The Research On Application Of Barycentric Interpolation

The interpolation is that according to the given values of discrete points to construct a specific function such that it has the same values at all points in the interval. Polynomial interpolation is the simplest interpolation, which is the basis of whole numerical approximation. It can be widely used in dealing with finding roots of equations, function approximation, numerical differentiation, numerical integration and numerical solution of differential equations and so on. But when the number of interpolating nodes is large, it may emerge the Runge phenomenon. It is well known that the classical rational interpolation sometimes gives better approximations than polynomial interpolation, but it is difficult to avoid and control poles. Constructing the barycentric rational function interpolation according to the conditions is not only meet them, but also can avoid the above limitations. Barycentric rational interpolation possess various advantages in comparison with classical rational interpolants, such as it has small calculation quantity, good numerical stability and adding a new data pair, it doesn't require renew computation all basis functions. Because it cause more and more people's attention, so I will do further research on it in this paper.First of all, for different interpolation weight, we can obtain different barycentric rational interpolation. It is key issue how to choose weight so that the interpolation error attain to the minimum value. In this paper, we use the barycentric interpolation with the optimal barycentric weight and choose the Chebyshev nodes as interpolating nodes to reduce the interpolation error.Then, it has not good method for computing the derivatives in previous reference, so we introduce a good method for computing the derivatives with barycentric rational interpolation.This method can be used for computing arbitrary order derivatives of the functions on any intervals. The result gives an excellent precision and the operation of this method is simple, so it has some practical value.Finally, a kind of blending rational interpolants was constructed by combination of traditional barycentric interpolation and Pade approximation. In order to obtain more accurate interpolation, Pade-type approximation based blending rational interpolation and perturbed Pade approximation based blending rational interpolation were studied. Numerical examples are given to show the effectiveness of the new methods.