Example 1  Investigate Romberg integration for approximating the integral  .  
Use the tolerances .  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 Romberg table using   tol = 0.001
What do the entries in the first column mean ?
What do the entries in the second column mean ?
Which entries in the table are used to determine if Romberg integration is converging ?

1 (c). The last entry in the table is  .  Let's find it.
    (This is possible because  j  is a global variable in the subroutine.)

1 (d). Look at 10 digits in  .

1 (e). Are all 10 digits correct ?   Why ?  
Be sure to support your answer to this question !
NO.  The subroutine was called with the accuracy  .  

1 (f). Determine how to call the Romberg integration subroutine so that it will achieve 10 digits of accuracy.   
You will need to experiment to find the solution.  Do it !
Report the answer with 10 digits of accuracy.
How many function evaluations were required to achieve this answer ?

We suspect the following answer might have 10 digits of accuracy.

Since the last row was row 7, the sequential trapezoidal rule used the following number of function calls.

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