Reliable Computing

, Volume 6, Issue 2, pp 207–218 | Cite as

Mathematical Function Software on the Web—Are Such Codes Useful for Verification Algorithms?

  • Werner Hofschuster
  • Walter Krämer
Article

Abstract

We give an overview on existing software implementations of special mathematical functions (erf(x), erfe(x), Γ(x), Bessel-Functions,...) which can be found on the web. We discuss the quality of the numerical results and their usability in an interval setting. We also point out whether it is an easy or difficult task to find reliable routines with approved (relative or absolute) error bounds. We will show which additional steps have to be performed to get worst-case error bounds for such routines.

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References

  1. 1.
    A Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std. 754-1985, New York, 1985.Google Scholar
  2. 2.
    Abramowitz, M. and Stegun, I. A.: Handbook of Mathematical Functions (Nat. Bur. Standards, Appl. Math. Series 55), U.S. Government Printing Office, Washington, D.C., 1964.Google Scholar
  3. 3.
    Adams, E. and Kulisch, U.: Scientific Computing with Automatic Result Verification (Mathematics in Science and Engineering 189), Academic Press, 1993.Google Scholar
  4. 4.
    Agarwal, R. C. et al.: New Scalar and Vector Elementary Functions for the IBM System/370, IBM J. Res. Develop. 30(2) (1986).Google Scholar
  5. 5.
    Bantle, A. and Krämer, W.: Ein Kalkül für verläβliche absolute und relative Fehlerabschätzungen, Preprint 98/5 des IWRMM, Universität Karlsruhe, 1998.Google Scholar
  6. 6.
    Black, Ch. M., Burton, R. B., and Miller, T. H.: The Need for an Industry Standard of Accuracy for Elementary-Function Programs, ACM Trans. on Math. Software 10(4) (1984), pp. 361-366.Google Scholar
  7. 7.
    Blomquist, F. and Krämer, W.: Algorithmen mit garantierten Fehlerschranken für die Fehler-und die komplementäre Fehlerfunktion, Preprint 97/3 des IWRMM, Universität Karlsruhe, 1997, FTP://iamk4515.mathematik.uni-karlsruhe.de in the directory/pub/iwrmm/preprints.Google Scholar
  8. 8.
    Braune, K., Krämer, W.: Standard Functions for Intervals with Maximum Accuracy, in: 11th IMACS World Congress, Proceedings, Vol. 1, 1985, pp. 167-170.Google Scholar
  9. 9.
    Gal, S.: Computing Elementary Functions: A New Approach for Achieving High Accuracy and Good Performance, in: Accurate Scientific Computations (Lecture Notes in Computer Science 235), Springer, New York, 1986, pp. 1-16.Google Scholar
  10. 10.
    Gal, S. and Bachelis, B.: An Accurate Elementary Mathematical Library for the IEEE Floating Point Standard, IBM Technical Report 88.223, IBM Israel, Technion City, Haifa, Israel, 1988.Google Scholar
  11. 11.
    Hammer, R. et al.: C++ Toolbox for Verified Computing I, Springer, 1995.Google Scholar
  12. 12.
    High Accuracy Arithmetic—Extendend Scientific Computation (ACRITH-XSC). Reference Manual, IBM, SC 33-6462-00, 1990.Google Scholar
  13. 13.
    High Accuracy Arithmetic Subroutine Library (ACRITH). Program Description and User's Guide, IBM, SC 33-6164-02, 3rd Edition, 1986.Google Scholar
  14. 14.
    Higham, N.J: Accuracy and Stability of Numerical Algorithms, SIAM, 1996.Google Scholar
  15. 15.
    Hofschuster, W. and Krämer, W.: A Computer Oriented Approach to Get Sharp Reliable Error Bounds, Reliable Computing 3(3) (1997), pp. 239-248.Google Scholar
  16. 16.
    Hofschuster, W. and Krämer, W.: A Fast Public Domain Interval Library in ANSI-C, in: Proceedings zur IMACS'97 in Berlin, Volume 2, 1997, pp. 395-400.Google Scholar
  17. 17.
