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A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms

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Abstract

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified \(\mathcal {O}(n\log n)\) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.

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References

  1. Pownuk, A., Neumaier, A.: Linear systems with large uncertainties, with applications to truss structures. Reliab. Comput. 13, 149–172 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Floudas, C.A.: Deterministic Global Optimization. Theory, Methods and Applications, Volume 37 of Nonconvex Optimization and its Applications. Kluwer, Dordrecht (2000)

    Google Scholar 

  3. Hansen, E.R., Walster, G.W.: Global Optimization using Interval Analysis, 2nd edn. Marcel Dekker, New York (2004)

    MATH  Google Scholar 

  4. Hladík, M.: Optimal value range in interval linear programming. Fuzzy Optim. Decis. Making 8(3), 283–294 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hladík, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561–574 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Stolfi, J., Comba, J.L.D.: Affine arithmetic and its applications to computer graphics Proceedings of the SIBGRAPI’93 VI Simpósio Brasileiro de Computação Gráfica e Processamento de Imagens (Recife, BR), pp 9–18 (1993)

  7. Kolev, L.: Automatic computation of a linear interval enclosure. Reliab. Comput. 7, 17–18 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kolev, L.: An improved interval linearization for solving nonlinear problems. Numer. Algorithms 37, 213–224 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kolev, L.: Parameterized solution of linear interval parametric systems. Appl. Math. Comput. 246(11), 229–246 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Kolev, L.: A new class of iterative interval methods for solving linear parametric systems. Reliable Computing (2016)

  11. Stolfi, J., de Figueiredo, L.H.: Self-Validated Numerical Methods and Applications. Brazilian Mathematics Colloquium monograph. IMPA, Rio de Janeiro, Brazil (1997)

  12. Stolfi, J., de Figueiredo, L.H.: An introduction to affine arithmetic. TEMA Tend Mat. Apl. Comput. 4(3), 297–312 (2003)

  13. Stolfi, J., de Figueiredo, L.H.: Affine arithmetic: concepts and applications. Numer. Algorithms 37(1–4), 147–158 (2004)

  14. Martin, 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)

    Article  MATH  Google Scholar 

  15. Messine, F.: New Affine Forms in Interval Branch and Bound Algorithms. Technical Report R2I 99-02, Université de Pau et des Pays de l’Adour (UPPA), France (1999)

  16. Miyajima, S.: On the Improvement of the Division of the Affine Arithmetic. Bachelor Thesis. PhD Thesis, Kashiwagi Laboratory, Waseda University, Japan (2000)

  17. Moore, Ramon E., Baker Kearfott, R., Cloud, M.J.: Introduction to Interval analysis. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  18. Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numerica 13, 271–369 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  20. Popova, E.D.: Improved enclosure for some parametric solution sets with linear shape. Comput. Math. Appl., 68(9) (2014)

  21. Popova, E.D.: Solvability of parametric interval linear systems of equations and inequalities. SIAM J. Matrix Anal. Appl. 36(2), 615–633 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Popova, E.D.: Generalization of a parametric fixed-point iteration PAMM Proceedings in Applied Mathematics and Mechanics, vol. 4, pp 680–68 (2004)

  23. Rump, S.M.: Verification methods for dense and sparse systems of equations. In: Herzberger, J. (ed.) Topics in Validated Computations, pp 63–135. Elsevier, Amsterdam (1994)

  24. Miyajima, M., Kashiwagim, S., Miyata, T.: On the best multiplication of the affine arithmetic. IEICE Transac. J86-A(2), 150–159 (2003)

    Google Scholar 

  25. Kashiwagi, M., Rump, S.M.: Implementation and improvements of affine arithmetic. Nonlinear Theory Appl., IEICE 6(3), 341–359 (2017)

    Google Scholar 

  26. Cormen, R.L., Rivest, T.H., Leiserson, C.E., Stein, C.: Introduction to Algorithms. The MIT Press, Cambridge, Massachusetts London, England 3rd edition (2009)

  27. Vu, B.V., Faltings X.-H., Sam-haroud, D.: A Generic Scheme for Combining Multiple Inclusion Representations in Numerical Constraint Propagation. Technical Report No. IC/2004/39 EPFL (2004)

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Authors and Affiliations

  1. AGH University of Science and Technology, Krakow, Poland

    Iwona Skalna

  2. Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Milan Hladík

Authors
  1. Iwona Skalna
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  2. Milan Hladík
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Correspondence to Iwona Skalna.

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Skalna, I., Hladík, M. A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms. Numer Algor 76, 1131–1152 (2017). https://doi.org/10.1007/s11075-017-0300-6

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  • DOI: https://doi.org/10.1007/s11075-017-0300-6

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