A new closed formula for the Hermite interpolating polynomial with applications on the spectral decomposition of a matrix

TL;DR

This paper introduces a new closed-form formula for Hermite interpolation that simplifies calculations and applies it to matrix spectral decomposition, leading to new identities and an implementation in MATLAB.

Contribution

It provides a novel explicit formula for the Hermite interpolating polynomial and its application to the spectral decomposition of matrices, with practical implementation.

Findings

New closed-form for Hermite interpolation requiring only polynomial derivatives

Closed formula for the semi-simple part of Jordan decomposition

Implementation of the formula in MATLAB

Abstract

We present a new closed form for the interpolating polynomial of the general univariate Hermite interpolation that requires only calculation of polynomial derivatives, instead of derivatives of rational functions. This result is used to obtain a new simultaneous polynomial division by a common divisor over a perfect field. The above findings are utilized to obtain a closed formula for the semi--simple part of the Jordan decomposition of a matrix. Finally, a number of new identities involving polynomial derivatives are obtained, based on the proposed simultaneous polynomial division. The proposed explicit formula for the semi--simple part has been implemented using the Matlab programming environment.