Interpolation and Approximation of Functions

Interpolation describes the problem of finding a curve that passes through a given set of real values at real data points, which are sometimes called abscissae or nodes. The theory of interpolation is important as a basis for numerical integration known also as quadrature. Approximation theory on the other hand seeks an approximation such that an error norm is minimized. The Newton interpolation formula has several advantages over the Lagrange formula. The degree of an interpolating polynomial can be increased by adding more points and more terms. The accuracy of an interpolation polynomial depends on how far the point of interest is from the middle of the interpolation points used. The Newton form of the interpolating polynomial can be viewed as one of a class of methods for generating successively higher order interpolation polynomials. The problem with polynomial interpolation is that with increasing degree the polynomial ‘wiggles’ from data point to data point.