Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock–Chebyshev subset interpolation

Abstract

Runge showed more than a century ago that polynomial interpolation of a function f( x), using points evenly spaced on x ∈ [ - 1 , 1 ] , could diverge on parts of this interval even if f( x) was analytic everywhere on the interval. Least-squares fitting of a polynomial of degree N to an evenly spaced grid with P points should improve accuracy if P ≫ N . We show through numerical experiments that such an overdetermined fit reduces but does not eliminate the Runge Phenomenon. More precisely, define β ≡ ( N + 1 ) / P . The least-squares fit will fail to converge everywhere on [−1, 1] as N → ∞ for fixed β if f( x) has a singularity inside a curve D ( β ) in the complex-plane. The width of the region enclosed by the convergence boundary D shrinks as β diminishes (i. e., more points for a fixed polynomial degree), but shrinks to the interval [−1, 1] only when β → 0 . We also show that the Runge Phenomenon can be completely defeated by interpolation on a “mock–Chebyshev” grid: a subset of ( N + 1 ) points from an equispaced grid with O ( N 2 ) points chosen to mimic the non-uniform N + 1 -point Chebyshev–Lobatto grid.