In our previous paper (Camargo, Numer. Algor., 85:591–606, 2020), we proved that the algorithms in a certain class of divided differences schemes are backward stable and, in particular, we proved that Neville’s algorithm for Lagrange interpolation is backward stable for extrapolation for monotonically ordered nodes. That proof was based on a very particular pattern of the signs of the components of the divided differences which, in the case of Neville’s algorithm for monotonically ordered nodes, is not satisfied when interpolation is considered instead of extrapolation. In this note we present a different argument that shows that Neville’s algorithm is backward stable on the whole real line for monotonically ordered nodes. Our reasoning is based on a pyramid algorithm for the computation of Lebesgue functions. We also explain that obtaining sharp upper bounds for the numerical error in the computation of Neville’s algorithm for generic sets of nodes is difficult.

在我们之前的论文中 (Camargo, Numer. Algor., 85:591–606, 2020)，我们证明了某类划分差分方案中的算法是向后稳定的，特别是我们证明了 Neville 的拉格朗日插值算法对于单调有序节点的外推是向后稳定的。该证明基于非常特殊的分割差分分量的符号模式，在单调有序节点的 Neville 算法的情况下，当考虑插值而不是外插时，该模式不满足。在本笔记中，我们提出了一个不同的论点，表明 Neville 算法在单调有序节点的整个实线上是向后稳定的。我们的推理基于用于计算 Lebesgue 函数的金字塔算法。