Skip to main content
Log in

Central orderings for the Newton interpolation formula

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others


  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. 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)

  3. Leja, F.: Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme. Ann. Polon. Math. 4, 8–13 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  4. Reichel, L.: Newton interpolation at Leja points. BIT 30, 332–346 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rivlin, T.J.: Chebyshev polynomials. From approximation theory to algebra and number theory. Pure and Applied Mathematics. Wiley, New York (1990)

    MATH  Google Scholar 

  6. Schönhage, A.: Fehlerfortpflanzung bei Interpolation. Numer. Math. 3, 62–71 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  7. Trefethen, L.N., Weideman, J.A.C.: Two results on polynomial interpolation in equally spaced points. J. Approx. Theory 65, 247–260 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Y. Khiar.

Additional information

Communicated by Michael S. Floater.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carnicer, J.M., Khiar, Y. & Peña, J.M. Central orderings for the Newton interpolation formula. Bit Numer Math 59, 371–386 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification