Numerical Analysis

  • A. C. R. Newbery

Abstract

The basic responsibilities of the numerical analyst are:
  1. 1.

    To design computer programs for the solution of numerical problems.

     
  2. 2.

    When reliable computer programs are available, to select a program which is optimally matched to a given problem.

     
  3. 3.

    To design and implement tests which will verify that a given program is behaving according to specifications, and which will give a clear indication of any weaknesses in the program.

     
  4. 4.

    To provide error estimates associated with the computerized solution of any numerical problem.

     

Keywords

Expense Hull Sine Rounding Error Estima 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 18-1.
    Anderberg, M. R., Cluster Analysis for Applications, Academic Press, New York, 1973.Google Scholar
  2. 18-2.
    Baker, C. T. H., The Numerical Treatment of Integral Equations,Clarendon, Oxford, 1977.Google Scholar
  3. 18-3.
    Björck, A., “Solving Linear Least Squares Problems by Gram-Schmidt Orthogonalization,” Nordisk Tidskr. Informationsbehandlung, vol. 7, 1–21, 1967.Google Scholar
  4. 18-4.
    Booth, A. D., Numerical Methods, Butterworth, London, 1957.Google Scholar
  5. 18-5.
    Bulirsch, R., and Stoer, J., Numerical Treatment of Ordinary Differential Equations by Extrapolation Methods,“ Numer. Math., 8, 1–13, 1966.CrossRefGoogle Scholar
  6. 18-6.
    Bunch, J. R., and Parlett, B. N., “Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations,” SIAM J. Numer. Anal., 8, 639–655, 1971.CrossRefGoogle Scholar
  7. 18-7.
    Cody, W. J., “A Survey of Practical Rational and Polynomial Approximation of Functions,” SIAM Rev.,12, 400–423, 1970.Google Scholar
  8. 18-8.
    Collatz, L., The Numerical Treatment of Differential Equations,Springer, Berlin, 1960.Google Scholar
  9. 18-9.
    Crane, P. C., and Fox, P. A., A Comparative Study of Computer Programs for Integrating Differential Equations,Bell Labs, Murray Hill, N.J., 1969.Google Scholar
  10. 18-10.
    Curtis, J. H. (ed.), Proc. Symp. Appl. Math.,Amer. Math. Soc., vol. 6, McGraw-Hill, New York, 1956.Google Scholar
  11. 18-11.
    Davis, P. J., Interpolation and Approximation,Blaisdell, New York, 1963.Google Scholar
  12. 18-12.
    Davis, P. J., and Rabinowitz, P., Numerical Integration, Blaisdell, Waltham, Mass., 1967.Google Scholar
  13. 18-13.
    de Boor, C., A Practical Guide to Splines,Springer, New York, 1978.CrossRefGoogle Scholar
  14. 18-14.
    de Boor, C., Elementary Numerical Analysis,McGraw-Hill, New York, 1980Google Scholar
  15. 18-15.
    Dold, A., and Eckmann, E. (ed.), Lecture Notes in Mathematics, Conference on the Numerical Solution of Differential Equations,Dundee, Scotland, June 1969, Springer, New York, 1969.Google Scholar
  16. 18-16.
    Donelson, J., and Hansen, E., “Cyclic composite multistep predictor-corrector methods,” SIAM J. Numer. Anal.,8, 137–157, 1971.CrossRefGoogle Scholar
  17. 18-17.
    Enright, W. H., Hull, T. E., and Lindberg, B., “Comparing Numerical Methods for Stiff Systems of O.D.E.s.,” BIT,15, 10–48, 1975.CrossRefGoogle Scholar
  18. 18-18.
    Fletcher, R., Methods for the Solution of Optimization Problems, Technical Report HL. 70/5927, Atomic Energy Research Establishment, Harwell, England, 1970.Google Scholar
  19. 18-19.
    Forsythe, G. E., “Generation and use of orthogonal polynomials for data fitting with a digital computer,” SIAM J., 5, 74–88, 1957.Google Scholar
  20. 18-20.
    Forsythe, G. E., and Moler, C. B., Computer Solution of Linear Algebraic Systems, Prentice Hall, Englewood Cliffs, N.J., 1967.Google Scholar
