A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation

Two alternative algorithms based on known interpolating functions have been implemented for area calculations in the Lagrange method and the spline method and are described with numerical examples.

Abstract

In the trapezoidal method, linear interpolation between data points tends to overestimate or underestimate the area, depending on the concavity of the curve. In some instances, area estimates can be obtained by linear interpolation of logarithmically transformed data. Two alternative algorithms based on known interpolating functions have been implemented for area calculations. In the Lagrange method, the linear interpolations are replaced by cubic polynomial interpolations. In the spline method, the cubic functions are further modified so that the fitted curves are completely smooth. This report describes their computing procedures with numerical examples.