Skip to main content
SpringerLink
Log in

A robust implementation of the Carathéodory-Fejér method for rational approximation

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Best rational approximations are notoriously difficult to compute. However, the difference between the best rational approximation to a function and its Carathéodory-Fejér (CF) approximation is often so small as to be negligible in practice, while CF approximations are far easier to compute. We present a robust and fast implementation of this method in the Chebfun software system and illustrate its use with several examples. Our implementation handles both polynomial and rational approximation and substantially improves upon earlier published software.

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

Access this article

Log in via an institution

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. Adamjan, V., Arov, D., Krein, M.: Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem. Math. USSR Sb. 15, 31–73 (1971)

    Article  Google Scholar 

  2. Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743–1770 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carathéodory, C., Fejér, L.: Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz. Rend. Circ. Mat. Palermo 32, 218–239 (1911)

    Article  MATH  Google Scholar 

  4. Gutknecht, M.H.: Rational Carathéodory-Fejér approximation on a disk, a circle, and an interval. J. Approx. Theory 41(3), 257–278 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gutknecht, M.H., Trefethen, L.N.: Real polynomial Chebyshev approximation by the Carathéodory–Fejér method. SIAM J. Numer. Anal. 19(2), 358–371 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hayashi, E., Trefethen, L.N., Gutknecht, M.H.: The CF table. Constr. Approx. 6, 195–223 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM Rev. 21(4), 481–527 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Meinardus, G.: Approximation of Functions: Theory and Numerical Methods. Springer, Berlin (1967)

    MATH  Google Scholar 

  9. Newman, D.J.: Rational approximation to |x|. Mich. Math. J. 11, 11–14 (1964)

    Article  MATH  Google Scholar 

  10. Pachón, R., Platte, R.B., Trefethen, L.N.: Piecewise-smooth chebfuns. IMA J. Numer. Anal. 30, 898–916 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schmelzer, T., Trefethen, L.: Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45(2), 558–571 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Schur, I.: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. I. J. Reine Angew. Math. 147, 205–232 (1917)

    Article  Google Scholar 

  13. Schur, I.: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. II. J. Reine Angew. Math. 148, 122–145 (1918)

    Article  Google Scholar 

  14. Stahl, H.: Uniform rational approximation of |x|. In: Methods of Approximation Theory in Complex Analysis and Mathematical Physics, Leningrad, 1991. Lecture Notes in Math., vol. 1550, pp. 110–130. Springer, Berlin (1993)

    Chapter  Google Scholar 

  15. Takagi, T.: On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau. Jpn. J. Math. 1, 83–93 (1924)

    Google Scholar 

  16. Takagi, T.: Remarks on an algebraic problem. Jpn. J. Math. 2, 13–17 (1925)

    MathSciNet  Google Scholar 

  17. Trefethen, L.N.: Near-circularity of the error curve in complex Chebyshev approximation. J. Approx. Theory 31(4), 344–367 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Trefethen, L.N.: Rational Chebyshev approximation on the unit disk. Numer. Math. 37(2), 297–320 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Trefethen, L.N.: Chebyshev approximation on the unit disk. In: Werner, H., et al. (ed.) Computational Aspects of Complex Analysis, pp. 309–323. Reidel, Dordrecht (1983)

    Google Scholar 

  20. Trefethen, L.N.: Square blocks and equioscillation in the Padé, Walsh, and CF tables. In: Graves-Morris, P., Saff, E., Varga, R. (eds.) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol. 1105, Springer, Berlin (1984)

    Chapter  Google Scholar 

  21. Trefethen, L.N.: Matlab programs for CF approximation. In: Approximation Theory V, pp. 599–602. Academic Press, Boston (1986)

    Google Scholar 

  22. Trefethen, L.N., Gutknecht, M.H.: The Carathéodory–Fejér method for real rational approximation. SIAM J. Numer. Anal. 20(2), 420–436 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Trefethen, L., Weideman, J., Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Varga, R.S., Ruttan, A., Carpenter, A.J.: Numerical results on the best uniform rational approximations of the function |x| on the interval [−1,1]. Math. USSR Sb. 74(2), 271–290 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, 2020, Antwerpen, Belgium

    Joris Van Deun

  2. Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford, OX1 3LB, UK

    Lloyd N. Trefethen

Authors
  1. Joris Van Deun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lloyd N. Trefethen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joris Van Deun.

Additional information

Communicated by Hans Petter Langtangen.

Research supported by the EPSRC grant EP/E045847/1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van Deun, J., Trefethen, L.N. A robust implementation of the Carathéodory-Fejér method for rational approximation. Bit Numer Math 51, 1039–1050 (2011). https://doi.org/10.1007/s10543-011-0331-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-011-0331-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation