Recursive algorithms for computing Birkhoff interpolation polynomials

TL;DR

This paper introduces recursive algorithms for Birkhoff interpolation that extend existing methods to broader problem classes, improving computational efficiency and storage compared to traditional approaches.

Contribution

It generalizes the recursive interpolation algorithm to Birkhoff problems using Schur complement theory, ensuring well-posedness and efficiency.

Findings

Reduces computational cost compared to Gaussian elimination

Decreases storage space requirements

Applicable to a broader class of interpolation problems

Abstract

As a generalization of Hermite interpolation problem, Birkhoff interpolation is an important subject in numerical approximation. This paper generalizes the existing Generalized Recursive Polynomial Interpolation Algorithm (GRPIA) that is used to compute the Hermite interpolation polynomial. Based on the theory of the Schur complement and the Sylvester identity, the proposed recursive algorithms are applicable to a broader class of Birkhoff interpolation problems, where each interpolation condition is given by the composition of an evaluation functional and a differential polynomial. The approach incorporates a judgment condition to ensure the problem's well-posedness and computes a lower-degree Newton-type interpolation basis (which is also a strongly proper interpolation basis) along with the corresponding interpolation polynomial. Following the numerical examples, we analyze and compare the computational process and complexity of the proposed algorithm against traditional interpolation methods based on Gaussian elimination, and thus demonstrate that the proposed recursive approach reduces both computational cost and storage space requirements.