Computing

, Volume 82, Issue 1, pp 77–102

A comparison of some methods for bounding connected and disconnected solution sets of interval linear systems

Article

Summary

Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously locating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide interval coefficients. In many cases, singular systems are admitted within the intervals of uncertainty of the coefficients, leading to unbounded solution sets with more than one disconnected component. This, combined with the fact that computing exact bounds on the solution set is NP-hard, limits the range of techniques available for bounding the solution sets for such systems. However, the componentwise nature and other properties make the interval Gauss–Seidel method suited to computing meaningful bounds in a predictable amount of computing time. For this reason, we focus on the interval Gauss–Seidel method. In particular, we study and compare various preconditioning techniques we have developed over the years but not fully investigated, comparing the results. Based on a study of the preconditioners in detail on some simple, specially–designed small systems, we propose two heuristic algorithms, then study the behavior of the preconditioners on some larger, randomly generated systems, as well as a small selection of systems from the Matrix Market collection.

AMS Subject Classifications

65F10 65G20 65K99 

Keywords

numerical linear algebra global optimization validated computing interval analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alefeld, G., Kreinovich, V., Mayer, G.: The shape of the symmetric solution set. In: Kearfott, R. B. and Kreinovich, V. (eds.) Applications of interval computations: Papers presented at an international workshop in El Paso, TX, February 23–25, 1995. Applied Optimization, vol. 3, Norwell, MA, pp. 61–80. Kluwer, Dordrecht (1996)Google Scholar
  2. 2.
    Alefeld G., Kreinovich V. and Mayer G. (2003). On the solution sets of particular classes of linear interval systems. J Comput Appl Math 152: 1–15 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boisvert R.F., Pozo R., Remington K., Barrett R. and Dongarra J.J. (1997). The Matrix Market: a web resource for test matrix collections. In: Boisvert, R.F. (eds) Quality of numerical software, assessment and enhancement, pp 125–137. Chapman and Hall, London Google Scholar
  4. 4.
    Gau, C.-Y., Stadtherr, M. A.: Reliable high-performance computing strategies for chemical process modelling. (1999) http://www.nd.edu/~markst/dallas99/slides213c.pdf
  5. 5.
    Hu, C.-Y.: Splitting preconditioners for the interval Newton method. Ph.D thesis, University of Southwestern Louisiana (1990)Google Scholar
  6. 6.
    Hu, C.-Y., Kearfott, R.B.: A pivoting scheme for the interval Gauss–Seidel method: Numerical experiments. In: Approximation, optimization and computing: Theory and applications, pp. 97–100. Elsevier, Amsterdam (1990)Google Scholar
  7. 7.
    Kearfott R.B. (1990). Preconditioners for the interval Gauss–Seidel method. SIAM J Numer Anal 27: 804–822 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kearfott R.B. (1996). Rigorous global search: continuous problems. Kluwer, Dordrecht MATHGoogle Scholar
  9. 9.
    Kearfott, R. B., Hongthong, S.: On preconditioners and splitting in the interval Gauss–Seidel method. Tech. report, University of Louisiana at Lafayette (2005) http://interval.louisiana.edu/preprints/2005_new_S_preconditioner.as_submitted.pdf
  10. 10.
    Kreinovich V., Lakeyev A., Rohn J. and Kahl P. (1998). Computational complexity and feasibility of data processing and interval computations. Kluwer, Dordrecht MATHGoogle Scholar
  11. 11.
    Leyffer S., Lopez-Calva G. and Nocedal J. (2006). Interior methods for mathematical programs with complementarity constraints. SIAM J Optim 17: 52–77 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Muhanna R.L. and Mullen R.L. (2001). Uncertainty in mechanics problems. J Eng Mech 127: 557–566 CrossRefGoogle Scholar
  13. 13.
    Neumaier A. (1990). Interval methods for systems of equations. Cambridge University Press, Cambridge MATHGoogle Scholar
  14. 14.
    Neumaier A. and Pownuk A. (2007). Linear systems with large uncertainties, with applications to truss structures. Reliable Comput 13: 149–172 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Novoa, M.: Theory of preconditioners for the interval Gauss-Seidel method and existence/uniqueness theory with interval newton methods. Tech. report, Deptartment of Mathematics, University of Southwestern Louisiana (1993)Google Scholar
  16. 16.
    Oettli W. (1965). On the solution set of a linear system with inaccurate coefficients. SIAM J Numer Anal 2: 115–118 MathSciNetGoogle Scholar
  17. 17.
    Oettli W. and Prager W. (1964). Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer Math 6: 405–409 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pryce J.D. and Corliss G.F. (2006). Interval arithmetic with containment sets. Computing 78: 251–276 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rohn J.: Solving systems of linear interval equations. In: Reliability in computing, perspectives in computing, pp. 171–182 Academic Press, New York (1988)Google Scholar
  20. 20.
    Rump, S. M.: INTLAB – INTerval LABoratory (INTLAB home page) (2005) http://www.ti3.tu-harburg.de/~rump/intlab/

Copyright information

© Springer-Verlag Wien 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LouisianaLafayetteUSA

Personalised recommendations