1. The implement of the induction method: km-sys.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 is an n by n matrice, A is an n by l matrice and B is an n by 1 matrix. The output is a set of solutions. Each solution is an array of length l, indicating G1,G2,...Gl.

For example, the command

KernelMethod(matrix([[(x-y)^2]]),matrix([[x,y]]),matrix([[y^2]]),x,y);
returns
{[x, -2*x]}

We may get extra equations by setting y to be some special values. This can be done by input the list [[y=y1, M1],[y=y2, M2], ...], where Mi is the transfer matrix of F(x,yi) and G(x). For example, if G(x)=F(x,0), then we can input KernelMethod(K,A,B,x,y,[[y=0,Id(n)]]), where Id(n) is the identity matrix of order n.

For example, the command

KernelMethod(matrix([[(x-y)^2]]),matrix([[x,y,x-y]]),matrix([[y^2]]),x,y,[[y=0,matrix([[1,-1,1]])]]);
returns
{[1-2*x, x-1, 3*x-1]}

For examples, see examples.mws

2. The implement of the elimination method: km-sys.mpl

You need first read the maple program for wu's method: wsolve.mpl

The usage is exactly the same as above. The output will be the algebraic equations that \$g_1, \ldots, g_l\$ satisfy. See the examples KM-wu-examples.mws

3. The examples in the paper: PaperEx.mws