Convergence of Linear Barycentric Rational Interpolation for Analytic Functions

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 choos...

[1]  Georges Klein.  An Extension of the Floater-Hormann Family of Barycentric Rational Interpolants , 2013, Math. Comput..

[2]  P. Davis.  Interpolation and approximation , 1965 .

[3]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[4]  Kai Hormann,et al.  Bounding the Lebesgue constant for Berrut's rational interpolant at general nodes , 2013, J. Approx. Theory.

[5]  Kai Hormann,et al.  On the Lebesgue constant of barycentric rational interpolation at equidistant nodes , 2012, Numerische Mathematik.

[6]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[7]  Jean-Paul Berrut,et al.  Rational functions for guaranteed and experimentally well-conditioned global interpolation , 1988 .

[8]  Simon J. Smith,et al.  Lebesgue constants in polynomial interpolation , 2006 .

[9]  R. Platte.  How fast do radial basis function interpolants of analytic functions converge , 2011 .

[10]  Herbert Stahl,et al.  Convergence of Rational Interpolants ∗ , 1996 .

[11]  Hans Wallin,et al.  Potential theory and approximation of analytic functions by rational interpolation , 1979 .

[12]  W. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

[13]  K. Zeller,et al.  Auswertung der Normen von Interpolationsoperatoren , 1966 .

[14]  Lloyd N. Trefethen.  Computing numerically with functions instead of numbers , 2007, SNC '07.

[15]  Arnold Schönhage,et al.  Fehlerfortpflanzung bei Interpolation , 1961 .

[16]  R. Langer.  Interpolation and Approximation by Rational Functions in the Complex Domain , 1937 .

[17]  Lloyd N. Trefethen,et al.  Barycentric Lagrange Interpolation , 2004, SIAM Rev..

[18]  Daan Huybrechs,et al.  On the Fourier Extension of Nonperiodic Functions , 2010, SIAM J. Numer. Anal..

[19]  Tobin A. Driscoll,et al.  Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation , 2005, SIAM J. Numer. Anal..

[20]  Kai Hormann,et al.  Barycentric rational interpolation at quasi-equidistant nodes , 2012 .

[21]  James F. Epperson.  On the Runge example , 1987 .

[22]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[23]  L. Trefethen,et al.  Two results on polynomial interpolation in equally spaced points , 1991 .

[24]  Lloyd N. Trefethen,et al.  Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples , 2011, SIAM Rev..

[25]  Jean-Paul Berrut,et al.  Exponential convergence of a linear rational interpolant between transformed Chebyshev points , 1999, Math. Comput..

[26]  Thomas Ransford,et al.  Potential Theory in the Complex Plane: Bibliography , 1995 .

[27]  Kai Hormann,et al.  Barycentric rational interpolation with no poles and high rates of approximation , 2007, Numerische Mathematik.

[28]  Kai Hormann,et al.  Università Della Svizzera Italiana Usi Technical Report Series in Informatics on the Lebesgue Constant of Berrut's Rational Interpolant at Equidistant Nodes , 2022 .