    Hofschuster, W. and Krämer, W.: Ein rechnergestützter Fehlerkalkül mit Anwendung auf ein genaues Tabellenverfahren, Preprint 96/5 des Instituts für Wissenschaftliches Rechnen und Mathematische Modellbildung (IWRMM), Universität Karlsruhe, 1996.Google Scholar
  18. 18.
    Hofschuster, W. and Krämer, W.: Fi_Lib, eine schnelle und portable Funktionsbibliothek für reelle Argumente und reelle Intervalle im IEEE-double-Format, Preprint 98/7 des IWRMM, Universität Karlsruhe, 227 Seiten, 1998.Google Scholar
  19. 19.
    Kearfott, B., Dawande, M., Du, K., and Hu, Ch.: INTLIB: A Portable Fortran-77 Elementary Function Library, Interval Computations 3 (1993), pp. 96-105.Google Scholar
  20. 20.
    Knüppel, O.: BIAS—Basic Interval Arithmetic Subroutines, TU Hamburg-Harburg, Bericht 93.3, 1993.Google Scholar
  21. 21.
    Krämer, W.: A Priori Worst Case Error Bounds for Floating-Point Computations, IEEE Transactions on Computers 47(7) (1998).Google Scholar
  22. 22.
    Krämer, W.: Berechnung der Gammafunktion für reelle Punkt-und Intervallargumente, Z. Angew. Math. Mech. 70 (1990), pp. 581-584.Google Scholar
  23. 23.
    Krämer, W.: Computation of Verified Bounds for Elliptic Integrals, in: Herzberger, J. and Atanassova, L. (eds), Proceedings of the International Symposium on Computer Arithmetic and Scientific Computation, Oldenburg 1991 (SCAN91), Elsevier Science Publishers (North-Holland).Google Scholar
  24. 24.
    Krämer, W.: Constructive Error Analysis, Journal of Universal Computer Science (JUCS) 4(2) (1998), pp. 147-163.Google Scholar
  25. 25.
    Krämer, W: Multiple-Precision Computations with Result Verification, in: Adams, E. and Kulisch, U. (eds), Scientific Computing with Automatic Result Verification, Academic Press, 1992, pp. 311-343.Google Scholar
  26. 26.
    Krämer, W.: Sichere und genaue Abschätzung des Approximationsfehlers bei rationalen Approximationen, Bericht 3/1996 des Instituts für Angewandte Mathematik, Universität Karlsruhe, 1996.Google Scholar
  27. 27.
    Krämer, W. and Barth, B.: Computation of Interval Bounds for Weierstrass' Elliptic Function, Computing Suppl. 9 (1993), Springer Verlag, pp. 147-159.Google Scholar
  28. 28.
    Krämer, W., Kulisch, U., and Lohner, R.: Numerical Toolbox for Verified Computing II, Theory, Algorithms, and PASCAL-XSC Programs, Springer, Berlin, to appear.Google Scholar
  29. 29.
    Kulisch, U. and Miranker, W.: Computer Arithmetic in Theory and Practice, Academic Press, 1981.Google Scholar
  30. 30.
    Lefèvre, V., Muller, J. M., and Tisserand, A.: Towards Correctly Rounded Transcendentals, in: Proceedings of the 13th IEEE Symp. on Computer Arithmetic, Asilomar, California, 1997, pp. 132-137.Google Scholar
  31. 31.
    Lozier, D. W.: Software Needs in Special Functions, Journal of Computational and Applied Mathematics 66 (1996), pp. 345-358.Google Scholar
  32. 32.
    Lozier, D. W. and Olver, F. W.: Numerical Evaluation of Special Functions, in: Proceedings of Symposia in Applied Mathematics, Vol. 48, 1994, pp. 79-125.Google Scholar
  33. 33.
    Luke, Y. L.: Mathematical Functions and Their Approximations, Academic Press, New York-San Francisco-London, 1975.Google Scholar
  34. 34.
    Luke, Y. L.: The Special Functions and Their Approximations, Volume II; Academic Press, New York-London, 1969.Google Scholar
  35. 35.
    Luther, W.: Highly Accurate Tables for Elementary Functions, BIT 35 (1995), pp. 352-360.Google Scholar
  36. 36.
    Luther, W. and Otten, W.: Computation of Standard Interval Functions in Multiple-Precision Interval Arithmetic, Interval Computations 4 (1994), pp. 78-99.Google Scholar
