Level-index arithmetic: An introductory survey

  • C. W. Clenshaw
  • F. W. J. Olver
  • P. R. Turner
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1397)


Arithmetic Operation Real Zero Computer Arithmetic Relative Precision Absolute Precision 
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]
    E.H. Bareiss, Resultant procedure and the mechanization of the Graeffe process, J.Assoc. Comput. Mach., 7 (1960), pp. 346–386.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J.L. Barlow and E.H. Bareiss, On roundoff error distributions in floating-point and logarithmic arithmetic, Computing, 34 (1985), pp. 325–341.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    F. Benford, The law of anomalous numbers, Proc. Am. Phil. Soc., 78 (1938), pp. 551–572.zbMATHGoogle Scholar
  4. [4]
    J.L. Blue, A portable Fortran program to find the Euclidean norm of a vector, ACM Trans. Math. Software, 4 (1978), pp. 15–23.MathSciNetCrossRefGoogle Scholar
  5. [5]
    C.W. Clenshaw, A note on the summation of Chebyshev series, Math. Comp., 9 (1955), pp. 118–120.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    C.W. Clenshaw, D.W. Lozier, F.W.J. Olver and P.R. Turner, Generalized exponential and logarithmic functions, Comput. Math. Appl., 12B (1986), pp. 1091–1101.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C.W. Clenshaw and F.W.J. Olver, An unrestricted algorithm for the exponential function, SIAM J. Numer. Anal., 17 (1980), pp. 310–331.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    C.W. Clenshaw and F.W.J. Olver, Beyond floating-point, J. Assoc. Comput. Mach., 31 (1984), pp. 319–328.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    C.W. Clenshaw and F.W.J. Olver, Level-index arithmetic operations, SIAM J. Numer. Anal., 24 (1987) pp. 470–485.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C.W.Clenshaw and P.R.Turner, The symmetric level-index system, IMA J. Numer. Anal. [In press].Google Scholar
  11. [11]
    C.W.Clenshaw and P.R.Turner, Root-squaring using level-index arithmetic, [Manuscript].Google Scholar
  12. [12]
    W.J. Cody Jr., and W. Waite, Software Manual for the Elementary Functions, Prentice-Hall, Englewood Cliffs, N.J., 1980.zbMATHGoogle Scholar
  13. [13]
    M.G. Cox, The numerical evaluation of B-splines, J. Inst. Maths Applics., 10 (1972), pp. 134–149.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    P.J. Davis, Leonhard Euler's integral: a historical profile of the Gamma function, Amer. Math. Monthly, 66 (1959), pp. 849–869.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J.W. Demmel, Underflow and the reliability of numerical software, SIAM J. Sci. Statist. Comp., 5 (1984), pp. 887–919.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    J.W. Demmel, On error analysis in arithmetic with varying relative precision, Proc. 8th Symposium on Computer Arithmetic, M.J. Irwin and R. Stefanelli, eds., IEEE Computer Society Press, Washington, D.C., 1987, pp. 148–152.Google Scholar
  17. [17]
    A. Feldstein and R. Goodman, Loss of significance in floating-point subtraction and addition, IEEE Trans. Comp., 31 (1982), pp. 328–335.CrossRefzbMATHGoogle Scholar
  18. [18]
    A. Feldstein and J.F. Traub, Asymptotic behavior of vector recurrences with applications, Math. Comp., 31 (1977), pp. 180–192.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Feldstein and P.R. Turner, Overflow, underflow, and severe loss of significance in floating-point addition and subtraction, IMA J. Numer. Anal., 6 (1986), pp. 241–251.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    B.J. Flehinger, On the probability that a random number has leading digit A, Amer. Math. Monthly, 73 (1966), pp. 1056–1061.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    R. Goodman, Some models of error in floating-point multiplication. Computing, 27 (1981), pp. 227–236.CrossRefzbMATHGoogle Scholar
  22. [22]
