[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

On the Approximation of Interval Functions

  • Conference paper
Applied Parallel Computing. State of the Art in Scientific Computing (PARA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3732))

Included in the following conference series:

  • 1476 Accesses

Abstract

Many problems in interval arithmetic in a natural way lead to a quantifier elimination problem over the reals. By studying closer the precise form of the latter we show that in some situations it is possible to obtain a refined complexity analysis of the problem. This is done by structural considerations of the special form of the quantifiers and its implications for the analysis in a real number model of computation. Both can then be used to obtain as well new results in the Turing model. We exemplify our approach by dealing with different versions of the approximation problem for interval functions.

Partially supported by the EU Network of Excellence PASCAL Pattern Analysis, Statistical Modelling and Computational Learning and by the Danish Natural Science Research Council SNF. This publication only reflects the author’s views.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM 43(6), 1002–1045 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)

    Google Scholar 

  3. Cucker, F., Matamala, M.: On digital nondeterminism. Mathematical Systems Theory 29, 635–647 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Cucker, F., Shub, M., Smale, S.: Complexity separations in Koiran’s weak model. Theoretical Computer Science 133, 3–14 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determimants. Birkhäuser, Basel (1994)

    Book  Google Scholar 

  6. Gustafson, S.Å., Kortanek, K.O.: Semi-infinte programming and applications. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: The State of the Art, pp. 132–157. Springer, Heidelberg (1983)

    Google Scholar 

  7. Koiran, P.: A weak version of the Blum-Shub-Smale model. In: 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 486–495 (1993)

    Google Scholar 

  8. Kreinovich, V., Lakeyev, A.V., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1997)

    Google Scholar 

  9. Kreinovich, V., Meer, K.: Complexity results for the range problem in interval arithmetic (in preparation)

    Google Scholar 

  10. Koshelev, M., Longpré, L., Taillibert, P.: Optimal Enclusure of Quadratic Interval Functions. Reliable Computing 4, 351–360 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lakeyev, A.V., Noskov, S.I.: A description of the set of solutions of a linear equation with interval defined operator and right-hand side. Russian Acad. Sci. Dokl. Math. 47(3), 518–523 (1993)

    MathSciNet  Google Scholar 

  12. Meer, K.: On the complexity of quadratic programming in real number models of computation. Theoretical Computer Science 133, 85–94 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Meer, K.: On a refined analysis of some problems in interval arithmetic using real number complexity theory. Reliable Computing 10(3), 209–225 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Plaisted, D.A.: New NP-hard and NP-complete polynomial and integer divisibility problems. Theoretical Computer Science 31, 125–138 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rohn, J.: Enclosing solutions of linear interval equations is NP-hard. Computing 53, 365–368 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Sturmfels, B.: Introduction to resultants. Application of Computational Algebraic Geometry. In: Cox, D.A., Sturmfels, B. (eds.) Proc. of Symposia in Applied Mathematics, vol. 53, pp. 25–39. American Mathematical Society, Providence (1998)

    Google Scholar 

  17. Woźniakowski, H.: Why does information-based complexity use the real number model? Theoretical Computer Science 219, 451–465 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Syddansk Universitet, Campusvej 55, 5230, Odense M, Denmark

    Klaus Meer

Authors
  1. Klaus Meer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Science Department, University of Tennessee, 37996-3450, Knoxville, TN, USA

    Jack Dongarra

  2. Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800, Lyngby, Denmark

    Kaj Madsen

  3. Informatics & Mathematical Modeling, Technical University of Denmark, DK-2800, Lyngby, Denmark

    Jerzy Waśniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Meer, K. (2006). On the Approximation of Interval Functions. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds) Applied Parallel Computing. State of the Art in Scientific Computing. PARA 2004. Lecture Notes in Computer Science, vol 3732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11558958_19

Download citation

  • DOI: https://doi.org/10.1007/11558958_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29067-4

  • Online ISBN: 978-3-540-33498-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation