Consider the following closed type quadrature:
We would like to experiment this quadrature with the Scientific Workplace (which uses Maple as a tool for computation). But first we need to make the following adjustments for computation purpose. We define the right and left endpoints as follows: and which correspond to un,k and respectively.
We define our first closed type quadrature as follows:
We note that the first term of Q1(n), (1/2)a(n,1)f(r(n,1)), is a tail term to take care of functions with a singularity, and the second term of Q1(n), denoted by Q(n) is a trapezoidal sum. Thus, we may call the quadrature, Q1(n), to be the adaptive trapezoidal sum. We shall use the combination of Q1(n) and Q(n) to come up with the rule for Richardson extrapolation integration as follows
Example: Consider the function , if , and f(0)=0. (We notice that f has a singularity at x=0.) Use Q1(n) to approximate If we use Evaluate numerically with Scientific Workplace under ''Maple'', we get the following numeric results:
By using Maple V R4 on R(n), we obtained the following info:
We observed that the Ricahrdson extrapolation gives better estimate, the answer above is accurate up to 4 digits. We note that when we increase n, we will be warned of the existence of the singularity at x=0 . For a maple worksheet on this quadrature, click here or go to the Appendix. To further investigate the convergence or divergence of this integral, we could write a separate computer program to run our quadrature.