Numerical Algorithms

, Volume 82, Issue 3, pp 895–908

# Optimal interval length for the collocation of the Newton interpolation basis

Original Paper

## 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 PL of the monic Newton basis and a normalized Newton basis, so that PL is the lower triangular Pascal matrix. In contrast to L, PL does not depend on the interval length, and we show that the Skeel condition number of the (n + 1) × (n + 1) lower triangular Pascal matrix is 3n. The -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.

## Keywords

Newton interpolation formula Divided differences Condition number Pascal matrix

## Mathematics Subject Classification (2010)

41A05 65F35 15A12

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