| Interpolation, an ancient mathematical methods from the production practices is an important means of function approximation. After polynomial interpolation which is the most basic interpolation method, rational spline interpolation becomes another effective method. It is mainly used in data fitting, function approximation, numerical differentiation, integration and other issues. But in reality, it is also required that experimental data has some good properties (positive, monotone, convexity). For example monthly rainfall must be positive to have an actual meaning. Therefore, it required to maintain the original nature of the data when we use interpolation curve to represent the data. Thus it is very significant to study shape-preserving rational spline interpolation.The paper contains following tasks:In chapter1, this paper give a brief depict of the background and the study status of shape-preserving rational spline, and the study significance of this article.In chapter2, the paper puts forward three kinds of low degree shape-preserving rational interpolation, and gives its monotonic analysis and the estimation method of its derivative value.In chapter3, a rational cubic spline function (3/1) involving two shape parameters is presented in chapter three, and is applied to the interpolation problem with the data which type is a parameter equation function. By adjusting the shape of parameters, it can interactively modify the shape of the interpolation curve and meet the relevant requirement for shape preserving. Then, we deduce a sufficient condition for sign preserving and monotonicity preserving. The error estimation and the relevant numerical experiments of the spline interpolation is given in the end.In chapter4, a rational cubic spline function (3/2) involving two shape parameters is presented in chapter four. We deduce a sufficient condition for sign preserving and monotonicity preserving and give the error estimation and numerical experiments. |
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