New Recursive Divided-Difference Interpolation Method

Abstract

Abstract This article is devoted to the direct determination of the coefficients of an interpolation polynomial in the monomial basis, proposing a new recursive method based on divided differences. The proposed method does not require an explicit expansion of the Newton interpolation polynomial into the monomial basis, nor does it require constructing and solving a Vandermonde system of linear equations. The method is based on the sequential recursive reconstruction of the divided-differences table. When a column of divided differences yields a constant value, this value is determined as the corresponding coefficient of the polynomial. The contributions of the coefficients found so far are then subtracted from the interpolation values, and the same process is repeated for the remaining lower-degree polynomial. As a result, all coefficients of the polynomial are determined sequentially, starting from the highest-degree term down to the constant term. The mathematical foundations of the proposed algorithm are rigorously proven through lemmas and theorems. In particular, the necessary and sufficient condition for the common value obtained in a column of recursive divided differences to be equal to the corresponding coefficient of the polynomial is established. The numerical example provided confirms the correctness of the proposed algorithm and demonstrates that the polynomial coefficients can be recursively determined with high accuracy based on the interpolation data. The proposed approach offers a new computational interpretation of divided differences and forms an alternative recursive computational scheme for determining polynomial coefficients.

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