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Diff for /OpenXM/src/k097/lib/minimal/minimal-test.k between version 1.4 and 1.16

version 1.4, 2000/06/14 07:44:05 version 1.16, 2000/08/09 03:45:27
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 /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal-test.k,v 1.3 2000/06/09 08:04:54 takayama Exp $ */  /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal-test.k,v 1.15 2000/08/02 05:14:31 takayama Exp $ */
 load["minimal.k"];  load["minimal.k"];
 def test5() {  
   local a,b,c,cc,v;  
   a = Sannfs3_laScala2("x^3-y^2*z^2");  
   b = a[0];  
   v = [x,y,z];  
   c = Skernel(b[0],v);  
   c = c[0];  
   sm1_pmat([c,b[1],v]);  
   Println("-----------------------------------");  
   cc = sm1_res_div(c,b[1],v);  
   sm1_pmat(sm1_gb(cc,v));  
   c = Skernel(b[1],v);  
   c = c[0];  
   cc = sm1_res_div(c,b[2],v);  
   sm1_pmat(sm1_gb(cc,v));  
   return(a);  
 }  
 def test6() {  
   local a,b,c,cc,v;  
   a = Sannfs3("x^3-y^2*z^2");  
   b = a[0];  
   v = [x,y,z];  
   c = Skernel(b[0],v);  
   c = c[0];  
   sm1_pmat([c,b[1],v]);  
   Println("-------ker = im for minimal ?---------------------");  
   cc = sm1_res_div(c,b[1],v);  
   sm1_pmat(sm1_gb(cc,v));  
   c = Skernel(b[1],v);  
   c = c[0];  
   cc = sm1_res_div(c,b[2],v);  
   sm1_pmat(sm1_gb(cc,v));  
   Println("------ ker=im for Schreyer ?------------------");  
   b = a[3];  
   c = Skernel(b[0],v);  
   c = c[0];  
   sm1_pmat([c,b[1],v]);  
   cc = sm1_res_div(c,b[1],v);  
   sm1_pmat(sm1_gb(cc,v));  
   c = Skernel(b[1],v);  
   c = c[0];  
   cc = sm1_res_div(c,b[2],v);  
   sm1_pmat(sm1_gb(cc,v));  
   return(a);  
 }  
   
 /* May 23, Tue */  
 def test7() {  
   local a,b,c,cc,v;  
   a = Sannfs3_laScala2("x^3-y^2*z^2");  
   b = a[0];  
   v = [x,y,z];  
   c = Skernel(b[0],v);  
   c = c[0];  
   sm1_pmat([c,b[1],v]);  
   Println("-------ker = im for minimal ?---------------------");  
   cc = sm1_res_div(c,b[1],v);  
   sm1_pmat(sm1_gb(cc,v));  
   c = Skernel(b[1],v);  
   c = c[0];  
   cc = sm1_res_div(c,b[2],v);  
   sm1_pmat(sm1_gb(cc,v));  
   Println("------ ker=im for Schreyer ?------------------");  
   b = a[3];  
   c = Skernel(b[0],v);  
   c = c[0];  
   sm1_pmat([c,b[1],v]);  
   cc = sm1_res_div(c,b[1],v);  
   sm1_pmat(sm1_gb(cc,v));  
   c = Skernel(b[1],v);  
   c = c[0];  
   cc = sm1_res_div(c,b[2],v);  
   sm1_pmat(sm1_gb(cc,v));  
   return(a);  
 }  
   
 def sm1_resol1(p) {  def sm1_resol1(p) {
   sm1(" p resol1 /FunctionValue set ");    sm1(" p resol1 /FunctionValue set ");
 }  }
   
   
 def test8() {  def test8() {
   local p,pp,ans,b,c,cc,ww,ww2;    local p,pp,ans,b,c,cc,ww,ww2;
   f = "x^3-y^2*z^2";    f = "x^3-y^2*z^2";
Line 118  def test8() {
Line 41  def test8() {
    SisComplex(a):     SisComplex(a):
 */  */
   
