A New Approach to Newton-Type Polynomial Interpolation with Parameters

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.