    R. Goodman and A. Feldstein, Effect of guard digits and normalisation options on floating-point multiplication, Computing, 18 (1977), pp. 93–106.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    R. Goodman, A. Feldstein and J. Bustoz, Relative error in floating-point multiplication, Computing, 35 (1985), pp. 137–139.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    J.B. Gosling, Design of arithmetic units for digital computers, MacMillan, London, 1980.CrossRefGoogle Scholar
  25. [25]
    H. Hamada, URR: Universal representation of real numbers, New Gencration Computing, OHM-Sha, Springer-Verlag 1 (1983), pp. 205–209.Google Scholar
  26. [26]
    H. Hamada, A new real number representation and its operation, Proc. 8th Symposium on Computer Arithmetic, M.J. Irwin and R. Stefanelli, eds., IEEE Computer Society Press, Washington D.C., 1987, pp. 153–157.Google Scholar
  27. [27]
    J.M. Hammersley, Probability and arithmetic in science, IMA Bulletin, 21 (1985), pp. 114–120.Google Scholar
  28. [28]
    R.W. Hamming, On the distribution of numbers, Bell Systems Tech. J., 49 (1970), pp. 1609–1625.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J.G. Hayes, Curved knot lines and surfaces with ruled segments, Numerical Analysis: Proceedings of the 9th Biennial Conference, Dundee 1981, G.A. Watson, ed., Lecture Notes in Mathematics No. 912, Springer-Verlag, Berlin, 1982, pp. 140–156.Google Scholar
  30. [30]
    P. Henrici, Applied and Computational Complex Analysis, Vol. 3, J. Wiley and Sons, New York, 1986.zbMATHGoogle Scholar
  31. [31]
    F.B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956.zbMATHGoogle Scholar
  32. [32]
    T.E. Hull, Precision control, exception handling and a choice of numerical algorithms, Lecture Notes in Mathematics No. 912, Numerical Analysis, G.A. Watson ed., Springer-Verlag, Berlin, pp. 169–178, 1982.CrossRefGoogle Scholar
  33. [33]
    T.E. Hull and M.S. Cohen, Toward an ideal computer arithmetic, Proc. 8th Symposium on Computer Arithmetic, M.J. Irwin and R. Stefanelli, eds. IEEE Computer Society Press, Washington, D.C., 1987, pp. 131–138.Google Scholar
  34. [34]
    T.E. Hull et al, Numerical Turing, ACM SIGNUM Newsletters, 20, No. 3 (1985), pp. 26–32.CrossRefGoogle Scholar
  35. [35]
    K. Hwang and F.A. Briggs, Computer Architecture and Parallel Processing, McGraw-Hill, New York, 1984.zbMATHGoogle Scholar
  36. [36]
    IBM, High-accuracy arithmetic, General Information Manual, GC33-6163-1, second edition, IBM Corporation, Mechanicsburg, PA, 1984.Google Scholar
  37. [37]
    IEEE Standard 754, Binary floating-point arithmetic, The Institute of Electrical and Electronic Engineers, New York, 1985.Google Scholar
  38. [38]
    M.J. Irwin and R. Stefanelli, eds., Proc. 8th Symposium on Computer Arithmetic, IEEE Computer Society, Washington D.C., 1987.Google Scholar
  39. [39]
    H. Kneser, Reelle analytische Lösungen der Gleichung φ(φ(x))=e x und verwandte Funktionalgleichungen, J.Reine Angew. Math., 187 (1950) pp. 56–67.MathSciNetGoogle Scholar
  40. [40]
    D.E. Knuth, The art of computer programming: Volume 2, Semi numerical algorithms, Addison-Wesley, Reading, Mass., 1969.Google Scholar
  41. [41]
    P. Kornerup and D.W. Matula, Finite precision lexicographic continued fraction number systems, Proc. 7th Symposium on Computer Arithmetic, K. Hwang, ed., IEEE Computer Society Press, Washington, D.C., 1985, pp. 207–214.Google Scholar
  42. [42]
    P. Kornerup and D.W. Matula, A bit-serial arithmetic unit for rational arithmetic, Proc. 8th Symposium on Computer Arithmetic, M.J. Irwin and R. Stenanelli, eds., IEEE Computer Society Press, Washington, D.C., 1987, pp. 204–211.Google Scholar
  43. [43]
    U.W. Kulisch and W.L. Miranker, The arithmetic of the digital computer: a new approach, SIAM Review, 28 (1986), pp. 1–40.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    D.W.Lozier and F.W.J.Olver, Closure and precision in level-index arithmetic, [Manuscript].Google Scholar
  45. [45]
