Defeating the Runge phenomenon for equispaced polynomial interpolation via Tikhonov regularization

Abstract

Runge showed that polynomial interpolation using an equispaced grid often diverges as the degree N of the interpolating polynomial f X 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 - times a smoothness norm. The Tikhonov parameter - 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.