The interpolation method of Sprague-Karup

摘要

The usual interpolation method is that of Lagrange. The disadvantage of the method is that in the given points the derivatives of the interpolating polynomials are not equal one to the other. In the method of Hermite, polynomials of a higher degree are used, whose derivatives in the given points are supposed to be equal to the derivatives of the function at the given points. This means that those derivatives must be known.If those derivatives are not known, then in the given points the derivatives may be replaced by approximative values, e.g. based on the interpolating polynomials of Lagrange. Such a method has been described by T. B. Sprague (1880) and in a simplified form by J. Karup (1898). In this paper the formulae are derived. Both methods are illustrated with an example. Some properties and theorems are stated. Tables to simplify the computational work are given. Subroutines for these interpolation methods will be published in a next article.