Quadratures

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 u_n,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:

Q₁(300)=-2.720856531

Q₁(400)=-2.720938148

Q₁(430)=-2.720950937

By using Maple V R4 on R(n), we obtained the following info:

R(300)=-2.721249539

R(400)=-2.721164891

R(430)=-2.721149108

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.