A geometrical interpretation of the 2n-th central difference

Many algorithms used for data smoothing, data classification and error detection require the calculation of the distance from a point to the polynomial interpolating its 2n neighbors (n on each side). This computation, if performed naively, would require the solution of a system of equations and could create numerical problems. This note shows that if the data is equally spaced, then this calculation can be performed using a simple recursion formula.