The implement of the induction method: km-sys.mpl

Function

*KernelMethod*solve the equation system The basic input is*K(x,y)F(x,y)=A(x,y)G(x)+B(x,y)* where*KernelMethod(K,A,B,x,y)**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

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

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

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

The examples in the paper: PaperEx.mws