Open type in two dimensions

We mention an open type quadrature in two dimensions (see [YC[) by using two uniformly regular matrices, c_nk, d_ml, and denote them by c(n,k) and d(m,l) for computation purpose. Now set , and consider the function if , and , and g(x,y)=0 if x=y=0. First, we define the followings:

Next we define the following open quadrature:

g(l(n,k),r(m,l))+g(r(m,l),l(n,k))+g(l(n,k),l(m,l))))

We obtain the following information:

Q(20,20)=3.94397632

Q(30,30)=3.97226585

Q(40,40)=3.98311667

For the Maple workshet on this open type quadrature, click here or go to the Appendix section. We could speed up the rate of convergence for this type of function by considering the following closed type quadrature.