Romberg Integration

Background for Romberg Integration.  To approximate the integral     by generating a table of approximations, and using    as the final answer.  
The approximations    are stored in a special lower triangular matrix.  The elements    of the first column are computed using the sequential trapezoidal rule
based on    subintervals of  ;  then   is computed using Romberg's rule.  

Elements of row  j  are   .

The algorithm is terminated when   .

The subroutine Romberg is "dynamic" in the following sense.  At the start, we initialize the array with the command   and it contains one row and one element  ,  in which we place one element  .  Next, the increment command,    is used to make  , and the Append command,  ,   is invoked which adds a second row to  , which is initialized with zeros,  .  Then the TrapRule subroutine is called to perform the sequential trapezoidal rule and fill in the first entry    and Romberg's rule is used to fill in the second entry  .  And so it goes, the sequential trapezoidal rule is used to fill in the first entry in succeeding rows and  Romberg's rule fills in rest of the entries in that row.  The algorithm is terminated when  .  

Mathematica Subroutine (Romberg Integration).