Polynomial interpolation to analytic functions can be very accurate, depending on the distribution of the interpolation nodes. However, in equispaced nodes and the like, besides being badly conditioned, these interpolants fail to converge even in exact arithmetic in some cases. Linear barycentric rational interpolation with the weights presented by Floater and Hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases. This has been proven for differentiable functions and indicated in several experiments for analytic functions. So far, these rational interpolants have been used mainly with a constant parameter usually denoted by $d$, the degree of the blended polynomials, which leads to small condition numbers but to merely algebraic convergence. With the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes. Moreover, we present suggestions for how to choose $d$ in order to observe fast and stable convergence, even in equispaced nodes where stable geometric convergence is provably impossible. We demonstrate our results with several numerical examples.


  1. barycentric rational interpolation
  2. potential theory
  3. level curves
  4. analyticity

MSC codes

  1. 65D05
  2. 65E05
  3. 41A20

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2560 - 2580
ISSN (online): 1095-7170


Submitted: 6 February 2012
Accepted: 29 July 2012
Published online: 23 October 2012


  1. barycentric rational interpolation
  2. potential theory
  3. level curves
  4. analyticity

MSC codes

  1. 65D05
  2. 65E05
  3. 41A20



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