Abstract
A new backward stable algorithm (Algorithm 2) for polynomial interpolation based on the Lagrange and the Newton interpolation forms is proposed. It is shown that the Aitken algorithm and the scheme of the divided differences can be significantly less accurate than the proposed unconditionally stable Algorithm 2. Numerical examples that illustrate the advantage of a new algorithm are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Baglama D. Calvetti L. Reichel (1998) ArticleTitleFast Leja points ETNA 7 124–140 Occurrence Handle0912.65004 Occurrence Handle1667643
Å. Björck V. Pereyra (1970) ArticleTitleSolution of Vandermonde systems of equations Math. Comput. 24 IssueID112 893–903 Occurrence Handle10.2307/2004623
Brent, R. P.: Stability of fast algorithms for structured linear systems. Technical Report TR-CS-97-18, ANU, September 1997.
D. Calvetti L. Reichel (2003) ArticleTitleOn the evaluation of polynomial coefficients Numer. Algorith. 33 153–161 Occurrence Handle1035.65156 Occurrence Handle10.1023/A:1025555803588 Occurrence Handle2005559
G. Dahlquist Å. Björck (1974) Numerical methods Prentice-Hall Englewood Cliffs, NJ
N. J. Higham (1996) Accuracy and stability of numerical algorithms SIAM Philadelphia Occurrence Handle0847.65010
J. Jankowska M. Jankowski (1981) Przeglad metod i algorytmów numerycznych, cz. 1 WNT Warszawa
W. Kahan I. Farkas (1963) ArticleTitleAlgorithm 167-calculation of confluent divided differences Comm. ACM 6 164–165 Occurrence Handle10.1145/366349.366444
F. Leja (1957) ArticleTitleSur certaines suits liées aux ensemble plan et leur application à la representation conforme Ann. Polon. Math. 4 8–13 Occurrence Handle0089.08303 Occurrence Handle100726
A. McCurdy K. C. Ng B. N. Parlett (1984) ArticleTitleAccurate computation of divided differences of the exponential function Math. Comput. 43 IssueID168 501–528 Occurrence Handle0561.65009 Occurrence Handle10.2307/2008291 Occurrence Handle758198
H. J. Rack M. Reimer (1982) ArticleTitleThe numerical stability of evaluation schemes for polynomials based on the Lagrange interpolation form BIT 22 101–107 Occurrence Handle0477.65007 Occurrence Handle10.1007/BF01934399 Occurrence Handle654746
L. Reichel (1990) ArticleTitleNewton interpolation at Leja points BIT 30 332–346 Occurrence Handle0702.65012 Occurrence Handle10.1007/BF02017352 Occurrence Handle1039671
Smoktunowicz, A.: Stability issues for special algebraic problems. Ph.D. thesis, University of Warsaw, 1981 (in Polish).
A. Smoktunowicz (2002) ArticleTitleBackward stability of Clenshaw’s algorithm BIT 42 IssueID3 600–610 Occurrence Handle1019.65004 Occurrence Handle10.1023/A:1022001931526 Occurrence Handle1931888
Smoktunowicz, A., Kosowski, P., Wróbel, I.: How to overcome the numerical instability of the scheme of divided differences? arXiv.org/pdf/math.NA/0407195, 2004.
A. Smoktunowicz I. Wróbel (2005) ArticleTitleOn improving the accuracy of Horner’s and Goertzel’s algorithms Numer. Algorith. 38 243–258 Occurrence Handle1075.65034 Occurrence Handle10.1007/s11075-004-4570-4
H. Tal-Ezer (1991) ArticleTitleHigh degree polynomial interpolation in Newton form SIAM J. Sci. Stat. Comput. 12 648–667 Occurrence Handle0728.65009 Occurrence Handle10.1137/0912034 Occurrence Handle1093210
W. Werner (1984) ArticleTitlePolynomial interpolation: Lagrange versus Newton Math. Comput. 43 IssueID167 205–217 Occurrence Handle0566.65009 Occurrence Handle10.2307/2007406
J. H. Wilkinson (1963) Rounding errors in algebraic processes. Notes on Applied Science No. 32 Her Majesty's Stationary Office London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Smoktunowicz, A., Wróbel, I. & Kosowski, P. A new efficient algorithm for polynomial interpolation. Computing 79, 33–52 (2007). https://doi.org/10.1007/s00607-006-0185-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-006-0185-z