Interpolating and Extrapolating

  • Éric WalterEmail author


Interpolating is not always a good idea and one should be especially careful when extrapolating. These operations are nevertheless sometimes very useful. Classical methods are described for the univariate and multivariate case. Conditions that make very high-degree polynomial interpolation a viable option are explained. Pros and cons of rational interpolation are presented. Richardson’s extrapolation principle is described. It is used, for instance, in numerical integration and differentiation as well as for solving differential equations. Kriging, a multivariate interpolation method initially developed in the context of mining, receives special attention. It is increasingly used in computer experiments to build surrogate models for functions that are very costly to evaluate.


Input Factor Polynomial Interpolation Interpolation Point Vandermonde Matrix Rational Interpolation 
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.


  1. 1.
    Trefethen, L.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2013)zbMATHGoogle Scholar
  2. 2.
    Trefethen, L.: Six myths of polynomial interpolation and quadrature. Math. Today 47, 184–188 (2011)MathSciNetGoogle Scholar
  3. 3.
    Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments (with discussion). Stat. Sci. 4(4), 409–435 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Farouki, R.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. Des. 29, 379–419 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Berrut, J.P., Trefethen, L.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Higham, N.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24(4), 547–556 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    de Boor, C.: Package for calculating with B-splines. SIAM J. Numer. Anal. 14(3), 441–472 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)CrossRefGoogle Scholar
  9. 9.
    de Boor, C.: A Practical Guide to Splines, revised edn. Springer, New York (2001)Google Scholar
  10. 10.
    Kershaw, D.: A note on the convergence of interpolatory cubic splines. SIAM J. Numer. Anal. 8(1), 67–74 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Wahba, G.: Spline Models for Observational Data. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar
  12. 12.
    Cressie, N.: Statistics for Spatial Data. Wiley, New York (1993)Google Scholar
  13. 13.
    Krige, D.: A statistical approach to some basic mine valuation problems on the Witwatersrand. J. Chem. Metall. Min. Soc. 52, 119–139 (1951)Google Scholar
  14. 14.
    Chilès, J.P., Delfiner, P.: Geostatistics. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Wackernagel, H.: Multivariate Geostatistics, 3rd edn. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  16. 16.
    Vazquez, E., Walter, E.: Estimating derivatives and integrals with Kriging. In: Proceedings of 44th IEEE Conference on Decision and Control (CDC) and European Control Conference (ECC), pp. 8156–8161. Seville, Spain (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire des Signaux et SystèmesCNRS-SUPÉLEC-Université Paris-SudGif-sur-YvetteFrance

Personalised recommendations