Abstract
Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.
Highlights
Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. e interpolation function of most classical approaches is unique to the given data
In the last several years, many researchers have been focusing on this subject and have obtained interesting results. e existing approaches can be divided into polynomial and rational interpolation methods, both of which are applicable to numerical approximation [1, 2], image interpolation [3,4,5,6], and arc structuring and surface modeling [7,8,9,10]
By selecting appropriate parameters and different coefficients, the value of the spline interpolation function can be modified at any point in the interpolant interval, under the condition that the values of the interpolant points are fixed, so that the geometric surfaces can be adjusted. e drawback is that the computation is complicated. e bivariate rational interpolation with parameters, based only on the function values, has been studied in [29]
Summary
Le Zou ,1,2,3 Liangtu Song, Xiaofeng Wang ,1 Thomas Weise, Yanping Chen, and Chen Zhang. Zhan et al [22] developed a nonlocal and local image interpolation model based on nonlocal bounded variation regularization and local total variation and obtained better performance He et al [24] proposed an image inpainting algorithm by using continued fractions rational interpolation. Based on function values and partial derivatives of the function being interpolated, Duan et al [31] proposed a new bivariate rational interpolation, which had a simple and explicit rational mathematical representation with parameters. In [34], the authors presented a weighted bivariate rational bicubic spline interpolation based on function values, which is C1-continuous for any positive parameters. To overcome the above shortcomings, we propose a novel Newton interpolation polynomial only based on (partial) divided differences by introducing one or multiple parameters, which can be seen as a new approach to interpolation method of Newton polynomials.
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