Kybernetika 54 no. 2, 279-288, 2018

A numerical study of Newton interpolation with extremely high degrees

Michael Breuß, Friedemann Kemm and Oliver VogelDOI: 10.14736/kyb-2018-2-0279


In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort. However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation. This is an experimental account of the analytic result that the limit distribution of FL points and Chebyshev nodes is the same when letting the number of points go to infinity. Since the FL construction is easy to perform and allows to add interpolation nodes on the fly in contrast to the use of Chebyshev nodes, our study suggests that Newton interpolation with FL points is currently the most efficient numerical technique for polynomial interpolation. Moreover, we give numerical evidence that any reasonable function can be approximated up to machine accuracy by Newton interpolation with FL points if desired, which shows the potential of this method.


polynomial interpolation, Newton interpolation, interpolation nodes, Chebyshev nodes, Leja ordering, fast Leja points


65-05, 65D05, 97N50


  1. K. E. {Atkinson}: An Introduction to Numerical Analysis. Second edition. John Wiley and Sons, Inc., New York 1989.   CrossRef
  2. J. Baglama, D. Calvetti and L. Reichel: Iterative methods for the computation of a few eigenvalues of a large symmetric matrix. BIT 36 (1996), 3, 400-421.   DOI:10.1007/bf01731924
  3. J. Baglama, D. Calvetti and L. Reichel: Fast Leja points. ETNA, Electron. Trans. Numer. Anal. 7 (1998), 124-140.   CrossRef
  4. D. Calvetti and L. Reichel: Adaptive Richardson iteration based on Leja points. J. Comput. Appl. Math. 71 (1996), 2, 267-286.   DOI:10.1016/0377-0427(96)87162-7
  5. D. Calvetti and L. Reichel: On the evaluation of polynomial coefficients. Numer. Algorithms 33 (2003), 1-4, 153-161.   DOI:10.1023/a:1025555803588
  6. C. de Boor: A Practical Guide to Splines. Revised edition. Springer-Verlag, Inc., New York 2001.   CrossRef
  7. A. Eisinberg and G. Fedele: On the inversion of the Vandermonde matrix. Appl. Math. Comput. 174 (2006), 2, 1384-1397.   DOI:10.1016/j.amc.2005.06.014
  8. W. Gautschi: Numerical Analysis. An Introduction. Birkhäuser, Boston 1997.   CrossRef
  9. N. J. Higham: Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM J. Matrix Anal. Appl. 11 (1990), 1, 23-41.   DOI:10.1137/0611002
  10. W. G. Horner: A new method of solving numerical equations of all orders, by continuous approximation. In: Philosophical Transactions of the Royal Society of London, 1819, pp. 308-335.   DOI:10.1098/rstl.1819.0023
  11. I. P. Natanson: Konstruktive Funktionentheorie. Mathematische Lehrbücher und Monographien. I. Abteilung, Bd. VII., Akademie-Verlag. XIV, 515 S., 2. Abb. (1955), Berlin 1955.   CrossRef
  12. L. Reichel: Newton interpolation at Leja points. BIT 30 (1990), 2, 332-346.   DOI:10.1007/bf02017352
  13. C. Runge: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Schlömilch Z. 46 (1901), 224-243.   CrossRef
  14. H. Tal-Ezer: High degree polynomial interpolation in Newton form. SIAM J. Sci. Stat. Comput. 12 (1991), 3, 648-667.   DOI:10.1137/0912034
  15. L. N. Trefethen: Approximation Theory and Approximation Practice. PA: Society for Industrial and Applied Mathematics (SIAM), Philadelphia 2013.   CrossRef