Skip to main content
Log in

Barycentric rational interpolation with asymptotically monitored poles

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We present a method for asymptotically monitoring poles to a rational interpolant written in barycentric form. Theoretical and numerical results are given to show the potential of the proposed interpolant.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comput. 47, 285–299 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berrut, J.-P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berrut, J.-P., Mittelmann, H.D.: Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval. Comput. Math. Appl. 33, 77–86 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berrut, J.-P., Baltensperger, R., Mittelmann, H.D.: Recent developments in barycentric rational interpolation. Trends and applications in constructive approximation. Internat. Ser. Numer. Math. (ISNM) 151, 27–51 (2005)

    Article  MathSciNet  Google Scholar 

  5. Berrut, J.-P., Trefethen, L.N.: Barycentric lagrange interpolation. SIAM Rev. 46, 501–517 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Higham, N.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Salzer, H.E.: Lagrangian interpolation at the Chebyshev points x n,ν = cos(νπ/n); some unnoted advantages. Comput. J. 15, 156–159 (1972)

    MATH  MathSciNet  Google Scholar 

  8. Henrici, P.: Essentials of Numerical Analysis with Pocket Calculator Demonstrations. Wiley, New York (1982)

    MATH  Google Scholar 

  9. Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2001)

    Google Scholar 

  11. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  12. Berrut, J.-P.: The barycentric weights of rational interpolation with prescribed poles. J. Comput. Appl. Math. 86, 45–52 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Berrut, J.-P., Mittelmann, H.D.: Rational interpolation through the optimal attachment of poles to the interpolating polynomial. Numer. Algorithms 23, 315–328 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Baltensperger, R., Berrut, J.-P., Dubey, Y.: The linear rational pseudospectral method with preassigned poles. Numer. Algorithms 33, 53–63 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315-331 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Baltensperger, R., Berrut, J.-P., Noël, B.: Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Math. Comput. 68, 1109–1120 (1999)

    Article  MATH  Google Scholar 

  17. Kosloff, D., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an \(\mathcal{O}(N^{-1})\) time step restriction. J. Comput. Phys. 104, 457–469 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Baltensperger, R., Berrut, J.-P.: The linear rational collocation method. J. Comput. Appl. Math. 134, 243–258 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Berrut, J.-P., Baltensperger, R.: The linear rational pseudospectral method for boundary value problems. BIT 41, 868–879 (2001)

    Article  MathSciNet  Google Scholar 

  20. Bayliss, A., Turkel, E.: Mappings and accuracy for Chebyshev pseudo-spectral approximations. J. Comput. Phys. 101, 349–359 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Berrut, J.-P., Mittelmann, H.D.: Adaptive point shifts in rational approximation with optimized denominator. J. Comput. Appl. Math. 164–165, 81–92 (2004)

    Article  MathSciNet  Google Scholar 

  22. Berrut, J.-P., Mittelmann, H.D.: Optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems. J. Comput. Phys. 204, 292–301 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tee, T.W., Trefethen, L.N.: A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28, 1798–1811 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hale, N., Tee, T.W.: Conformal maps to multiply-slit domains and applications. SIAM J. Sci. Comput. 31, 3195–3215 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Berrut, J.-P., Mittelmann, H.D.: Point shift in rational interpolation with optimized denominator. In: Algorithms for Approximation IV: Proceedings of the 2001 International Symposium, pp. 420–427 (2002)

  26. Tee, T.W.: An adaptive rational spectral method for differential equations with rapidly varying solutions. Ph.D. Thesis, University of Oxford (2006)

  27. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1964)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Richard Baltensperger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baltensperger, R. Barycentric rational interpolation with asymptotically monitored poles. Numer Algor 57, 67–81 (2011). https://doi.org/10.1007/s11075-010-9415-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-010-9415-8

Keywords

Mathematics Subject Classifications (2010)

Navigation