Newton Interpolation with Extremely High Degrees by Leja Ordering and Fast Leja Points

In this paper we perform a numerical study of Newton polynomial interpolation. We explore the Leja ordering of Chebyshev knots and the Fast Leja knots introduced by Reichel. In all previous publications we are aware of, the degree of interpolation polynomials in use is in the order of a few hundreds. We show that it is possible to employ degrees of up to one million or higher without a numerical stability problem or excessive computation times. We also show experimentally that Leja ordering and Fast Leja points enable stable and meaningful interpolation of functions that are just continuous or even discontinuous.