 def test8a() {  
   local p,pp,ans,b,c,cc,ww, ans_all;  
   f = "x^3-y^2*z^2";  
   p = Sannfs(f,"x,y,z");  
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");  
   ww = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];  
   /* Removed "x",1, ... ===> It causes an error. I do not know the reason.*/  
   Sweyl("x,y,z",ww);  
   pp = Map(p,"Spoly");  
   /* return(pp); */  
   /* pp =  
      [y*Dy-z*Dz , -2*x*Dx-3*y*Dy+1 , 2*x*Dy*Dz^2-3*y*Dx^2 ,  
       2*x*Dy^2*Dz-3*z*Dx^2 , 2*x*z*Dz^3-3*y^2*Dx^2+4*x*Dz^2 ]  
   */  
   ans_all = Sschreyer(pp);  
   ans = ans_all[0];  
   /* ans = sm1_resol1([pp,"x,y,z",ww]); */  
   /* Schreyer is in ans. */  
   
   v = [x,y,z];  
   b = ans;  
   Println("------ ker=im for Schreyer ?----- wrong method!!!-----------");  
   c = Skernel(b[0],v);  
   c = c[0];  
   sm1_pmat([c,b[1],v]);  
   cc = sm1_res_div(c,b[1],v);  
   sm1_pmat(sm1_gb(cc,v));  
   c = Skernel(b[1],v);  
   c = c[0];  
   cc = sm1_res_div(c,b[2],v);  
   sm1_pmat(sm1_gb(cc,v));  
   return(ans);  
 }  
   
 /* Comparing two constructions */  
 def test9() {  
   local p,pp,ans,b,c,cc,ww,ww2,ans_all,ans2;  
   f = "x^3-y^2*z^2";  
   p = Sannfs(f,"x,y,z");  
   ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];  
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");  
   Sweyl("x,y,z",ww2);  
   pp = Map(p,"Spoly");  
   ans = sm1_resol1([pp,"x,y,z",ww2]);  
   
   f = "x^3-y^2*z^2";  
   p = Sannfs(f,"x,y,z");  
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");  
   ww = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];  
   Sweyl("x,y,z",ww);  
   pp = Map(p,"Spoly");  
   ans_all = Sschreyer(pp);  
   ans2 = ans_all[0];  
   
   return([ans,ans2]);  
   
 }  
   
 /* Check if the complex by Sschreyer() is exact or not in our example? */  
 def test10() {  
   local p,pp,ans,b,c,cc,ww,ww2,ans_all,ans2, r;  
   f = "x^3-y^2*z^2";  
   p = Sannfs(f,"x,y,z");  
   ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];  
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");  
   Sweyl("x,y,z",ww2);  
   pp = Map(p,"Spoly");  
   ans = sm1_resol1([pp,"x,y,z",ww2]);  
   
   f = "x^3-y^2*z^2";  
   p = Sannfs(f,"x,y,z");  
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");  
   ww = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];  
   Sweyl("x,y,z",ww);  
   pp = Map(p,"Spoly");  
   ans_all = Sschreyer(pp);  /* Schreyer by LaScala-Stillman */  
   ans2 = ans_all[0];  
   
   sm1(" /gb.verbose 1 def ");  
   
   ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];  
   Sweyl("x,y,z",ww2);  
   ans2 = ReParse(ans2);  
   r= IsExact_h(ans2,[x,y,z]);  
   Print(r);  
   
   return([r,[ans,ans2]]);  
   
 }  
   
 def test11() {  def test11() {
   local  a;    local  a;
   a = test_ann3("x^3-y^2*z^2");    a = test_ann3("x^3-y^2*z^2");
   return(a);    return(a);
 }  }
 /* f should be a string. */  /* f should be a string. */
   /* a=test_ann3("x^3+y^3+z^3");
   It returns the following resolution in 1.5 hours.  June 14, 2000.
    [
     [
       [    x*Dx+y*Dy+z*Dz-3*h^2 ]
       [    -z*Dy^2+y*Dz^2 ]
       [    -z*Dx^2+x*Dz^2 ]
       [    -y*Dx^2+x*Dy^2 ]
     ]
     [
       [    0 , -x , y , -z ]
       [    z*Dx^2-x*Dz^2 , x*Dy , x*Dx+z*Dz-3*h^2 , z*Dy ]
       [    y*Dx^2-x*Dy^2 , -x*Dz , y*Dz , x*Dx+y*Dy-3*h^2 ]
       [    0 , Dx^2 , -Dy^2 , Dz^2 ]
       [    z*Dy^2-y*Dz^2 , x*Dx+y*Dy+z*Dz-2*h^2 , 0 , 0 ]
     ]
     [
       [    -x*Dx+3*h^2 , y , -z , 0 , -x ]
       [    Dy^3+Dz^3 , Dy^2 , -Dz^2 , x*Dx+y*Dy+z*Dz , -Dx^2 ]
     ]
    ]
   */
 def test_ann3(f) {  def test_ann3(f) {
   local a,v,ww2,ans2;    local a,v,ww2,ans2;
   a = Sannfs3_laScala2(f);    a = Sannfs3(f);
   ans2 = a[0];    ans2 = a[0];
   v = [x,y,z];    v = [x,y,z];
   ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];    ww2 = [["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]];
Line 224  def test_ann3(f) {
Line 79  def test_ann3(f) {
   ans2 = ReParse(ans2);    ans2 = ReParse(ans2);
   r= IsExact_h(ans2,[x,y,z]);    r= IsExact_h(ans2,[x,y,z]);
   Println(r);    Println(r);
   return([r,ans2]);    return([r,ans2,a]);
 }  }
 def test11a() {  def test11a() {
   local a,v,ww2,ans2;    local a,v,ww2,ans2;
Line 283  def test12() {
Line 138  def test12() {
   ans2 = ReParse(ans2); /* DO NOT FORGET! */    ans2 = ReParse(ans2); /* DO NOT FORGET! */
   r= IsExact_h(ans2,[x,y,z]);    r= IsExact_h(ans2,[x,y,z]);
   Println(r);    Println(r);
   Println("It may stop by non-exact statement. The code of Sminimal_v (non-LaScala-Stillman contains bugs.");  
   return([r,ans2]);    return([r,ans2]);
 }  }
   
