Piecewise interpolation. w(x) is the function to be interpolated between the end points (or nodes) x_i and x_i + 1. We assume that we know w and its derivatives at the end points: w_i = w(x_i), w_i + 1 = w(x_i + 1), , . The local interpolation function, H_i(x), is a cubic Hermite polynomial. It is a third degree polynomial defined between x_i and x_i + 1, from the four relations: , , , . The arrows indicate the slopes of H(x) at the end points, or, equivalently, the derivatives , . See Sect. 4.2 for detailed explanations.