Skip to main content
SpringerLink
Log in

Numerical differentiation procedures for non-exact data

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The numerical differentiation of data divides naturally into two distinct problems:

  1. (i)

    the differentiation of exact data, and

  2. (ii)

    the differentiation of non-exact (experimental) data.

In this paper, we examine the latter. Because methods developed for exact data are based on abstract formalisms which are independent of the structure within the data, they prove, except for the regularization procedure of Cullum, to be unsatisfactory for non-exact data. We therefore adopt the point of view that satisfactory methods for non-exact data must take the structure within the data into account in some natural way, and use the concepts of regression and spectrum analysis as a basis for the development of such methods. The regression procedure is used when either the structure within the non-exact data is known on independent grounds, or the assumptions which underlie the spectrum analysis procedure [viz., stationarity of the (detrended) data] do not apply. In this latter case, the data could be modelled using splines. The spectrum analysis procedure is used when the structure within the nonexact data (or a suitable transformation of it, where the transformation can be differentiated exactly) behaves as if it were generated by a stationary stochastic process. By proving that the regularization procedure of Cullum is equivalent to a certain spectrum analysis procedure, we derive a fast Fourier transform implementation for regularization (based on this equivalence) in which an acceptable value of the regularization parameter is estimated directly from a time series formulation based on this equivalence. Compared with the regularization procedure, which involvesO(n 3) operations (wheren is the number of data points), the fast Fourier transform implementation only involvesO(n logn).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anderssen, R. S., Bloomfield, P.: A time series approach to numerical differentiation. Technometrics (in press)

  2. Anderssen, R. S., Seneta, E.: On smoothing techniques for the removal of periodic noise of known period. Math. Geol.3, 157–170 (1971)

    Google Scholar 

  3. Bickley, W. G.: Formulae for numerical differentiation. Math. Gaz.25, 19–27 (1941)

    Google Scholar 

  4. Birkhoff, G., de Boor, Carl: Error bounds for spline interpolation. J. Math. and Mech.13, 827–835 (1964)

    Google Scholar 

  5. Björck, Åke, Pereyra, Victor: Solution of Van der Monde systems of equations. Math. Comp.24, 893–903 (1970)

    Google Scholar 

  6. de Boor, C., Rice, J. R.: Least squares cubic spline approximation, I—Fixed Knots, II—Variable Knots. Computer Science Department Tech. Report #20 and 21, Purdue University, April, 1968

  7. Box, G. E. P., Jenkins, G. M.: Time series analysis, forecasting and control, San Francisco. Holden-Day 1970

    Google Scholar 

  8. Brodskii, M. L.: Estimation of the remainder term in formulas of numerical differentiation [Russian], Uspekhi Mat. Nauk13 (6), 73–77 (1958)

    Google Scholar 

  9. Burchard, H. G.: Splines (with optimal knots) are better. Applicable Anal. (1973), to appear

  10. Chakravarti, P. C.: Truncation error in interpolation and numerical differentiation, Numer. Math.3, 279–284 (1961)

    Article  Google Scholar 

  11. Cooley, J. W., Tukey, J. W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp.19, 297–301 (1965)

    Google Scholar 

  12. Cullum, J.: Regularization and numerical differentiation. IBM Research Report RC2605, Sept. 1969

  13. Cullum, J.: Numerical differentiation and regularization. SIAM J. Numer. Anal.8, 254–265 (1971)

    Article  Google Scholar 

  14. Dolgopolova, T. F., Ivanov, V. K.: On numerical differentiation. USSR Comp. Math. and Math. Phys.6 (4), 223–232 (1966)

    Article  Google Scholar 

  15. Doob, J. L.: Stochastic processes. New York: John Wiley 1953

    Google Scholar 

  16. Engels, H.: Zur Anwendung der Richardson-Extrapolation auf die numerische Differentiation. Computing8, 255–272 (1971)

    Google Scholar 

  17. Galimberti, G., Pereyra, V.: Numerical differentiation and the solution of multidimensional Van der Monde systems. Math. Comp.24, 357–364 (1970)

    Google Scholar 

  18. Gentleman, W. M., Sande, G.: Fast Fourier transforms—for fun and profit. Proc. Fall Joint Computer Conference (1966), 563–578

  19. Grenander, U., Rosenblatt, M.: Statistical analysis of stationary time series. New York: John Wiley 1957

    Google Scholar 

  20. Hamming, R. W.: Numerical methods for scientists and engineers. New York: McGraw-Hill 1963

    Google Scholar 

  21. Hamming, R. W.: The frequency approach to numerical analysis. In: Studies in numerical analysis: Papers presented to Cornelius Lanczos. London: Academic Press 1973

