Numerical Integrations in Maple and Mathematica

The facts about some of the algorithms set up by Maple and Mathematica are as follows:

1. They are not reliable for treating functions with singularities in higher dimensions.
2. We know that the double integral of the function if x²+y²>0 and f(x,y)=0 if x²+y²=0 in the region does not exist and yet the value of its repeated integrals is 0. Both Maple V Release 4 and Mathematica 2.01 give the ''wrong'' answer 0.
3. Both Mathematica and Maple can't handle singularities which lie on the diagonal of a region, examples will be given later.

These facts motivate me to develop new algorithms which shall use my theoretical integration backgrounds.

 
• Numerical integration and theoretical integration
• Uniform regular matrices
• Quadratures
• Open type in two dimensions
• Closed type in two dimensions
• Singularities lie on a diagonal line
• Another Example
• Remarks