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A robust implementation of the Carathéodory-Fejér method

By Joris van Deun and Lloyd N. Trefethen


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

Topics: Numerical analysis
Publisher: Springer
Year: 2010
OAI identifier:

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  10. (1924). On an algebraic problem related to an analytic theorem of Carathe´odory and Feje´r and on an allied theorem of Landau.
  11. (2010). Piecewisesmooth chebfuns.
  12. (1964). Rational approximation to |x|.
  13. (1984). Rational Carathe´odory-Feje´r approximation on a disk, a circle, and an interval.
  14. (1981). Rational Chebyshev approximation on the unit disk.
  15. Real polynomial Chebyshev approximation by the Carathe´odory–Feje´r method.
  16. (1925). Remarks on an algebraic problem.
  17. (1984). Square blocks and equioscillation in the Pade´, Walsh, and CF tables.
  18. (2006). Talbot quadratures and rational approximations. doi
  19. (1983). The Carathe´odory– Feje´r method for real rational approximation.
  20. (1990). The CF table.
  21. (1911). U¨ber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und u¨ber den Picard-Landauschen Satz.
  22. (1917). U¨ber Potenzreihen, die im Innern des Einheitskreises beschra¨nkt sind.
  23. (1991). Uniform rational approximation of |x|. In Methods of approximation theory in complex analysis and mathematical physics

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