Cardinal B-Splines

The Cardinal B Spline is an interpolating function that we used to construct a charge mesh. It works in a similar manner to Lagrange interpolating polynomials⁷, but has several important advantages. The B-Spline does a more accurate interpolation, the charge mesh constructed using a given order of B-Spline will more accurately represent the actual charge distribution than would an interpolation made using Lagrange polynomials of the same order. The other advantage is that they are differentiable. This is important because differentiating the potential function we construct will allow one to easily calculate forces on a particle. Because the kinetics equations used in a molecular dynamics simulation use forces, not potentials, this is a very useful property. Lagrange polynomials are only defined at specific points, and thus are not differentiable. The B-Spline is recursively defined such that it is differentiable a number of times equal to the order of the function minus one.