BIT Numerical Mathematics

, Volume 51, Issue 4, pp 1039-1050

First online:

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

  • Joris Van DeunAffiliated withDepartment of Mathematics and Computer Science, Universiteit Antwerpen Email author 
  • , Lloyd N. TrefethenAffiliated withMathematical Institute, University of Oxford

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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.


Carathéodory-Fejér approximation Near-best rational approximation Chebfun

Mathematics Subject Classification (2000)

41A50 41A20