Advertisement

Numerical Algorithms

, Volume 10, Issue 2, pp 261–287 | Cite as

Using Markov's interval arithmetic to evaluate Bessel-Ricatti functions

  • M. C. Bartholomew-Biggs
  • S. Zakovic
Article

Abstract

This paper deals with the application of interval arithmetic to Bessel-Ricatti functions. The extended interval arithmetic we have used is due to Markov and involves a check on monotonicity of functions in an attempt to try and get sharper bounds on computed intervals. The results we obtain are compared with those from Hansen's methods (based on bounding the Taylor series remainder) and those from Moore's technique of subdividing intervals. Two techniques are considered for evaluating derivatives of functions — one uses hand-coded derivatives and the other uses automatic differentiation. Numerical results are given, using Fortran 90 implementations of interval and automatic differentiation arithmetic.

Keywords

Taylor Series Sharp Bound Interval Arithmetic Automatic Differentiation Compute Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Markov, Some applications of extended interval arithmetic to interval iterations, Computing, Suppl. 2 (1980) 69–84.Google Scholar
  2. [2]
    R.E. Moore,Methods and Applications of Interval Analysis (SIAM, Philadelphia, 1979).Google Scholar
  3. [3]
    S. Brown, Programming interval arithmetic in Fortran 90, C++ and Ada, Final Year Project, University of Hertfordshire (May 1993).Google Scholar
  4. [4]
    E. Hansen,Global Optimization Using Interval Analysis (Marcel Dekker, 1992).Google Scholar
  5. [5]
    M. Bartholomew-Biggs, Z.J. Ulanowski and S. Zakovic, A parameter estimation problem with multiple solutions arising in laser difractometry, Numerical Optimisation Centre, Technical Report No. 281 (January 1994).Google Scholar
  6. [6]
    M. Bartholomew-Biggs, Using a Fortran 90 module for automatic differentiation, Numerical Optimisation Centre, Working Paper (May 1994).Google Scholar
  7. [7]
    G.F. Corliss and L.B. Rall, Computing the range of derivatives, in:Computing Arithmetic: Scientific Computation and Programming Languages, eds. Kaucher, Kulisch and Ullrich (B.G. Teubner Verlag, Stuttgart, 1987).Google Scholar
  8. [8]
    A. Griewank, On automatic differentiation, in:Mathematical Programming 88 (Kluwer Academic, 1989).Google Scholar
  9. [9]
    L.B. Rall,Automatic Differentiation—Techniques and Applications, Springer Lecture Notes in Computer Science, Vol. 120 (1981).Google Scholar
  10. [10]
    C. Jansson and O. Knuppel, A global minimization method—The multi-dimensional case, Berichte 92.1, Berichte des Forschungsschwerpunktes Informations und Kommunikationstechnik, Technische Universität Hamburg (1992).Google Scholar

Copyright information

© J. C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • M. C. Bartholomew-Biggs
    • 1
  • S. Zakovic
    • 1
  1. 1.Numerical Optimization CentreUniversity of HertfordshireHatfieldEngland

Personalised recommendations