Adaptive point shifts in rational approximation with optimized denominator

  • Authors:

  • Jean-Paul Berrut;Hans D. Mittelmann

  • Affiliations:

  • Département de Mathématiques, Université de Fribourg, CH-1700 Fribourg/Pérolles, Switzerland;Department of Mathematics, Arizona State University, Tempe, AZ

  • Venue:
  • Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
  • Year:
  • 2004

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Abstract

Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equation, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.