 def test13() {  def test13() {
   Println("test13 try to construct a minimal free resolution");    Println("test13 try to construct a minimal free resolution");
   Println("of a GKZ system [[1,2]]. 6/12, 2000.");    Println("of a GKZ system [[1,2]]. 6/12, 2000.");
     ans2 = GKZ([[1,2]],[0]);
      /* Be careful!! It resets the grade to module1, not module1v */
   ww2 = [["x1",-1,"x2",-1,"Dx1",1,"Dx2",1]];    ww2 = [["x1",-1,"x2",-1,"Dx1",1,"Dx2",1]];
   Sweyl("x1,x2",ww2);    Sweyl("x1,x2",ww2);
   ans2 = GKZ([[1,2]],[0]);  
   ans2 = ReParse(ans2[0]);    ans2 = ReParse(ans2[0]);
     Println(ans2);
   return(Sminimal(ans2));    return(Sminimal(ans2));
 }  }
   
 def test14() {  def test14() {
   Println("test14 try to construct a minimal free resolution");    Println("test14 try to construct a minimal free resolution");
   Println("of a GKZ system [[1,2,3]]. 6/12, 2000.");    Println("of a GKZ system [[1,2,3]]. 6/12, 2000.");
     ans2 = GKZ([[1,2,3]],[0]);
        /* It stops by the strategy error.
           July 26, 2000. It works fine after fixing a bug in resol.c */
   ww2 = [["x1",-1,"x2",-1,"x3",-1,"Dx1",1,"Dx2",1,"Dx3",1]];    ww2 = [["x1",-1,"x2",-1,"x3",-1,"Dx1",1,"Dx2",1,"Dx3",1]];
   Sweyl("x1,x2,x3",ww2);    Sweyl("x1,x2,x3",ww2);
   ans2 = GKZ([[1,2,3]],[0]);  /* It stops by the strategy error. */  
   ans2 = ReParse(ans2[0]);    ans2 = ReParse(ans2[0]);
   return(Sminimal(ans2));    return(Sminimal(ans2));
 }  }
Line 315  def test14a() {
Line 173  def test14a() {
   ans2 = [x1*Dx1+2*x2*Dx2+3*x3*Dx3 , Dx1^2-Dx2*h , -Dx1*Dx2+Dx3*h ,    ans2 = [x1*Dx1+2*x2*Dx2+3*x3*Dx3 , Dx1^2-Dx2*h , -Dx1*Dx2+Dx3*h ,
           Dx2^2-Dx1*Dx3 ];            Dx2^2-Dx1*Dx3 ];
   ans2 = ReParse(ans2);    ans2 = ReParse(ans2);
   return(Sminimal(ans2,"homogenized"));    return(Sminimal(ans2,["homogenized"]));
 }  }
   
