Shape-Preserving Barycentric Rational Interpolation

The interpolation is that according to the given values of discrete points to construct a simple continuously function such that it have the same function values of at all the given points exactly. Polynomial interpolants are used as the basic ways of approximation in nearly all areas of numerical analysis. Examples that the polynomial interpolant does not converge uniformly were given by Runge, which limited application of the interpolation polynomial. It is well known that the classical rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points. But it is difficult to avoid poles. unattainable points and infinite inverse differences for Thiele-type continued fraction interpolation. Barycentric rational interpolation was presented by W.Werner, which possess various advantages in comparison with classical continued fraction rational interpolants, such as barycentric rational interpolants have small calculation quantity, good numerical stability, no poles and unattainable points by choosing weights regardless of the distribution of the points. In this dissertation, the study about shape preserving of the barycentric rational interpolation is carried out. Furthurly. two bivariate rational interpolation over triangular grids are constructed, the first one is bivariate barycentric rational interpolation; the second one is a blending interpolation based on Newton interpolation and barycentric rational interpolation. It is key to issue how to choose weights to make the interpolation preserve shape. The best interpolation weights for barycentric rational interpolation are obtained based on the optimization model. Numerical examples are given to show the effectiveness of our methods.