    S. Matsui and M. Iri, An overflow/underflow-free floating-point representation of numbers, J.Inform. Process., 4 (1981), pp. 123–133.Google Scholar
  46. [46]
    D.W. Matula and P. Kornerup, Foundations of finite precision rational arithmetic, Foundations of Numerical Computation (Computer-Orientated Numerical Analysis), G. Alefeld and R.D. Grigorieff, eds., Computing, Suppl., 2 (1980), pp. 85–111.Google Scholar
  47. [47]
    F.W.J. Olver, The evaluation of zeros of high-degree polynomials, Phil. Trans. Royal Soc. A, 244 (1952), pp. 385–415.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    F.W.J. Olver, A new approach to error arithmetic, SIAM J. Numer. Anal. 15 (1978), pp. 368–393.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    F.W.J. Olver, Further developments of rp and ap error analysis, IMA J. Numer. Anal., 2 (1982), pp. 249–274.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    F.W.J. Olver, A closed computer arithmetic, Proc. 8th Symposium on Computer Arithmetic, M.J. Irwin and R. Stefanelli, eds., IEEE Computer Society Press, Washington, D.C., 1987, pp. 139–143.Google Scholar
  51. [51]
    F.W.J.Olver, Roundings errors in algebraic processes — in level-index arithmetic, Proceedings of a Conference on Reliable Numerical Computation (in memoriam of J.H.Wilkinson), M.G.Cox and S.Hammarling, eds., Oxford University Press. [In press].Google Scholar
  52. [52]
    F.W.J. Olver and P.R. Turner, Implementation of level-index using partial table look-up, Proc. 8th Symposium on Computer Arithmetic, M.J. Irwin and R. Stefanelli, eds., IEEE Computer Society Press, Washington, D.C., 1987, pp. 144–147.Google Scholar
  53. [53]
    R.S. Pinkham, On the distribution of first significant digits, Ann. Math. Stat., 32 (1981), pp. 1223–1230.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    R.A. Raimi, On the distribution of first significant figures, Amer. Math. Monthly, 76 (1969), pp. 342–348.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    A. Ralston, A first course in numerical analysis, McGraw-Hill, New York, 1965.zbMATHGoogle Scholar
  56. [56]
    C.W. Schelin, Calculator function approximation, Amer. Math. Monthly 90 (1983), pp. 317–325.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    P.H. Sterbenz, Floating-point Computation, Prentice-Hall, Englewood Cliffs, N.J., 1974.Google Scholar
  58. [58]
    D.W. Sweeney, An analysis of floating-point addition, IBM Systems J., 4(1965), pp. 31–42.CrossRefGoogle Scholar
  59. [59]
    P.R. Turner, The distribution of leading significant digits, IMA J. Numer. Anal. 2 (1982), pp. 407–412.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    P.R. Turner, Further revelations on l.s.d., IMA J. Numer. Anal. 4 (1984), pp. 225–231.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    P.R. Turner, Towards a fast implementation of level-index arithmetic, Bull. Inst. Math. Appl., 22 (1986), pp. 188–191.MathSciNetzbMATHGoogle Scholar
  62. [62]
    P.R. Turner, The use of level-index arithmetic to avoid overflow/underflow in floating-point computation, University of Lancaster, Tech. Rep., 1986.Google Scholar
  63. [63]
    J. Volder, The CORDIC computing technique, IRE Trans. Computers EC8 (1959), pp. 330–334.CrossRefGoogle Scholar
  64. [64]
    J.S. Walther, A unified algorithm for elementary functions, AFIPS Conference Proc., 38 (1971), pp. 379–395.Google Scholar
  65. [65]
    S. Waser and M.J. Flynn, Introduction to Arithmetic for Digital Systems Designers, Holt, Rinehart and Winston, New York, 1982.Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • C. W. Clenshaw
    • 1
  • F. W. J. Olver
    • 2
  • P. R. Turner
    • 3
    • 4
  1. 1.Mathematics DepartmentUniversity of LancasterLancasterUK
  2. 2.Institute for Physical Science and TechnologyUniversity of MarylandUSA
  3. 3.Mathematics DepartmentUniversity of LancasterLancasterUK
  4. 4.Mathematics DepartmentUS Naval AcademyAnnapolisUSA

Personalised recommendations