A Numerical Study of Newton Interpolation with Extremely High Degrees

TL;DR

This study demonstrates that Newton interpolation with Fast Leja points offers a competitive and efficient alternative to Chebyshev nodes for high-degree polynomial interpolation, with practical advantages in node addition and approximation accuracy.

Contribution

It provides the first extensive numerical comparison showing FL points' effectiveness and efficiency in high-degree interpolation, highlighting their practical advantages over Chebyshev nodes.

Findings

FL points yield approximation quality comparable to Chebyshev nodes.

Newton interpolation with FL points can achieve machine accuracy.

FL points allow easy, on-the-fly node addition.

Abstract

In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort. However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation. This is an experimental account of the analytic result that the limit distribution of FL points and Chebyshev nodes is the same when letting the number of points go to infinity. Since the FL construction is easy to perform and allows to add interpolation nodes on the fly in contrast to the use of Chebyshev nodes, our study suggests that Newton interpolation with FL points is currently the most efficient numerical technique for polynomial interpolation. Moreover, we give numerical evidence that any reasonable function can be approximated up to machine accuracy by Newton interpolation with FL points if desired, which shows the potential of this method.