Cubic Spline Quadrature

Background for Cubic Spline.

    Suppose that    are  n+1  points, where  .  The function    is called a cubic spline if there exists  n  cubic polynomials    with coefficients    that satisfy the properties:

I.        
    for  .  

II.        for  .  
    The spline passes through each data point.

III.        for  .
    The spline forms a continuous function over [a,b].

IV.        for  .
    The spline forms a smooth function.

IV.        for  .
    The second derivative is continuous.

Natural Spline.  There exists a unique cubic spline with the free boundary conditions    and  .

Cubic Spline Quadrature.  Integrate the natural cubic spline over the interval [a,b].

Algorithm Natural Cubic Spline.  To construct and evaluate the cubic spline interpolant for the   data points  ,  using the free boundary conditions    and  . Then integrate the natural cubic spline for a quadrature method.

Mathematica Subroutine (Natural Cubic Spline).

Execute the following large group of cells: