Barycentric Rational Hermite Interpolation

Interpolation is one of the fundamental techniques of approximation theory. Polynomial interpolation is the basic of the whole numerical approximations, however, the high order interpolation which bring into the Runge phenomenon restricts its applications. Rational interpolation is suitable for approximating the function which has poles and it possesses a faster convergence than polynomial. However some rational interpolation, for example, the classical Thiele type's osculatory continued fraction interpolation may be revealed the following problems:The reverse divided differences are not existent, poles and unattainable points can not be avoided and the locations of the poles can not be controlled and so on. Loosing the restrictions on the degree of denominator and numerator, the barycentric rational Hermite interpolation which is constructed on certain conditions can not only satisfy the interpolation condition, but also avoid the unwanted poles. The barycentric rational interpolation and barycentric rational Hermite interpolation possess a quite better stability. In this paper, the composite barycentric rational Hermite interpolation with high-accuracy is constructed based on univariate pade-type approximation and univariate barycentric rational Hermite interpolation; New composite barycentric rational Hermite interpolation is constructed based on osculatory rational continued fraction interpolation and univariate barycentric rational Hermite interpolation; A new bivariate barycentric rational Hermite interpolation is constructed based on univariate barycentric rational Hermite interpolation; A composite bivariate barycentric rational Hermite interpolation is constructed based on pade-type approximation and bivariate barycentric rational Hermite interpolation. Choosing the different weights one may obtain the different univariate or bivariate barycentric rational Hermite interpolation. The poles and the unattainable points of the barycentric rational Hermite interpolation may be avoided through doing a proper choice for the interpolation weights. In this paper, the optimization model which is used for computing the optimal interpolation weights is given with respect to how to choose the weights so that the interpolation error is minimal. Lots of numerical examples are present to show the effectiveness of our new methods.