Université de Fribourg

Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm

Berrut, Jean-Paul ; Marchi, S. De ; Elefante, Giacomo ; Marchetti, F.

In: Applied Mathematics Letters, 2020, vol. 103, p. 106196

In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater–Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in De Marchi et al. (2020). Numerical tests show that it yields an accurate approximation of discontinuous functions.

Consortium of Swiss Academic Libraries

Linear barycentric rational quadrature

Klein, Georges ; Berrut, Jean-Paul

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

Université de Fribourg

The linear barycentric rational method for a class of delay Volterra integro-differential equations

Abdi, Ali ; Berrut, Jean-Paul ; Hosseini, Seyyed Ahmad

In: Journal of Scientific Computing, 2018, vol. 75, no. 3, p. 1757-1775

A method for solving delay Volterra integro-differential equations is introduced. It is based on two applications of linear barycentric rational interpolation, barycentric rational quadrature and barycentric rational finite differences. Its zero–stability and convergence are studied. Numerical tests demonstrate the excellent agreement of our implementation with the predicted convergence...

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

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...