Example 1.  Construct the natural cubic spline for the 7 points    
This requires that the spline has the second derivative endpoint constraints   .  

Solution 1.

The B-spline construction requires the two nodes   and   that are out of the range of the abscissas    of the given data points.
Thus we will construct the   in the following special way, with a formula for calculating the equal spacing nodes.

Now start with the list structure for the data points

The ordinates are obtained by taking the second element in the transpose of  .

The list structure or vector form of is different from the subscript notation used in describing the points.
To use the subscript notation we use the following formula for locating ordinates.

Solve the system of equations for the coefficients.

Now form the spline.  It is assumed that the cell defining the basic B-spline function B[x] has been evaluated as well as the functions for    and  .
Since the index on the summation ranges from  i = -1  to  i = n+1 = 7  there are  9  terms in the sum forming the B-spline.  

Let's look at the graph.

(c) John H. Mathews 2004