Optimal interval length for the collocation of the Newton basis

TL;DR

This paper investigates how the length of the interval affects the conditioning of Newton interpolation matrices, finding that an interval length of 3 minimizes the asymptotic growth of the condition number.

Contribution

It analyzes the influence of interval length on the conditioning of Newton basis collocation matrices and identifies the optimal interval length for minimal growth rate.

Findings

Condition number of Pascal matrix is 3^n.

Optimal interval length for minimal condition number growth is 3.

Conditioning depends on interval length, affecting interpolation stability.

Abstract

It is known that the Lagrange interpolation problem at equidistant nodes is ill-conditioned. We explore the influence of the interval length in the computation of divided differences of the Newton interpolation formula. Condition numbers are computed for lower triangular matrices associated to the Newton interpolation formula at equidistant nodes. We consider the collocation matrices $L$ and $P_L$ of the monic Newton basis and a normalized Newton basis, so that $P_L$ is the lower triangular Pascal matrix. In contrast to $L$, $P_L$ does not depend on the interval length, and we show that the Skeel condition number of the $(n+1)\times (n+1)$ lower triangular Pascal matrix is $3^n$. The $\infty$-norm condition number of the collocation matrix $L$ of the monic Newton basis is computed in terms of the interval length. The minimum asymptotic growth rate is achieved for intervals of length 3.