    Google Scholar 

  22. Hannan, E. J.: Time series analysis. London: Methuen 1960

    Google Scholar 

  23. Hannan, E. J.: Multiple time series. New York: John Wiley 1970

    Google Scholar 

  24. Hestenes, M.: Calculus of variations and optimal control theory. New York: John Wiley 1966

    Google Scholar 

  25. de Hoog, F., Weiss, R.: High order methods for first kind Volterra equations. SIAM J. Numer. Anal.10, 647–664 (1973)

    Article  Google Scholar 

  26. de Hoog, F., Weiss, R.: On the solution of Volterra integral equations of the first kind. Numer. Math.21, 22–32 (1973)

    Google Scholar 

  27. Hunter, D. B.: An iterative method of numerical differentiation. Computer J.3, 270–271 (1960)

    Google Scholar 

  28. Jenkins, G. M., Watts, D. G.: Spectral analysis and its application. San Francisco: Holden-Day 1968

    Google Scholar 

  29. Khintchine, A. Ya.: Korrelationstheorie der stationaren stochastischen Prozesse. Math. Ann.109, 604–616 (1934)

    Google Scholar 

  30. Kranzer, H. C.: An error formula for numerical differentiation. Numer. Math.5, 439–442 (1963)

    Article  Google Scholar 

  31. Lyness, J. N.: Numerical approximations based on the theory of complex variable. Proc. 22nd Nat. Conf. ACM Publication P-67, (1967), pp. 125–133

  32. Lyness, J. N.: Differentiation formulas for analytic functions. Math. Comp.22, 352–362 (1968)

    Google Scholar 

  33. Lyness, J. N., Moler, C. B.: Numerical differentiation of analytic functions. SIAM J. Numer. Anal.4, 202–210 (1966)

    Article  Google Scholar 

  34. Lyness, J. N., Moler, C. B.: Van der Monde systems and numerical differentiation. Numer. Math.8, 458–464 (1966)

    Google Scholar 

  35. Mood, A. M., Graybill, F. A.: Introduction to the theory of statistics, 2nd edition. New York: McGraw-Hill 1963

    Google Scholar 

  36. Pallaghy, C. K., Lüttge, U.: Light-induced and H+-ion fluxes and bioelectric phenomena in mesophyll cells of Atriplex spongiosa. Zeit. für Pflanz.62, 417–425 (1970)

    Google Scholar 

  37. Powell, M. J. D.: The local dependence of least squares cubic splines. SIAM J. Numer. Anal.6, 398–413 (1969)

    Article  Google Scholar 

  38. Ralston, A.: A first course in numerical analysis. New York: McGraw-Hill 1965

    Google Scholar 

  39. Salzer, H. E.: Optimal points for numerical differentiation. Numer. Math.2, 214–227 (1960)

    Article  Google Scholar 

  40. Salzer, H. E.: Equally-weighted formulas for numerical differentiation. Numer. Math.4, 381–392 (1963)

    Article  Google Scholar 

  41. Schönhage, A.: Optimale Punkte für Differentiation und Integration. Numer. Math.5, 303–331 (1963)

    Article  Google Scholar 

  42. Secrest, Don: Error bounds for interpolation and differentiation by the use of spline functions. SIAM J. Numer. Anal.2, 440–447 (1965)

    Article  Google Scholar 

  43. Strom, Torsten: An inclusion formula for derivatives. BIT11, 196–198 (1971)

    Google Scholar 

  44. Tsuda, T.: Numerical differentiation of functions of very many variables. Numer. Math.18, 327–335 (1972)

    Article  Google Scholar 

  45. Tukey, J. W.: An introduction to the calculations of numerical spectrum analysis. In: The spectral analysis of time series, ed. B. Harris. New York: John Wiley 1967

    Google Scholar 

  46. Vedeneev, E. P., Zhidkov, N. P.: The application of the method of regularization to the differentiation of a function of one variable, given as data [Russian]. Numerical Methods and Programming3, 255–261 (1969)

    Google Scholar 

  47. Wiener, N.: Generalised harmonic analysis. Acta Math.55, 117–258 (1930)

    Google Scholar 

  48. Wilkinson, J. H., Reinsch, C.: Linear Algebra, Die Grundlehren der mathematischen Wissenschaften, Band 186. Berlin-Heidelberg-New York: Springer 1971

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Centre, ANU, P.O. Box 4, 2600, Canberra, ACT, Australia

    R. S. Anderssen

  2. Statistics, Fine Hall, 08540, Princeton, NJ, USA

    P. Bloomfield

Authors
  1. R. S. Anderssen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Bloomfield
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was performed with the support of (a) the Office of Naval Research under contract 0014-67-A-0151-0017 and (b) the U.S. Army Research Office—Durham, under contract DA-31-124-ARO-D-215. An earlier version appeared in preprint form as Technical Report 13, Series 2, Department of Statistics, Princeton University, April, 1972.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anderssen, R.S., Bloomfield, P. Numerical differentiation procedures for non-exact data. Numer. Math. 22, 157–182 (1974). https://doi.org/10.1007/BF01436965

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01436965

Keywords

Search

Navigation