The ordinary cubic spline simetimes makes unpleasant wiggles, for example, around an outlier, or around a step-like feature of the tabulated function (read the Akima sub-spline chapter in the lecture notes). Here is yet another attempt to reduce the wiggles by building a sub-spline.

Consider a data set {x_i, y_i}_i=1,..,n which represents a tabulated function.

Implement a sub-spline of this data using the following algorithm:

1. For each inner point x_i=2,..,n-1, build a quadratic interpolating polynomial through the points x_i-1, x_i, x_i+1 and estimate, using this polynomial, the first derivative, p_i, of the function at the point x_i;
2. For the the first and the last points estimate p₁ and p_n from the the same polynomial that you used to estimate correspondingly p₂ and p_n-1.
3. Now you have a data set {x_i, y_i, p_i}_i=1,..,n where the function is tabulated together with its derivative: build a cubic sub-spline for it,

   S_i(x)= y_i +b_i(x-x_i) +c_i(x-x_i)² +d_i(x-x_i)³,

   where for each interval the three coefficients b_i, c_i, d_i are determined by the three conditions,

   S_i(x_i+1)=y_i+1,
   S'_i(x_i)=p_i,
   S'_i(x_i+1)=p_i+1.