  37. 37.
    Luther, W. and Otten, W.: Reliable Computation of Elliptic Functions, Journal of Universal Computer Science (J.UCS) 4(1) (1998), pp. 25-33 (http://www.iicm.edu/jucs_4_1/reliable_computation_of_elliptic).Google Scholar
  38. 38.
    MacLeod, A. J.: Table-Based Tests for Bessel Function Software, Advances in Computational Mathematics 2 (1994), pp. 251-260.Google Scholar
  39. 39.
    Markstein, P.W.: Computation of Elementary Functions on the IBM RISC System/6000 Processor, IBM J. Res. Develop. 34(1) (1990).Google Scholar
  40. 40.
    Moshier, S. L. B.: Methods and Programs for Mathematical Functions, Ellis Horwood Limited, Chichester, 1989.Google Scholar
  41. 41.
    Muller, J. M.: Elementary Functions: Algorithms and Implementation, Birkhäuser, Boston, 1997.Google Scholar
  42. 42.
    Oberhettinger, F.: Tabellen zur Fourier Transformation, Springer, Berlin, 1957.Google Scholar
  43. 43.
    Priest, D.: Fast Table-Driven Algorithms for Interval Functions, Proceedings of the 13th IEEE Symp. on Computer Arithmetic, Asilomar, California, 1997, pp. 168-174.Google Scholar
  44. 44.
    Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.: Numerical Recipes in Pascal—The Art of Scientific Computing, Cambridge University Press, 1989.Google Scholar
  45. 45.
    Schulte, M. J. and Stine, J.: Symmetric Bipartite Tables for Accurate Function Approximation, in: Proceedings of the 13th IEEE Symp. on Computer Arithmetic, Asilomar, California, 1997, pp. 175-183.Google Scholar
  46. 46.
    Schulte, M. J. and Swartzlander, E. E. Jr.: Design for Exactly Rounded Elementary Functions, IEEE Transactions on Computers C-44 (1994), pp. 964-973.Google Scholar
  47. 47.
    Schulte, M. J. and Swartzlander, E. E. Jr.: Exact Rounding of Certain Elemantary Functions, IEEE Transactions on Computers (1993), pp. 138-145.Google Scholar
  48. 48.
    Tang, P. T. P.: Table-Driven Implementation of the Expml Function in IEEE Floating-Point Arithmetic, ACM Trans. on Math. Software 18(2) (1992), pp. 211-222.Google Scholar
  49. 49.
    Tang, P. T. P.: Table-Driven Implementation of the Exponential Function in IEEE Floating-Point Arithmetic, ACM Trans. on Math. Software 15(2) (1989), pp. 144-157.Google Scholar
  50. 50.
    Tang, P. T. P.: Table-Driven Implementation of the Logarithm Function in IEEE Floating-Point Arithmetic, ACM Trans. on Math. Software 16(4) (1990), pp. 378-400.Google Scholar
  51. 51.
    Tang, P. T. P.: Table-Lookup Algorithms for Elementary Functions and Their Error Analysis, in: Proceedings of 10-th Symposium on Computer Arithmetic ARITH, IEEE Computer Society Press, 1991, pp. 232-236.Google Scholar
  52. 52.
    Thompson, W. J.: Atlas for Computing Mathematical Functions, John Wiley & Sons, New York, 1997.Google Scholar
  53. 53.
    Werner, K.: Calculation of the Inverse Weierstraβ Function in an Arbitrary Machine Arithmetic, in: Alefeld, G., Frommer, A., and Lang, B. (eds), Scientific Computing and Validated Numerics, Proceedings of SCAN-95, Akademie Verlag, Berlin, 1996.Google Scholar
  54. 54.
    Zhang, S. and Jianming, J.: Computation of Special Functions, John Wiley & Sons, New York, 1996.Google Scholar
  55. 55.
    Ziv, A.: Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded Last Bit, ACM Trans. on Math. Software 17(3) (1991), pp 410-423.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Werner Hofschuster
    • 1
  • Walter Krämer
  1. 1.Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung (IWRMM)Universität KarlsruheKarlsruheGermany

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