I alluded to Lebesgue constants within the earlier publish with out giving them a reputation. There I mentioned that the sure on order n interpolation error has the shape
the place h is the spacing between interpolation factors and δ is the error within the tabulated values. The fixed c depends upon the perform f being interpolated, and to a lesser extent on n. The fixed λ is impartial of f however depends upon n and on the relative spacing between the interpolation nodes. This publish will look nearer at λ.
Given a set of n + 1 nodes T
outline
Then the Lebesgue perform is outlined by
and the Lebesgue fixed for the grid is the utmost worth of the Lebesgue perform
![Lambda_n(T)=max_{xin[a,b]} lambda_n(x)](https://www.johndcook.com/lebesgueconst4.svg){kind=small}
The values of Λ are troublesome to compute, however there are good asymptotic expressions for Λ when the grid is evenly spaced:
When the grid factors are on the roots of a Chebyshev polynomial then
The earlier publish talked about the circumstances n = 11 and n = 29 for evenly spaced grids. The corresponding values of Λ are roughly 155 and 10995642. So eleventh order interpolation is amplifying the rounding error within the tabulated factors by an element of 155, which could be acceptable. However twenty ninth order interpolation is amplifying the rounding error by an element of over 10 million.
The corresponding values of Λ for Chebyshev-spaced nodes are 2.58 and three.17. Chebyshev spacing is clearly higher for high-order interpolation, which you could have that choice.
