Recursive algorithms for computing Birkhoff interpolation polynomials
Xue Jiang, Yuanhe Li, Zhe Li
TL;DR
This work addresses Birkhoff interpolation, where each condition is a composition of an evaluation functional and a differential polynomial, by generalizing the Generalized Recursive Polynomial Interpolation Algorithm (GRPIA). It develops two recursive algorithms based on the Schur complement and the Sylvester identity to construct a lower-degree Newton-type basis and the corresponding interpolation polynomial, provided a strongly proper interpolation space exists. Algorithm 1 yields a strongly proper Newton-type basis and interpolant under a non-singularity judgment, but may not always produce minimal degree. Algorithm 2 extends to general differential-polynomial conditions and can achieve degree reduction by reorganizing the interpolation conditions. Together, the methods reduce computational cost relative to Gaussian-elimination-based approaches and broaden applicability to non-ideal Birkhoff interpolation problems, with practical impact in efficient construction of interpolation polynomials for complex operator-based constraints.
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. Compared with existing algorithms that rely on Gaussian elimination to compute the interpolation basis, our recursive approach significantly reduces the computational cost.