1. The implement of the first method: kernel.mpl
Function KernelMethod solve the equation system
K(x,y)F(x,y)=A(x,y)G(x)+B(x,y)
The basic input is
KernelMethod(K,A,B,x,y)
where K,A are n by n matrices and B is an n by 1 matrix. The output is a set of solutions. Each solution is an array of length n, indicating G1,G2,...Gn.
For example, the command
KernelMethod(matrix([[z*x^2-x+z]]),matrix([[z]]),matrix([[-x]]),z,x);
returns
{[-1/2*(-1+(1-4*z^2)^(1/2))/z^2]}
As mentioned in Remark 3, we may get extra equations by setting v=1. This can be done by adding an extra parameter [v=1].
The K,A,B corresponding to Sections 4.1 and 4.2 can be generated by the functions K01(n),A01(n),B01(n) and K02(n),A02(n),B02(n) respectively. For the examples in the paper, see paperex.mws.

2. The implement of the second method: kernel-wu.mpl
We use the Maple package wsolve to eliminate extra variables. The input is the same as above and the output are the equations that Gi satisfy. For examples, see eqns.mws.