DEFEATING THE RUNGE PHENOMENON FOR EQUISPACED POLYNOMIAL INTERPOLATION VIA TIKHONOV REGULARIZATION

Runge showed that polynomial interpolation using an equispaced grid often diverges as the degree N of the interpolating polynomial f(N) increases, even when f(x) is analytic over the whole interval. We suppress Runge divergence by defining the approximating polynomial as the minimizer of the sum of the interpolation residual plus a constant alpha times a smoothness norm. The Tikhonov parameter alpha can be determined easily through the method of the L-shaped curve. The resulting "Tikhonov approximant" is at least as accurate as the truncated, N(th) degree Chebyshev series for the same function.