  21. 18-21.
    Fox, L. (ed.), Numerical Solution of Ordinary and Partial Differential Equations, Pergamon, London, 1962.Google Scholar
  22. 18-22.
    Garabedian, H. L. (ed.), Approximation of Functions, Proc. Symp. on Approximation of Functions, Elsevier, N.Y., 1964.Google Scholar
  23. 18-23.
    Gautschi, W., “Computational Aspects of Three-Term Recurrence Relations,” SIAM Rev.,9, 24–82, 1967.CrossRefGoogle Scholar
  24. 18-24.
    Gentleman, W. M., “An Error Analysis of Goertzel’s (Watt’s) Method for Computing Fourier Coefficients,” Comput. J., 12, 160–165, 1969.CrossRefGoogle Scholar
  25. 18-25.
    Greville, T. N. E. (ed.), Theory and Applications of Spline Functions, Academic Press, New York, 1969.Google Scholar
  26. 18-26.
    Hammersley, J. M., and Handscomb, D. C., Monte Carlo Methods,Methuen, London, 1964.CrossRefGoogle Scholar
  27. 18-27.
    Hamming, R. W., Introduction to Applied Numerical Analysis, McGraw-Hill, New York, 1971.Google Scholar
  28. 18-28.
    Hart, J. F. (et al.), Computer Approximations,Wiley, New York, 1968.Google Scholar
  29. 18-29.
    Hayes, J. G. (ed.), Numerical Approximation to Functions and Data, Athlone Press, London, 1970.Google Scholar
  30. 18-30.
    Henrici, P., Applied and Computational Complex Analysis,Wiley, New York, 1974.Google Scholar
  31. 18-31.
    Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962.Google Scholar
  32. 18-32.
    Hildebrand, F. B., Introduction to Numerical Analysis,McGraw-Hill, New York, 1956.Google Scholar
  33. 18-33.
    Hopper, M. J., Harwell Subroutine Library Document R.7477, Atomic Research Establishment, 1973.Google Scholar
  34. 18-34.
    Householder, A. S., The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964.Google Scholar
  35. 18-35.
    Hull, T. E., et al., “Comparing Numerical Methods for Ordinary Differential Equations,” SIAM J. Numer. Anal.,9, 603–637, 1972.CrossRefGoogle Scholar
  36. 18-36.
    IMSL, International Mathematical and Statistical Libraries, Inc., Houston, 1979.Google Scholar
  37. 18-37.
    Isaacson, E., and Keller, H. B., Analysis of Numerical Methods,Wiley, New York, 1966.Google Scholar
  38. 18-38.
    Jacobs, D. (ed.), Numerical Software Needs and Availability,Academic Press, New York, 1978.Google Scholar
  39. 18-39.
    Kowalik, J., and Osborne, M. R., Methods for Unconstrained Optimization Problems, Elsevier, New York, 1968.Google Scholar
  40. 18-40.
    Krogh, F. T., “Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation,” Math. Comput., 24, 185–190, 1970.CrossRefGoogle Scholar
  41. 18-41.
    Lambert, J. D., Computational Methods in Ordinary Differential Equations Wiley, New York, 1973.Google Scholar
  42. 18-42.
    Lancaster, P., Lambda-Matrices and Vibrating Systems, Pergamon, Oxford, 1966.Google Scholar
  43. 18-43.
    Lanczos, C., Applied Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1956.Google Scholar
  44. 18-44.
    Lapidus, L., and Seinfeld, J. H., Numerical Solution of Ordinary Differential Equations, Academic Press, N.Y., 1971.Google Scholar
  45. 18-45.
    Lyness, J. N., and Kaganove, J. J., “A Technique for Comparing Automatic Quadrature Routines,” Comput. J., 20, 170–177, 1977.CrossRefGoogle Scholar
  46. 18-46.
    Mihelcic, M., “A()-Stable Cyclic Composite Multistep Methods of Order 5,” Computing,20, 267–272, 1978.CrossRefGoogle Scholar
  47. 18-47.
    Moore, R. E., Interval Analysis, Prentice-Hall, Englewood-Cliffs, N.J., 1966.Google Scholar
  48. 18-48.
    NAG (Numerical Algorithms Group), Mathematical Subroutine Library, Oxford University Computing Centre, 1973.Google Scholar
