Example 4.  Error analysis.  The Taylor series method of order    allegedly has a Final Global Error (FGE) of order  .  Hence, the error at the right endpoint should appear to decrease by when the number of sub-intervals is doubled. Use the D.E. in exercise 1 and investigate this behavior for    sub-intervals of  .  
Solution  4.

Notice. The subroutine TayorMeth  stores the values in a list starting with subscript 1 and ending with subscript  . We need only check this last point with the value obtained from the analytic solution.  

Does the error decrease in the fashion ?  i.e.   should be , etc.
Do the following ratios tend to 16 ?  

 {-79.3339, 14.5167, 15.8344, 15.9399}

(c) John H. Mathews 2003