Sun Aug 18 18:15:30 2002 run on Windows
nc_setup({k,n,NN,KK},{NN*n-n*NN=NN,KK*k-k*KK=KK},left);
p1 := (n-k+1)*NN - (n+1);
p1 := - k*nn + n*nn - n + nn - 1
p2 := (k+1)*KK -(n-k);
p2 := k*kk + k - n + kk
l_g:=nc_groebner ({p1,p2});
l_g := {k*nn - n*nn + n - nn + 1,
k*kk + k - n + kk,
n*nn*kk - n*kk - n + nn*kk - kk - 1}
nc_preduce(p1+p2,l_g);
0
nc_divide (k*p1+p2,p1);
{k,k*kk + k - n + kk}
nc_divide (k*p1+p2,2*p1);
{k,2*k*kk + 2*k - 2*n + 2*kk}
nc_divide (2*k*k*p1 + k*p1 + p2,2*p1);
2
{2*k + k,
2*k*kk + 2*k - 2*n + 2*kk}
nc_factorize (p1*p2);
{ - k*nn + n*nn - n + nn - 1,
k*kk + k - n + kk}
nc_setup({k,n,NN,KK},{NN*n-n*NN=NN,KK*k-k*KK=KK},right);
nc_factorize (p1*p2);
{ - k*nn + n*nn - n + nn - 1,
k*kk + k - n + kk}
% applications to shift operators
nc_setup({n,NN},{NN*n-n*NN=1},left);
n*NN;
n*nn
nc_factorize(ws);
{n,nn}
nc_setup({n,NN},{NN*n-n*NN=1},right);
n*NN;
n*nn
nc_factorize(ws);
{n,nn}
nc_setup({NN,n},{NN*n-n*NN=1},right);
n*NN;
nn*n - 1
nc_factorize(ws);
{n,nn}
nc_setup({NN,n},{NN*n-n*NN=1},left);
n*NN;
nn*n - 1
nc_factorize(ws);
{n,nn}
% Applications to partial differential equations
nc_setup({x,Dx},{Dx*x-x*Dx=1});
p:= 2*Dx^2 + x* Dx^3 + 3*x*Dx + x^2*Dx^2 + 14 + 7*x*Dx;
2 2 3 2
p := x *dx + x*dx + 10*x*dx + 2*dx + 14
nc_factorize p;
2
{x*dx + 2,x*dx + dx + 7}
right_factor(p,1);
2 2 3 2
x *dx + x*dx + 10*x*dx + 2*dx + 14
% no factor of degr 1
right_factor(p,2);
2
x*dx + dx + 7
left_factor(p,2);
x*dx + 2
nc_setup({x,Dx},{Dx*x-x*Dx=1});
q := x**2*dx**2 + 2*x**2*dx + x*dx**3 + 2*x*dx**2
+ 8*x*dx + 16*x + 2*dx**2 + 4*dx$
nc_factorize q;
2
{x*dx + dx + 7,
x,
dx + 2}
right_factor(q,1);
dx + 2
right_factor(q,1,{x});
2 2 2 3 2 2
x *dx + 2*x *dx + x*dx + 2*x*dx + 8*x*dx + 16*x + 2*dx + 4*dx
% no such right factor
right_factor(q,1,{dx});
dx + 2
% looking for factor with degree bound for an individual variable
q := x**6*dx + x**5*dx**2 + 12*x**5 + 10*x**4*dx + 20*x**3
+ x**2*dx**3 - x**2*dx**2 + x*dx**4 - x*dx**3 + 8*x*dx**2
- 8*x*dx + 2*dx**3 - 2*dx**2$
right_factor(q,dx);
6 5 2 5 4 3 2 3 2 2 4 3
x *dx + x *dx + 12*x + 10*x *dx + 20*x + x *dx - x *dx + x*dx - x*dx
2 3 2
+ 8*x*dx - 8*x*dx + 2*dx - 2*dx
right_factor(q,dx^2);
4 2
x + dx - dx
% some coefficient sports
nc_setup({NN,n},{NN*n-n*NN=1},left);
q:=(n*nn)^2;
2 2
q := nn *n - 3*nn*n + 1
nc_factorize q;
{n,
nn,
n,
nn}
nc_preduce(q,{c1+c2*n + c3*nn + c4*n*nn});
2 2 2 2 2 2
(c3 *c4)*nn + (2*c1*c3*c4 - 2*c2*c3 + c3*c4 )*nn + (c2 *c4)*n
2 2 2
+ (2*c1*c2*c4 - 2*c2 *c3 - c2*c4 )*n + (c1 *c4 - 2*c1*c2*c3 + c2*c3*c4)
nc_divide(q,n);
2
{nn *n - 3*nn,1}
nc_cleanup;
end;
Time for test: 179828 ms, plus GC time: 4245 ms