Central orderings for the Newton interpolation formula

  • J. M. Carnicer
  • Y. KhiarEmail author
  • J. M. Peña


The stability properties of the Newton interpolation formula depend on the order of the nodes and can be measured through a condition number. Increasing and Leja orderings have been previously considered (Carnicer et al. in J Approx Theory, 2017.; Reichel in BIT 30:332–346, 1990). We analyze central orderings for equidistant nodes on a bounded real interval. A bound for conditioning is given. We demonstrate in particular that this ordering provides a more stable Newton formula than the natural increasing order. We also analyze of a central ordering with respect to the evaluation point, which provides low bounds for the conditioning. Numerical examples are included.


Newton interpolation formula Conditioning Central ordering 

Mathematics Subject Classification

65D05 65F35 41A05 41A10 



This work has been partially supported by the Spanish Research Grant MTM2015-65433-P (MINECO/FEDER), BES-2013-065398B (MINECO), by Gobierno the Aragón and Fondo Social Europeo.


  1. 1.
    Carnicer, J.M., Khiar, Y., Peña, J.M.: Optimal stability of the Lagrange formula and conditioning of the Newton formula, to appear in J. Approx. Theory.
  2. 2.
    Corless, R.M., Watt, S.M.: Bernstein bases are optimal, but, sometimes, Lagrange bases are better. In: Proc. SYNASC (Symbolic and Numeric Algorithms for Scientific Computing), pp. 141–152, Timisoara (2004)Google Scholar
  3. 3.
    Leja, F.: Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme. Ann. Polon. Math. 4, 8–13 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Reichel, L.: Newton interpolation at Leja points. BIT 30, 332–346 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rivlin, T.J.: Chebyshev polynomials. From approximation theory to algebra and number theory. Pure and Applied Mathematics. Wiley, New York (1990)zbMATHGoogle Scholar
  6. 6.
    Schönhage, A.: Fehlerfortpflanzung bei Interpolation. Numer. Math. 3, 62–71 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Trefethen, L.N., Weideman, J.A.C.: Two results on polynomial interpolation in equally spaced points. J. Approx. Theory 65, 247–260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada/IUMAUniversidad de ZaragozaZaragozaSpain

Personalised recommendations