Skip to main content
Log in

On the Condition Number of the Newton Interpolation on the Unit Disk

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

The stability of representations of univariate Lagrange interpolation polynomials in the complex plane is measured through a condition number. We study the growth of the condition number of the Newton formula for Lagrange interpolation. We prove that the condition number of Newton’s formula at the first n points of a Leja sequence for the closed unit disk \(\overline{{\mathbb {D}}}\) is bounded by \((n^3+2n-3)/3\) from above and by n/2 from below. We also point out that the condition number corresponding to any \(n+1\) distinct points on the unit circle is greater than \(n^c\), where \(0<c<1\) is an absolute constant.

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.

Similar content being viewed by others

Data availability

The manuscript has no associated data.

References

  1. Andrievskii, V., Nazarov, F.: A simple upper bound for Lebesgue constants associated with Leja points on the real line. J. Approx. Theory 275, Paper No. 105699, 13 pp (2022)

  2. Beck, J.: The modulus of polynomials with zeros on the unit circle: a problem of Erdös. Ann. Math. 2(134), 609–651 (1991)

    Article  MathSciNet  Google Scholar 

  3. Bialas-Ciez, L., Calvi, J.-P.: Pseudo Leja sequences. Ann. Mat. Pura e Appl. 191, 53–75 (2012)

    Article  MathSciNet  Google Scholar 

  4. Calvi, J.-P., Phung, V.M.: On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation. J. Approx. Theory 163, 608–622 (2011)

    Article  MathSciNet  Google Scholar 

  5. Camargo, A.P.: On the condition number of Newton interpolation: Chebyshev points and monotonically ordering. J. Approx. Theory 287, Paper No. 105876, 13 pp (2023)

  6. Camargo, A. P.: On the numerical stability of Newton’s formula for Lagrange interpolation. J. Comput. Appl. Math. 365 112369, 12 pp (2020)

  7. Carnicer, J.M., Khiar, Y., Peña, J.M.: Optimal stability of the Lagrange formula and conditioning of the Newton formula. J. Approx. Theory 238, 52–66 (2019)

    Article  MathSciNet  Google Scholar 

  8. Carnicer, J.M., Khiar, Y., Peña, J.M.: Central orderings for the Newton interpolation formula. BIT 59(2), 371–386 (2019)

    Article  MathSciNet  Google Scholar 

  9. Carnicer, J.M, Khiar, Y., Peña, J.M: Inverse central ordering for the Newton interpolation formula. Numer. Algorithms 90, no. 4, 1691–1713 (2022)

  10. Erdös, P.: Problems and results on diophantine approximation. Compos. Math. 16, 52–66 (1964)

    MathSciNet  Google Scholar 

  11. Fischer, B., Reichel, L.: Newton interpolation in Fejér and Chebyshev points. Math. Comp. 53(187), 265–278 (1989)

    MathSciNet  Google Scholar 

  12. Dinh, H.L., Le, N.C., Phung, V.M, Nguyen, V.M.: On the conditioning of the Newton formula for Lagrange interpolation. J. Math. Anal. Appl. 505, no. 1, Paper No. 125473, 14 pp (2022)

  13. Erdös, P., Turán, P.: On interpolation. III. Interpolatory theory of polynomials, Ann. Math. (2), 41, 510–553 (1940)

  14. Linden, C.: The modulus of polynomials with zeros on the unit circle. Bull. Lond. Math. Soc. 9, 65–69 (1977)

    Article  MathSciNet  Google Scholar 

  15. Ounaïes, M.: A sharp bound on the Lebesgue constant for Leja points in the unit disk. J. Approx. Theory 213, 70–77 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Totik, V.: The Lebesgue constants for Leja points are subexponential. J. Approx. Theory 287, Paper No. 105863, 15 pp (2023)

Download references

Acknowledgements

We are grateful to the referees for their careful reading and constructive comments. This research is funded by Thuongmai University, Hanoi, Vietnam.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Phung Van Manh.

Additional information

Communicated by Lothar Reichel.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tung, P.T., Cuong, L.N. & Van Manh, P. On the Condition Number of the Newton Interpolation on the Unit Disk. Comput. Methods Funct. Theory 24, 427–440 (2024). https://doi.org/10.1007/s40315-023-00497-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-023-00497-1

Keywords

Mathematics Subject Classification

Navigation