mathematics of Computation 49 (179) (1987), pp. 187–202.

    Google Scholar 

  3. Kearfott, R. B.: Some Tests of Generalized Bisection, ACM Transactions on Mathematical Software 13 (3) (1987), pp. 197–220.

    Google Scholar 

  4. Krawczyk, R. and Neumaier, A.: Interval Slopes for Rational Functions and Associated Centered Forms, SIAM J. Numerical Analysis 22 (1985), pp. 604–616.

    Google Scholar 

  5. Lawrence, J. D.: A Catalog of Special Plane Curves, Dover, New York, 1972.

    Google Scholar 

  6. MATLAB User Guide, version 5.3, The MathWorks Inc., MA, 2000.

  7. Moore, R. E.: Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.

    Google Scholar 

  8. Morgan, A. P.: Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice Hall, Englewood Cliffs, 1987.

    Google Scholar 

  9. Morgan, A. and Shapiro, V.: Box Bisection for Solving Second Degree Systems and the Problem of Clustering, ACM Trans. Math. Software 13 (1987), pp. 152–167.

    Google Scholar 

  10. Nataraj, P. S. V. and Sardar, G.: Template Generation for Continuous Transfer Functions Using Interval Analysis, Automatica 36 (2000), pp. 111–119.

    Google Scholar 

  11. Neumaier, A.: Personal communication.

  12. Neumaier, A.: The Enclosure of Solutions of Parameter Dependent Systems of Equations, in: Moore, R. E. (ed.), Reliability in Computing: The Role of Interval Methods in Scientific Computations, Academic Press, 1988.

  13. Rump, S. M.: INTLAB-INTerval LABoratory, in: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, 1999.

  14. Ushida, A. and Chua, L. O.: Tracing Solution Curves of Nonlinear Equations with Sharp Turning Points, Circuit Theory and Applications 12 (1984), pp. 1–21.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Systems and Control Engineering Group, IIT Bombay, 400 076, India

    Paluri S. V. Nataraj & Airani Kalathil Prakash

Authors
  1. Paluri S. V. Nataraj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Airani Kalathil Prakash
    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

Nataraj, P.S.V., Prakash, A.K. A Parallelized Version of the Covering Algorithm for Solving Parameter-Dependent Systems of Nonlinear Equations. Reliable Computing 8, 123–130 (2002). https://doi.org/10.1023/A:1014750005366

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014750005366

Keywords

Search

Navigation