 def test15() {  def test15() {
   Println("test15 try to construct a minimal free resolution");    Println("test15 try to construct a minimal free resolution");
   Println("of a GKZ system [[1,2,3]] by the order filt. 6/12, 2000.");    Println("of a GKZ system [[1,2,3]] by the order filt. 6/12, 2000.");
   ww2 = [["Dx1",1,"Dx2",1,"Dx3",1]];    ww2 = [["Dx1",1,"Dx2",1,"Dx3",1]];
   Sweyl("x1,x2,x3",ww2);  
   ans2 = GKZ([[1,2,3]],[0]);    ans2 = GKZ([[1,2,3]],[0]);
     Sweyl("x1,x2,x3",ww2);
   ans2 = ReParse(ans2[0]);    ans2 = ReParse(ans2[0]);
   return(Sminimal(ans2));    a = Sminimal(ans2);
     Println("Minimal Resolution is "); sm1_pmat(a[0]);
     Sweyl("x1,x2,x3");
     ans3 = ReParse(a[0]);
     r= IsExact_h(ans3,[x1,x2,x3]);
     Println(r);
     return(a);
 }  }
   
 def test15b() {  def test15b() {
Line 335  def test15b() {
Line 199  def test15b() {
   Sweyl("x1,x2,x3",ww2);    Sweyl("x1,x2,x3",ww2);
   ans2 = [Dx1^2-Dx2*h , -Dx1*Dx2+Dx3*h , Dx2^2-Dx1*Dx3 ];    ans2 = [Dx1^2-Dx2*h , -Dx1*Dx2+Dx3*h , Dx2^2-Dx1*Dx3 ];
   ans2 = ReParse(ans2);    ans2 = ReParse(ans2);
   return(Sminimal(ans2,"homogenized"));    return(Sminimal(ans2,["homogenized"]));
 }  }
   
   def test15c() {
     Println("test15c try to construct a minimal free resolution ");
     Println("of a GKZ system [[1,2,3]] by -1,1");
     ww2 = [["Dx1",1,"Dx2",1,"Dx3",1,"x1",-1,"x2",-1,"x3",-1]];
     ans2 = GKZ([[1,2,3]],[0]);
     Sweyl("x1,x2,x3",ww2);
     ans2 = ReParse(ans2[0]);
     a = Sminimal(ans2);
     Println("Minimal Resolution is "); sm1_pmat(a[0]);
     Sweyl("x1,x2,x3");
     ans3 = ReParse(a[0]);
     r= IsExact_h(ans3,[x1,x2,x3]);
     Println(r);
     return(a);
   }
 def test16() {  def test16() {
   Println("test16 try to construct a minimal free resolution");    Println("test16 try to construct a minimal free resolution");
   Println("of a GKZ system [[1,2,3,5]] by the order filt. 6/12, 2000.");    Println("of a GKZ system [[1,2,3,5]] by the order filt. 6/12, 2000.");
Line 361  def test16b() {
Line 240  def test16b() {
   return(Sminimal(ans3));    return(Sminimal(ans3));
 }  }
   
   
   def test17() {
      a=Sannfs3("x^3-y^2*z^2");
      b=a[0]; w = ["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1];
      Sweyl("x,y,z",[w]); b = Reparse(b);
      c=Sinit_w(b,w);
      Println("Resolution (b)----");
      sm1_pmat(b);
      Println("Initial (c)----");
      sm1_pmat(c);
      Println(IsExact_h(c,"x,y,z"));
   }
   
   def test_if_v_strict(resmat,w,v) {
      local b,c,g;
      Sweyl(v,[w]); b = Reparse(resmat);
      Println("Degree shifts ");
      Println(SgetShifts(b,w));
      c=Sinit_w(b,w);
      Println("Resolution (b)----");
      sm1_pmat(b);
      Println("Initial (c)----");
      sm1_pmat(c);
      Println("Exactness of the resolution ---");
      Println(IsExact_h(b,v));
      Println("Exactness of the initial complex.---");
      Println(IsExact_h(c,v));
      g = Sinvolutive(b[0],w);
      /* Println("Involutive basis ---");
         sm1_pmat(g);
         Println(Sinvolutive(c[0],w));
         sm1(" /gb.verbose 1 def "); */
      Println("Is same ideal?");
      Println(IsSameIdeal_h(g,c[0],v));
   }
   def test17b() {
      a=Sannfs3("x^3-y^2*z^2");
      b=a[0]; w = ["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1];
      test_if_v_strict(b,w,"x,y,z");
      return(a);
   }
   
   def test18() {
      a=Sannfs2("x^3-y^2");
      b=a[0]; w = ["x",-1,"y",-1,"Dx",1,"Dy",1];
      test_if_v_strict(b,w,"x,y");
      return(a);
   }
   
   def test19() {
     Println("test19 try to construct a minimal free resolution and check if it is v-strict.");
     Println("of a GKZ system [[1,2,3]] by -1,1");
     ww2 = ["Dx1",1,"Dx2",1,"Dx3",1,"x1",-1,"x2",-1,"x3",-1];
     ans2 = GKZ([[1,2,3]],[0]);
     Sweyl("x1,x2,x3",[ww2]);
     ans2 = ReParse(ans2[0]);
     a = Sminimal(ans2);
     Println("Minimal Resolution is "); sm1_pmat(a[0]);
     b = a[0];
     test_if_v_strict(b,ww2,"x1,x2,x3");
     return(a);
   }
   
