Solving the Problem of Runge Phenomenon by Pseudoinverse Cubic Spline

Abstract

The Runge phenomenon illustrates that equidistant polynomial interpolation of the Runge function will cause wild oscillation near the endpoints of the interpolation interval as the order of the interpolation polynomial increases. In this paper, the pseudo inverse cubic spline (PCS) is presented to accurately approximate the Runge function at equidistant interpolation nodes and solve the problem of Runge phenomenon. PCS is constructed by employing the right pseudo inverse to figure out the minimum norm solution of the cubic spline's second-order derivatives. Thus, unlike the traditional cubic splines that additionally rely on endpoint constraints, PCS only employs the information of interpolation nodes. By PCS, the Runge function is effectively approximated without causing oscillation. Numerical experiments substantiate the efficacy and accuracy of PCS.