Cubic (sub-)spline for data with derivatives

Introduction

In practice a function is often tabulated together with its derivative (for example, after numerical integration of a second order ordinary differential equation). The table is then given as {x_i, y_i, y'_i}_i=1,..,n , where y_i is the value of the tabulated function, and y'_i is the value of the derivative of the tabulated function at the point x_i.

Project

Given a data-set, {x_i, y_i, y'_i}_i=1,..,n , build a cubic sub-spline,

S(x) _{x∈[xi,xi+1]} =S_i(x)

where

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

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)=y'_i,
S'_i(x_i+1)=y'_i+1.

See the subsection "Akima sub-spline interpolation" for the inspiration.

Extra

1. Derivative and integral of the spline.
2. Make the second derivative of the spline continuous by increasing the order of the spline to four, for example, in the form

   S_i(x)= y_i +b_i(x-x_i) +c_i(x-x_i)² +d_i(x-x_i)³ +e_i(x-x_i)²(x-x_i+1)² ,

   and choosing the coefficients e_i such that the spline has continuous second derivative.