Barycentric rational interpolation.

Tato bakálářská práce se zabývá odvozením barycentrického tvaru polynomiální interpolace a následně pak barycentrického tvaru racionální interpolace. Dále ukazuje nevýhody klasické racionální interpolace a zabývá se eliminací těchto nevýhod v jejím barycentrickém tvaru. Také ukazuje některé konkrétní metody řešení barycentrické racionální interpolace.This bachelor thesis deals with deducing of barycentric form of polynomial interpolation and then barycentric form of rational interpolation. It shows disadvantages of classic rational interpolation and deals with elimination of these disadvantages in its barycentric form. It also shows some specific methods of barycentric rational interpolation.

Barycentric interpolation and exact integration formulas for the finite volume element method

This contribution concerns with the construction of a simple and effective technology for the problem of exact integration of interpolation polynomials arising while discretizing partial differential equations by the finite volume element method on simplicial meshes. It is based on the element-wise representation of the local shape functions through barycentric coordinates (barycentric interpolation) and the introducing of classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over the geometrical shapes defined by a barycentric dual mesh. Numerical examples are presented that illustrate the validity of the technolog

Stability of barycentric interpolation formulas

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or "first barycentric" formula dating to Jacobi in 1825. This difference in stability confirms the theory published by N. J. Higham in 2004 (IMA J. Numer. Anal., v. 24) and has practical consequences for computation with rational functions