Numerical Algorithms

, Volume 63, Issue 4, pp 601–614

Algorithm for min-range multiplication of affine forms

Open Access
Original Paper
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Abstract

Affine arithmetic produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as round-off errors. Elementary operations on affine forms are redefined so they result in affine forms. Affine-linear operations result straightforwardly in affine forms. Non-linear operators, such as multiplication, must be approximated by affine forms. Choosing the appropriate approximation is a big challenge. The reason is that different approximations may be more accurate for specific purposes. This paper presents an efficient method for computing the minimum range (min-range) affine approximation of the product of arbitrary affine forms that do not contain zero properly. Numerical experiments are carried out to demonstrate the essential features of the proposed approach, especially its usefulness for bounding ranges of functions for global optimisation and for finding roots of functions.

Keywords

Affine arithmetic Multiplication Min-range approximation Range bounding Global optimisation Roots of functions 

References

  1. 1.
    Alefeld, G.: Inclusion methods for systems of nonlinear equations—the interval Newton method and modifications. In: Herzberger, J. (ed.) Topics in Validated Computations. Elsevier, Amsterdam (1994)Google Scholar
  2. 2.
    Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: Proc. SIBGRAPI’93–VI Simpósio Brasileiro de Computação Gráfica e Processamento de Imagens (Recife, BR), pp. 9–18 (1993)Google Scholar
  3. 3.
    de Figueiredo, L.H., Stolfi, J.: Adaptive enumeration of implicit surfaces with affine arithmetic. Comput. Graph. Forum 15, 287–296 (1996)CrossRefGoogle Scholar
  4. 4.
    de Figueiredo, L.H., Stolfi, J.: Self-validated numerical methods and applications. In: Brazilian Mathematics Colloquium Monograph, IMPA. Rio de Janeiro, Brazil (1997)Google Scholar
  5. 5.
    de Figueiredo, L.H., Stolfi, J.: An introduction to affine arithmetic. TEMA Tend. Mat. Apl. Comput. 4(3), 297–312 (2003)MathSciNetMATHGoogle Scholar
  6. 6.
    de Figueiredo, L.H., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37(1–4), 147–158 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kolev, L.V.: Optimal multiplication of G-intervals. Reliab. Comput. 13, 399–408 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kolev, L.V.: A new method for global solution of systems of nonlinear equations. Reliab. Comput. 4(1), 1–21 (1998)CrossRefGoogle Scholar
  9. 9.
    Zhang, Q., Martin, R.R.: Polynomial evaluation using affine arithmetic for curve drawing. In: Proc. Eurographics UK 2000 Conf., pp. 49–56. Eurographics UK, Abingdon (2000)Google Scholar
  10. 10.
    Martin, R.R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G.: Comparison of interval methods for plotting algebraic curves. Comput. Aided Geom. Des. 19(7), 553–587 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)MATHGoogle Scholar
  12. 12.
  13. 13.
  14. 14.

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.AGH University of Science and TechnologyKrakowPoland

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