The Research On Theories And Methods Of 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 means of approximation in nearly all areas of numerical analysis. Examples of meromorphic functions for which the polynomial interpolant does not converge uniformly were given by Meray and later 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 and control poles and there is sometimes 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 calculated amount, good numerical stability, poles and unattainable points are prevented through choose weights, regardless of the distribution of the points. In this paper, based on polynomial interpolation and barycentric rational interpolation, new bivariate blending rational interpolation are constructed and the error estimation is given. Furthurly, new bivariate barycentric rational interpolation is constructed based on univariate barycentric rational interpolation. It is key issue how to choose weights so that the interpolation error attain to the minimum value. The best interpolation weights for barycentric rational interpolation are obtained based on an optimization model. At last, new composite barycentric rational interpolion with high-accuracy is proposed, regardless of the distribution of the points. Numerical examples are given to show the effectiveness of our methods.