High-accuracy reconstruction of a function f(x) when only df(x)/dx or d²f(x)/dx²is known at discrete measurement points

Abstract

The task of reconstructing a function <i>f(x)</i> from a set of measurements of the first or second derivative is considered. Methods for numerical integration are briefly outlined and discussed with regard to their potential application to the current task. Furthermore, cubic spline interpolation of the data followed by integration of the interpolation spline is proposed, and the influence of noise on the accuracy of the reconstructed function <i>f(x) </i>is examined. Results of numerical simulations for several test problems are presented for both noiseless and noisy data, and the efficiency of different methods is compared. Best results were obtained by cubic spline interpolation and subsequent integration of the spline function. The examples presented demonstrate that the reconstruction of a function <i>f(x) </i>from a set of measurements of the first or second derivative can be performed with high accuracy.