Shape Preserving Rational Spline Interpolation Based On Approximation Derivatives

Interpolation is to construct a simple function such that it have the same function values at all the given points. Interpolation is the most basic method of numerical approximation. 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. However, polynomial interpolation may emerge the Runge phenomenon. In this case, spline interpolation is an effective method. Since the 1960s, the spline function method is widely used in data fitting, function approximation, numerical integration and differentiation, and gradually became a new branch of approximation theory and the main mathematical tool of computer-aided geometric design. Rational spline interpolation are presented as a result of preserving the shape of the interpolated function. In this dissertation, some rational spline interpolation are constructed based on approximation derivatives. It can be summarized as follows:First of all, approximation derivatives are obtained based on the three-point barycentric rational interpolation. Specifically, the derivatives of the interpolated function at all the nodes are approximated with the the derivative of those three-point barycentric rational interpolatants. New 3/1 shape-preserving interpolation rational spline are further constructed making use of the approximate derivatives.Then, in the case of the expression of the interpolated function was known, an optimization method is presented to obtain the best approximation derivatives. We take all the weights as the decision variables and take the absolute error of derivative of the interpolant as the objective function while take those constraints such as no unattainable points, no poles and the sum of all the absolute of weights is 1. The optimal weights are obtained by means of LINGO. The approximate derivatives are presented and new 3/3 shape-preserving rational spline interpolation is further constructed.Finally, the approximation derivatives are obtained based on the three-point Thiele-type continued fraction interpolation. Then new 3/1 shape-preserving rational spline interpolation is constructed. Given numerical examples show effectiveness of our method.Figure[11] table[7] reference[30]...