Barycentric-Remez algorithms for best polynomial approximation in the chebfun system

Abstract  The Remez algorithm, 75 years old, is a famous method for computing minimax polynomial approximations. Most implementations
of this algorithm date to an era when tractable degrees were in the dozens, whereas today, degrees of hundreds or thousands
are not a problem. We present a 21st-century update of the Remez ideas in the context of the chebfun software system, which
carries out numerical computing with functions rather than numbers. A crucial feature of the new method is its use of chebfun
global rootfinding to locate extrema at each iterative step, based on a recursive algorithm combining ideas of Specht, Good,
Boyd, and Battles. Another important feature is the use of the barycentric interpolation formula to represent the trial polynomials,
which points the way to generalizations for rational approximations. We comment on available software for minimax approximation
and its scientific context, arguing that its greatest importance these days is probably for fundamental studies rather than
applications.

• Content Type Journal Article
• DOI 10.1007/s10543-009-0240-1
• Authors
  
  
  • Ricardo Pachón, University of Oxford Computing Laboratory Parks Rd. Oxford OX1 3QD UK
  • Lloyd N. Trefethen, University of Oxford Computing Laboratory Parks Rd. Oxford OX1 3QD UK
  
  

• Journal BIT Numerical Mathematics
• Online ISSN 1572-9125
• Print ISSN 0006-3835

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Original source : http://www.springerlink.com/content/5n17515j262845...