Skip to main content
Springer Nature Link
Log in

Roundoff errors for polynomial evaluation by a family of formulae

  • Published:
Computing Aims and scope Submit manuscript

Abstract

A roundoff error analysis of formulae for evaluating polynomials is performed. The considered formulae are linear combinations of basis functions, which can be computed with high relative accuracy. We have taken into account that all steps but the last one can be computed to high relative accuracy. The exactness of the initial data is crucial for obtaining low error bounds. The Lagrange interpolation formula and related formulae are considered and numerical experiments are provided.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Berrut JP, Trefethen LN (2004) Barycentric Lagrange interpolation. SIAM Rev 46: 501–517

    Article  MATH  MathSciNet  Google Scholar 

  2. Demmel J, Gu M, Eisenstat S, Slapniar I, Veseli K, Drma Z (1999) Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl 299: 21–80

    Article  MATH  MathSciNet  Google Scholar 

  3. Farin G (2002) Curves and Surfaces of CAGD. A practical guide, 5th edn. Academic Press, San Diego

    Google Scholar 

  4. Farouki RT, Rajan VT (1988) Algorithms for polynomials in Bernstein form. Comput. Aided Geom Des 5: 1–26

    Article  MATH  MathSciNet  Google Scholar 

  5. Higham NJ (2002) Accuracy and stability of numerical algorithms. SIAM, Philadelphia

    MATH  Google Scholar 

  6. Higham NJ (2004) The Numerical Stability of barycentric Lagrange interpolation. IMA J Numer Anal 24: 547–556

    Article  MATH  MathSciNet  Google Scholar 

  7. Mainar E, Peña JM (1999) Error analysis of corner cutting algorithms. Numer Algorithms 22: 41–52

    Article  MATH  MathSciNet  Google Scholar 

  8. Peña JM, Sauer T (2000a) On the multivariate Horner scheme. SIAM J Numer Anal 37: 1186–1197

    Article  MATH  MathSciNet  Google Scholar 

  9. Peña JM, Sauer T (2000b) On the multivariate Horner scheme. II: Running error analysis. Computing 65: 313–322

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada/IUMA, University of Zaragoza, 50009, Zaragoza, Spain

    J. M. Carnicer & J. M. Peña

  2. Department of Mathematics, University of Dundee, Dundee, DD1 4HN, UK

    T. N. T. Goodman

Authors
  1. J. M. Carnicer
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. T. N. T. Goodman
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. J. M. Peña
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to J. M. Carnicer.

Additional information

Research partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carnicer, J.M., Goodman, T.N.T. & Peña, J.M. Roundoff errors for polynomial evaluation by a family of formulae. Computing 82, 199–215 (2008). https://doi.org/10.1007/s00607-008-0007-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-008-0007-6

Keywords

Mathematics Subject Classification (2000)