Algorithmic Differencing

  • Louis B. Rall
  • Thomas W. Reps


An algorithmic representation of a function is a step-by-step specification of its evaluation in terms of known operations and functions, such as a computer program. In addition to function values, the algorithmic representation can be used to compute related quantities such as derivatives of the function. A process similar to automatie (or algorithmic) differentiation will be applied to obtain differences and divided differences of functions. Advantages of this approach are that it often reduces the sometimes catastrophic cancellation errors in computation of differences and divided differences and provides numerical convergence of divided differences to derivatives.


Arithmetic Operation Standard Function Divided Difference Symmetric Polynomial Automatic Differentiation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. Bischof, A. Carle, G. F. Corliss, and A. Griewank, ADIFOR: Automatie differentiation in a source translator environment, in Proc. of Int. Symp. on Symb. and Alg. Comp. (ISSAC 1992), ACM, New York, 1992, pp. 294–302.Google Scholar
  2. 2.
    G. Bohlender, C. Ullrich, J. Wolff von Gudenberg, and L. B. Rall, Pascal-SC, A Computer Language for Scientific Computation, Academic Press, New York, 1987.zbMATHGoogle Scholar
  3. 3.
    C. de Boor, A multivariate divided difference, in Approximation Theory VIII, ed. by C. K. Chui and L. L. Schumaker, World Scientific, Singapore, 1995.Google Scholar
  4. 4.
    J. Cocke and J. T. Schwartz, Programming Languages and Their Compilers, Preliminary Notes, 2nd Revised Version, Courant Inst. of Math. Sei., New York University, New York, 1970.Google Scholar
  5. 5.
    G. F. Corliss and L. B. Rall, Adaptive, self-validating quadrature, SIAM J. Sei. Stat. Comput., 8 (1987), pp. 831–847.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    G. F. Corliss and L. B. Rall, Computing the range of derivatives, in Computer Arithmetic, Scientific Computation and Mathematical Modeling, ed. by E. Kaueher, S. M. Markov, and G. Mayer, J.C. Baltzer AG, Basel, 1991, pp. 195-212.Google Scholar
  7. 7.
    H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, New York, 1977.zbMATHGoogle Scholar
  8. 8.
    A. Griewank, Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation, SIAM, Philadelphia, 2000.zbMATHGoogle Scholar
  9. 9.
    W. Kahan and R. J. Fateman, Symbolic computation of divided differences, Unpublished Report, Dept. of Elec. Eng. and Comp. Sei., Univ. of CaliforniaBerkeley, 1985. See;-fateman/papers/divdif.pdf.Google Scholar
  10. 10.
    R. Klatte, U. Kuliseh, M. Neaga, D. Ratz, C. Ullrich, Pascal-XSC, Language Reference with Examples, Springer-Verlag, Berlin-New York, 1992.zbMATHGoogle Scholar
  11. 11.
    R. Klatte, U. Kuliseh, A. Wiethoff, C. Lawo, M. Rauch, C-XSC, A C++ Class Library for Extended Scientific Computation, Springer-Verlag, Berlin-New York, 1993.Google Scholar
  12. 12.
    U. W. Kulisch and W. L. Miranker, Computer Arithmetic in Theory and Practice, Academic Press, New York, 1981.zbMATHGoogle Scholar
  13. 13.
    U. W. Kulisch and W. L. Miranker (Eds.), A New Approach to Scientific Computation, Academic Press, New York, 1983.zbMATHGoogle Scholar
  14. 14.
    A. C. McCurdy, Accurate Computation of Divided Differences, Ph.D. diss. and Tech. Rep. UCB/ERL M80/28, Univ. of California-Berkeley, 1980.Google Scholar
  15. 15.
    W. E. Milne, Numerical Calculus, Princeton, 1949.Google Scholar
  16. 16.
    C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20 (1978), pp. 801–836.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.zbMATHCrossRefGoogle Scholar
  18. 18.
    G. Opitz, Steigungsmatrizen, Zeitschrift Angew. Math. Mech. 48 (1964), pp. T52-T54.Google Scholar
  19. 19.
    R. Paige and S. Koenig, Finite differencing of computable expressions, ACM Trans. Program. Lang. Syst. 4 (1982), pp. 402–454.zbMATHCrossRefGoogle Scholar
  20. 20.
    T. W. Reps and L. B. Rall, Computational Divided Differencing and DividedDifference Arithmetics, Tech. Rep. TR-1415, Comp. Sei. Dept., Univ. of Wisconsin-Madison, 2000.Google Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Louis B. Rall
    • 1
  • Thomas W. Reps
    • 1
  1. 1.University of Wisconsin-MadisonMadisonUSA

Personalised recommendations