Well-conditioned, stable and infinitely smooth interpolation in arbitrary nodes is by no means a trivial task, even in the univariate setting considered here; already the most important case, equispaced points, is not obvious. Certain approaches have nevertheless experienced significant developments in the last decades. In this paper we review one of them, linear barycentric rational interpolation, as well as some of its applications.
The barycentric rational interpolants introduced by Floater and Hormann in 2007 are "blends" of polynomial interpolants of fixed degree d. In some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants, which, e.g., are ill-conditioned and lead to Runge's phenomenon if the interpolation nodes are equispaced. For such nodes, however, the condition of Floater-Hormann interpolation deteriorates exponentially with increasing d. In this paper, an extension of the Floater-Hormann family with improved condition at equispaced nodes is presented and investigated. The efficiency of its applications such as the approximation of derivatives, integrals and antiderivatives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants.
Abstract. We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a composite version of this quadrature rule, loses one order of convergence but is much cheaper. Both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision.Key words. Volterra integral equations, direct quadrature method, linear barycentric rational interpolation, linear rational quadrature
In polynomial and spline interpolation the k-th derivative of the interpolant, as a function of the mesh size h, typically converges at the rate of O(h d+1−k ) as h → 0, where d is the degree of the polynomial or spline. In this paper we establish, in the important cases k = 1, 2, the same convergence rate for a recently proposed family of barycentric rational interpolants based on blending polynomial interpolants of degree d.Math Subject Classification: 65D05, 41A05, 41A20, 41A25
Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut's rational interpolant at equidistant nodes to the family of Floater-Hormann interpolants, which includes the former as a special case.Math Subject Classification: 65D05, 65F35, 41A05, 41A20
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