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Université de Fribourg

Recent advances in linear barycentric rational interpolation

Berrut, Jean-Paul ; Klein, Georges

In: Journal of Computational and Applied Mathematics, 2014, vol. 259, Part A, p. 95–107

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...

Université de Fribourg

The linear barycentric rational quadrature method for Volterra integral equations

Berrut, Jean-Paul ; Hosseini, S. A. ; Klein, Georges

In: SIAM Journal on Scientific Computing, 2014, vol. 36, no. 1, p. A105–A123

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...

Université de Fribourg

Applications of linear barycentric rational interpolation

Klein, Georges ; Berrut, Jean-Paul (Dir.)

Thèse de doctorat : Université de Fribourg, 2012 ; no. 1762.

This thesis is a collection of properties and applications of linear barycentric rational interpolation, mainly with the weights presented by Floater and Hormann in 2007. We are motivated by the counterintuitive and provable impossibility of constructing from equispaced data an approximation scheme that converges very rapidly to the approximated function and is simultaneously computationally...

Université de Fribourg

Linear rational finite differences from derivatives of barycentric rational interpolants

Klein, Georges ; Berrut, Jean-Paul

In: SIAM Journal on Numerical Analysis, 2012, vol. 50, no. 2, p. 643–656

Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a...

Université de Fribourg

Linear barycentric rational quadrature

Klein, Georges ; Berrut, Jean-Paul

In: Bit Numerical Mathematics, 2012, vol. 52, no. 2, p. 407-424

Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational...

Université de Fribourg

Convergence rates of derivatives of a family of barycentric rational interpolants

Berrut, Jean-Paul ; Floater, Michael S. ; Klein, Georges

In: Applied Numerical Mathematics, 2011, vol. 61, no. 9, p. 989-1000

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(hd+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...

Université de Fribourg

A formula for the error of finite sinc interpolation with an even number of nodes

Berrut, Jean-Paul

In: Numerical Algorithms, 2010, vol. 56, no. 1, p. 143-157

Sinc interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. We give a formula for the error committed when the function neither decreases rapidly nor is periodic, so that the sinc series must be truncated for practical purposes. To do so, we first complete a previous result for an odd number of points, before deriving a formula...

Université de Fribourg

First applications of a formula for the error of finite sinc interpolation

Berrut, Jean-Paul

In: Numerische Mathematik, 2009, vol. 112, no. 3, p. 341-361

In former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval. In the present work we demonstrate the relevance of the formula through several applications: correction of the interpolant through the insertion of derivatives to increase its order of convergence, improvement of the...

Université de Fribourg

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

Berrut, Jean-Paul ; Welscher, Annick

In: Applied and Computational Harmonic Analysis, 2007, vol. 23, no. 3, p. 307-320

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the...

Université de Fribourg

A formula for the error of finite sinc-interpolation over a finite interval

Berrut, Jean-Paul

In: Numerical Algorithms, 2007, vol. 45, no. 1-4, p. 369-374

Sinc-interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. It, however, requires that the interpolated function decreases rapidly or is periodic. We give an error formula for the case where neither of these conditions is satisfied.