  49. 18-49.
    National Energy Software Center, EISPACK, Argonne National Lab.Google Scholar
  50. 18-50.
    National Energy Software Center, UNPACK, Argonne National Lab., 1979.Google Scholar
  51. 18-51.
    Newbery, A. C. R., “Convergence of Successive Substitution Procedures,” Math. Comput., 21, 489–490, 1967.Google Scholar
  52. 18-52.
    Newbery, A. C. R., “Trigonometric Interpolation and Curve-Fitting,” Math. Comput., 24, 869–876, 1970.CrossRefGoogle Scholar
  53. 18-53.
    Oettli, W., and Prager, W., “Compatibility of Approximate Solutions of Linear Equations with Given Error Bounds for Coefficients and Right-hand Sides,” Numer. Math., 6, 405–409, 1964.CrossRefGoogle Scholar
  54. 18-54.
    Oliver, J., “The Numerical Solution of Linear Recurrence Relations,” Numer. Math.,11, 349–360, 1968.CrossRefGoogle Scholar
  55. 18-55.
    Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.Google Scholar
  56. 18-56.
    Pearson, Carl E., “On Non-linear Ordinary Differential Equations of Boundary Layer Type,” Studies in Appl. Math., 47, 351–358, 1968.Google Scholar
  57. 18-57.
    Powell, M. J. D., A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations, H. M. Stationery Office, London, 1968.Google Scholar
  58. 18-58.
    Powell, M. J. D., “On the Maximum Errors of Polynomial Approximations Defined by Interpolation and by Least Squares Criteria,” Comput. J., 9, 404–407, 1967.Google Scholar
  59. 18-59.
    Powell, M. J. D., “A Survey of Numerical Methods for Unconstrained Optimization,” SIAM Rev.,12, 79–97, 1970.CrossRefGoogle Scholar
  60. 18-60.
    Powell, M. J. D., “A Theorem on Rank-One Modifications to a Matrix and Its Inverse,” Comput. J., 12, 288–290, 1969.Google Scholar
  61. 18-61.
    Ralston, A., A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.Google Scholar
  62. 18-62.
    Ralston, A., and Wilf, H. S., Mathematical Methods for Digital Computers, Vol. II, Wiley, New York, 1967.Google Scholar
  63. 18-63.
    Reinsch, C. H., “Smoothing by Spline Functions,” Numer. Math., 10, 177–183, 1967.CrossRefGoogle Scholar
  64. 18-64.
    Secrest, D., “Numerical Integration of Arbitrarily Spaced Data and Estimation of Errors,” SIAM J. Numer. Anal., 2, 52–68, 1965.Google Scholar
  65. 18-65.
    Späth, H., Spline Algorithms for Curves and Surfaces, translated by W. D. Hoskins and H. W. Sager, Utilias Mathematica, Winnipeg, 1974.Google Scholar
  66. 18-66.
    Späth, H., Cluster Analysis Algorithms, translated by Ursula Bull, Ellis Horwood, New York, 1980.Google Scholar
  67. 18-67.
    Todd, J. (ed.), A Survey of Numerical Analysis,McGraw-Hill, New York, 1962.Google Scholar
  68. 18-68.
    Traub, J. F., Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, N.J., 1964.Google Scholar
  69. 18-69.
    Varga, R. S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962.Google Scholar
  70. 18-70.
    Wilkinson, J. H., The Algebraic Eigenvalue problem, Clarendon, Oxford, 1965.Google Scholar
  71. 18-71.
    Wilkinson, J. H., Rounding Errors in Algebraic Processes, H. M. Stationery Office, London, 1963.Google Scholar
  72. 18-72.
    Winograd, S., “A New Algorithm for Inner Product,” IEEE Trans. Computers, 693–694, 1968.Google Scholar

Bibliography

  1. Abramowitz, M., and Stegun, I. A. (ed.) Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.Google Scholar

Copyright information

© Van Nostrand Reinhold 1990

Authors and Affiliations

  • A. C. R. Newbery
    • 1
  1. 1.Dep’t. of Computer ScienceUniversity of KentuckyLexingtonUSA

Personalised recommendations