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:
Q1(400)=-2.720938148
Q1(430)=-2.720950937
By using Maple V R4 on R(n), we obtained the following info:
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.