Skip to main content
SpringerLink
Log in
Book cover

Interval Methods for Solving Nonlinear Constraint Satisfaction, Optimization and Similar Problems pp 5–16Cite as

Interval Calculus

  • Chapter
  • First Online:

  • 451 Accesses

Part of the book series: Studies in Computational Intelligence ((SCI,volume 805))

Abstract

This chapter introduces the interval calculus. Its history and motivation is reviewed. Basic ideas, notions and theorems are presented.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

References

  1. Benhamou, F., Granvilliers, L.: Combining local consistency, symbolic rewriting and interval methods. In: Artificial Intelligence and Symbolic Mathematical Computation, pp. 144–159 (1996)

    Google Scholar 

  2. Fedrizzi, M., Kacprzyk, J., Yager, R.R. (eds.): Advances in the Dempster-Shafer Theory of Evidence (1994)

    Google Scholar 

  3. Gardenes, E., Sainz, M.A., Jorba, L., Calm, R., Estela, R., Mielgo, H., Trepat, A.: Modal intervals. Reliab. Comput. 7(2), 77–111 (2001)

    Google Scholar 

  4. Goldsztejn, A.: Modal intervals revisited, part 1: a generalized interval natural extension. Reliab. Comput. 16, 130–183 (2012)

    Google Scholar 

  5. Granvilliers, L., Monfroy, E., Benhamou, F.: Symbolic-interval cooperation in constraint programming. In: Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, pp. 150–166. ACM (2001)

    Google Scholar 

  6. Gutowski, M.W.: Interval straight line fitting (2001). arXiv:math/0108163

  7. Gutowski, M.W.: Introduction to Interval Calculi and Methods (in Polish). BEL Studio, Warszawa (2004)

    Google Scholar 

  8. Gutowski, M.W.: Breakthrough in interval data fitting I. The role of Hausdorff distance (2009). arXiv:0903.0188

  9. Gutowski, M.W.: Breakthrough in interval data fitting II. From ranges to means and standard deviations (2009). arXiv:0903.0365

  10. Hansen, E., Walster, W.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (2004)

    Google Scholar 

  11. Hansen, P., Jaumard, B., Xiong, J.: Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions. J. Glob. Optim. 3(4), 421–437 (1993)

    Google Scholar 

  12. Hurwicz, L.: Optimality criteria for decision making under ignorance. In: Cowles Commission Discussion Paper, Statistics, 370 (1951)

    Google Scholar 

  13. IEEE: 754-2008–IEEE standard for floating-point arithmetic (2008). http://ieeexplore.ieee.org/document/4610935/

  14. IEEE: 1788-2015–IEEE standard for interval arithmetic (2015). http://standards.ieee.org/findstds/standard/1788-2015.html

  15. Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis. Springer, London (2001)

    Google Scholar 

  16. Kaucher, E.: Interval analysis in the extended interval space \(\mathbb{I}{\mathbb{R}}\). In: Fundamentals of Numerical Computation (Computer-Oriented Numerical Analysis), pp. 33–49. Springer (1980)

    Google Scholar 

  17. Kearfott, R.B.: Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht (1996)

    Google Scholar 

  18. Kearfott, R.B., Kreinovich, V.: Applications of Interval Computations, vol. 3. Springer Science & Business Media (2013)

    Google Scholar 

  19. Kearfott, R.B., Nakao, M.T., Neumaier, A., Rump, S.M., Shary, S.P., van Hentenryck, P.: Standardized notation in interval analysis. Vychislennyie Tiehnologii (Computational Technologies) 15(1), 7–13 (2010)

    Google Scholar 

  20. Kubica, B.J., Woźniak, A.: Applying an interval method for a four agent economy analysis. In: PPAM 2011 (9th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 7204, pp. 477–483 (2012)

    Google Scholar 

  21. Kulisch, U.: An axiomatic approach to rounded computations. Numerische Mathematik 18(1), 1–17 (1971)

    Google Scholar 

  22. Kulisch, U.: Computer Arithmetic and Validity-Theory. Implementation and Applications. De Gruyter, Berlin, New York (2008)

    Google Scholar 

  23. Kulisch, U.: An axiomatic approach to computer arithmetic with an appendix on interval hardware. In: PPAM 2011 (9th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 7204, pp. 484–495 (2012)

    Google Scholar 

  24. Kulpa, Z.: Diagrammatic representation of interval space in proving theorems about interval relations. Reliab. Comput. 3(3), 209–217 (1997)

    Google Scholar 

  25. Moore, R.E.: Interval Analysis. Prentice Hall (1966)

    Google Scholar 

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

    Google Scholar 

  27. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press (1990)

    Google Scholar 

  28. Pedamallu, C.S., Özdamar, L., Csendes, T.: Symbolic interval inference approach for subdivision direction selection in interval partitioning algorithms. J. Glob. Optim. 37(2), 177–194 (2007)

    Google Scholar 

  29. Sevastianov, P.: Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory. Inf. Sci. 177(21), 4645–4661 (2007)

    Google Scholar 

  30. Sevastjanov, P., Bartosiewicz, P., Tkacz, K.: A method for comparing intervals with interval bounds. In: PPAM 2011 (9th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 7204, pp. 496–503 (2012)

    Google Scholar 

  31. Sevastjanov, P., Tikhonenko, A.: Direct interval extension of TOPSIS method. In: PPAM 2011 (9th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 7204, pp. 504–512 (2012)

    Google Scholar 

  32. Sevastjanov, P.V., Róg, P., Venberg, A.V.: A constructive numerical method for the comparison of intervals. In: PPAM 2001 (4th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 2328, pp. 756–761 (2003)

    Google Scholar 

  33. Sevastjanow, P.: Interval comparison based on Dempster–Shafer theory of evidence. In: PPAM 2003 (5th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 3019, pp. 668–675 (2004)

    Google Scholar 

  34. Sharaya, I.A.: On the distributivity in classical interval arithmetic. Vychislennyie Tiehnologii (Computational Technologies) 2(1), 71–83 (1997). (in Russian)

    Google Scholar 

  35. Shary, S.P.: Finite-Dimensional Interval Analysis. Institute of Computational Technologies, SB RAS, Novosibirsk (2013)

    Google Scholar 

  36. Warmus, M.: Calculus of approximations. Bulletin de l’ Academie Polonaise de Sciences 4(5), 253–257 (1956)

    Google Scholar 

  37. Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice Hall, Englewood Cliffs, NJ (1963)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Informatics, Warsaw University of Life Sciences, Warsaw, Poland

    Bartłomiej Jacek Kubica

Authors
  1. Bartłomiej Jacek Kubica
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bartłomiej Jacek Kubica .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kubica, B.J. (2019). Interval Calculus. In: Interval Methods for Solving Nonlinear Constraint Satisfaction, Optimization and Similar Problems. Studies in Computational Intelligence, vol 805. Springer, Cham. https://doi.org/10.1007/978-3-030-13795-3_2

Download citation

Publish with us

Policies and ethics

Search

Navigation