Skip to main content
SpringerLink
Log in

Stable Lagrangian numerical differentiation with the highest order of approximation

  • Published:
Science in China Series A Aims and scope Submit manuscript

Abstract

Some asymptotic representations for the truncation error for the Lagrangian numerical differentiation are presented, when the ratio of the distance between each interpolation node and the differentiated point to step-parameter h is known. Furthermore, if the sampled values of the function at these interpolation nodes have perturbations which are bounded by ε, a method for determining step-parameter h by means of perturbation bound ε and order n of interpolation is provided to saturate the order of approximation. And all the investigations in this paper can be generalized to the set of quasi-uniform nodes.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Wang, X. H., Numerical Differentiation (in Chinese), Encyclopedia of China (Mathematics Section), Beijing-Shanghai: Encyclopedia of China Publishing House, 1988.

    Google Scholar 

  2. Wang, X. H., The remainder of numerical differentiation formula, Hangzhou Daxue Xuebao (in Chinese), 1978, 5(1): 1–10. An announcement of the results, Kexue Tongbao, 1979, 24(19): 859-872.

    Google Scholar 

  3. Wang, X. H., Lai, M. J., Yang, S. J., On the divided differences of the remainder in polynomial interpolation, J. Approx. Theory, 2004, 127: 193–197.

    Article  MathSciNet  Google Scholar 

  4. Groetsch, C. W., Differentiation of approximately specified functions, Amer. Math. Monthly, 1991, 98: 847–850.

    MATH  MathSciNet  Google Scholar 

  5. Groetsch, C. W., Optimal order of accuracy in Vasin's method for differentiation of noisy functions, J. Optim. Theory Appl., 1992, 74(2): 373–378.

    Article  MATH  MathSciNet  Google Scholar 

  6. Yang, H. Q., Li, Y. S., Stable approximation method for differentiation of approximate known function, Progress in Natural Science (in Chinese), 2000, 10(12): 1088–1093.

    Google Scholar 

  7. Kantorovich, L. V., Krylov, V. I., Approximate Methods of Higher Analysis, New York: Interscience Publishers Inc, 1958.

    Google Scholar 

  8. Dokken, T., Lyche, T., A divided difference formula for the error in Hermite interpolation, BIT, 1979, 19: 540–541.

    Article  MathSciNet  Google Scholar 

  9. Hopf, E., Über die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften, Dissertation Universität Berlin, 1926.

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Brodskii, M. L., On estimation of remainders in the numerical differentiation formulae, Usp. Mat. Nauk. (in Russian), 1958, 13(6): 73–77.

    MATH  MathSciNet  Google Scholar 

  13. Shadrin, A., Interpolation by Lagrange polynomials, B-splines and bounds of errors, Anal. Math., 1994, 20: 213–224.

    MATH  MathSciNet  Google Scholar 

  14. Shadrin, A. Error bounds for Lagrange interpolation, J. Approx. Theory, 1995, 80: 25–49.

    Article  MATH  MathSciNet  Google Scholar 

  15. Salzer, H. E., Some problems in optimally stable Lagrangian differentiation, Math. Comp., 1974, 28: 1105–1115.

    MATH  MathSciNet  Google Scholar 

  16. Rivlin, T. J., Optimally stable Lagrangian numerical differentiation, SIAM J. Numer. Analysis, 1975, 12(5): 712–725.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310028, China

    Wang Xinghua & Cui Feng

Authors
  1. Wang Xinghua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cui Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wang Xinghua.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, X., Cui, F. Stable Lagrangian numerical differentiation with the highest order of approximation. SCI CHINA SER A 49, 225–232 (2006). https://doi.org/10.1007/s11425-005-0022-4

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-005-0022-4

Keywords

Search

Navigation