The Numerical Stability of Regularized Barycentric Interpolation Formulae for Interpolation and Extrapolation

TL;DR

This paper analyzes the numerical stability and efficiency of regularized barycentric and modified Lagrange interpolation formulae, highlighting their stability properties, computational costs, and performance in noise reduction for interpolation and extrapolation.

Contribution

It provides a detailed stability and efficiency analysis of regularized interpolation formulae, introducing algorithms with specific computational complexities and demonstrating their advantages over classical methods.

Findings

Regularized modified Lagrange formulae are backward and forward stable for interpolation.

Regularized barycentric formulae are only forward stable and lose accuracy outside [-1,1].

Regularized methods outperform classical ones in noise reduction.

Abstract

The $\ell_2-$ and $\ell_1-$regularized modified Lagrange interpolation formulae over $[-1,1]$ are deduced in this paper. This paper mainly analyzes the numerical characteristics of regularized barycentric interpolation formulae, which are presented in [C. An and H.-N. Wu, 2019], and regularized modified Lagrange interpolation formulae for both interpolation and extrapolation. Regularized barycentric interpolation formulae can be carried out in $\mathcal{O}(N)$ operations based on existed algorithms [H. Wang, D. Huybrechs and S. Vandewalle, Math. Comp., 2014], and regularized modified Lagrange interpolation formulae can be realized in an algorithm of $\mathcal{O}(N\log N)$ operations. For interpolation, the regularized modified Lagrange interpolation formulae are blessed with backward stability and forward stability, whereas the regularized barycentric interpolation formulae are only provided with forward stability. For extrapolation, the regularized barycentric interpolation formulae meet loss of accuracy outside $[-1,1]$, but the regularized modified Lagrange interpolation formulae still work. Consistent results for extrapolation are also verified outside the Chebfun ellipse (a special Bernstein ellipse) in the complex plain. Finally, we illustrate that regularized interpolation formulae perform better than classical interpolation formulae without regularization in noise reduction.