  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.
