Example 1  Investigate cubic spline quadrature for approximating the integral  .  
Use  11, 21, 41 and 81 nodes.  Compare with the analytic or "true value" of the integral.

Solution 1.

1 (a). Plot the function over the interval  [0, 1.25].

1 (b). Construct the cubic spline for 11 nodes and use it for quadrature.

1 (c). Construct the cubic spline for 21 nodes and use it for quadrature.

1 (d). Construct the cubic spline for 41 nodes and use it for quadrature.

1 (e). Construct the cubic spline for 41 nodes and use it for quadrature.

1 (f). Compare the results from parts b-d.

m sample points
11
21
41
81

1 (g). Use Mathematica to find the analytic solution to the integral, i.e. the "true value" of the integral.

1 (h). How close did our last numerical approximation using Romberg integration come to the "true value" of the integral.

(c) John H. Mathews 2004