Sequential Barycentric Interpolation

Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


Polynomial interpolators may exhibit oscillating behaviour which often makes them inadequate for modelling functions. A well-known correction to this problem is to use Chebyshev design points. However, in a sequential strategy it is not very clear how to add points, while still improving polynomial interpolation. We present a sequential design alternative by allocating an extra observation where the difference between consecutive interpolators is largest. Our proposal is independent of the response and does not require distributional assumptions. In simulated examples, we show the good interpolation performance of our proposal and its asymptotical convergence to the Chebyshev distribution.


Design Point Design Region Large Sample Property Chebyshev Point Sequential Interpolation 


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  1. Berrut, J.-P. and L. N. Trefethen (2004). Barycentric Lagrange interpolation. SIAM Review 46, 501–517.MATHCrossRefMathSciNetGoogle Scholar
  2. Boyd, J. P. and J. R. Ong (2009). Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions. I. Single-interval schemes. Communications in Computational Physics 5, 484–497.MathSciNetGoogle Scholar
  3. Boyd, J. P. and F. Xu (2009). Divergence (Runge phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation. Applied Mathematics and Computation 210, 158–168.MATHCrossRefMathSciNetGoogle Scholar
  4. Bratley, P. and B. L. Fox (1988). ALGORITHM 659 Implementing Sobol’s quasirandom sequence generator. ACM Transactions on Mathematical Software 14, 88–100.MATHCrossRefGoogle Scholar
  5. Dette, H. (1993a). A note on E-optimal designs for weighted polynomial regression. Annals of Statistics 21, 767–771.MATHCrossRefMathSciNetGoogle Scholar
  6. Dette, H. (1993b). On a mixture of the D- and D1-optimality criterion in polynomial regression. Journal of Statistical Planning and Inference 35, 233–249.MATHCrossRefMathSciNetGoogle Scholar
  7. Epperson, J. F. (1987). On the Runge example. American Mathematical Monthly 94, 329–341.MATHCrossRefMathSciNetGoogle Scholar
  8. Fedorov, V. V. (1972). Theory of Optimal Experiments. New York: Academic Press.Google Scholar
  9. Fedorov, V. V. and W. G. Müller (2007). Optimum design for correlated fields via covariance kernel expansions. In J. Lopez-Fidalgo, J. M. Rodriguez-Diaz, and B. Torsney (Eds.), mODa 8–-Advances in Model-oriented Design and Analysis, pp. 57–66. Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
  10. Heiligers, B. (1998). E-optimal designs for polynomial spline regression. Journal of Statistical Planning and Inference 75, 159–172.MATHCrossRefMathSciNetGoogle Scholar
  11. Karlin, S. and W. J. Studden (1966). Optimal experimental designs. Annals of Mathematical Statistics 37, 783–815.MATHCrossRefMathSciNetGoogle Scholar
  12. Müller, W. G. (2001). Collecting Spatial Data, rev. edn. Heidelberg: Physica-Verlag.Google Scholar
  13. O’Hagan, A. (2006). Bayesian analysis of computer code outputs: a tutorial. Reliability Engineering & System Safety 37, 1290–1300.CrossRefGoogle Scholar
  14. Pázman, A. (1986). Foundations of Optimum Experimental Design. Dordrecht: Reidel.MATHGoogle Scholar
  15. Platte, R. B. and T. A. Driscoll (2005). Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM Journal on Numerical Analysis 43, 750–766.MATHCrossRefMathSciNetGoogle Scholar
  16. Pukelsheim, F. (1993). Optimal Design of Experiments. New York: Wiley.MATHGoogle Scholar
  17. Pukelsheim, F. and B. Torsney (1991). Optimal weights for experimental designs on linearly independent support points. Annals of Statistics 19, 1614–1625.MATHCrossRefMathSciNetGoogle Scholar
  18. Studden, W. J. (1968). Optimal designs on Tchebycheff points. Annals of Mathematical Statistics 39, 1435–1447.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Queen MaryUniversity of LondonLondonUK
  2. 2.Departamento de Estadística e Investigación OperativaUniversidad de ValladolidValladolidSpain

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