   /* Need more than 100M memory. 291, 845, 1266, 1116, 592 : Schreyer frame.
      I've not yet tried to finish the computation. */
   def test20() {
     w = ["Dx1",1,"Dx2",1,"Dx3",1,"Dx4",1,"x1",-1,"x2",-1,"x3",-1,"x4",-1];
     ans2 = GKZ([[1,1,1,1],[0,1,3,4]],[0,0]);
     Sweyl("x1,x2,x3,x4",[w]);
     ans2 = ReParse(ans2[0]);
     a = Sminimal(ans2);
     Println("Minimal Resolution is "); sm1_pmat(a[0]);
     b = a[0];
     /* test_if_v_strict(b,w,"x1,x2,x3,x4"); */
     return(a);
   }
   def test20b() {
     w = ["Dx1",1,"Dx2",1,"Dx3",1,"Dx4",1,"x1",-1,"x2",-1,"x3",-1,"x4",-1];
     ans2 = GKZ([[1,1,1,1],[0,1,3,4]],[1,2]);
     Sweyl("x1,x2,x3,x4",[w]);
     ans2 = ReParse(ans2[0]);
     a = Sminimal(ans2);
     Println("Minimal Resolution is "); sm1_pmat(a[0]);
     b = a[0];
     /* test_if_v_strict(b,w,"x1,x2,x3,x4"); */
     return(a);
   }
   
   def test21() {
      a=Sannfs3("x^3-y^2*z^2+y^2+z^2");
      /* a=Sannfs3("x^3-y-z");  for debug */
      b=a[0]; w = ["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1];
      test_if_v_strict(b,w,"x,y,z");
      Println("Degree shifts of Schreyer resolution ----");
      Println(SgetShifts(Reparse(a[4,0]),w));
      return(a);
   }
   def test21b() {
     local i,j,n,sss, maxR, ttt,ans,p;
     Println("The dimensions of linear spaces -----");
     /* sss is the SgetShifts of the Schreyer resol. */
     sss=
     [[    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ,
      [ -1, -1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3 ] ,
      [ 0, 1, -1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 3, 2, 2, 1, 4, 3, 3, 2, 0, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 1, 3, 3, 3, 3, 4 ] ,
      [ 1, 0, 2, 3, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 0, 3, 1, 3, 2, 3, 4 ] ,
      [ 1, 1 ]  ] ;
      maxR = 2; /* Maximal root of the b-function. */
     n = Length(sss);
     for (i=0; i<n; i++) {
       ttt = sss[i];
       ans = 0;
       for (j=0; j<Length(ttt); j++) {
         p = ttt[j] + maxR + 3; /* degree */
         if (p >= 0) {
           ans = ans + CancelNumber(p*(p-1)*(p-2)/(3*2*1));
           /* Add the number of monomials */
         }
       }
       Print(ans); Print(", ");
     }
     Println(" ");
   }
   def test22() {
      a=Sannfs3("x^3+y^3+z^3");
      b=a[0]; w = ["x",-1,"y",-2,"z",-3,"Dx",1,"Dy",2,"Dz",3];
      test_if_v_strict(b,w,"x,y,z");
      return(a);
   }
   
   def FillFromLeft(mat,p,z) {
     local m,n,i,j,aa;
     m = Length(mat); n = Length(mat[0]);
     aa = NewMatrix(m,n+p);
     for (i=0; i<m; i++) {
       for (j=0; j<p; j++) {
         aa[i,j] = z; /* zero */
       }
       for (j=0; j<n; j++) {
         aa[i,j+p] = mat[i,j];
       }
     }
     return(aa);
   }
   
   def FillFromRight(mat,p,z) {
     local m,n,i,j,aa;
     m = Length(mat); n = Length(mat[0]);
     aa = NewMatrix(m,n+p);
     for (i=0; i<m; i++) {
       for (j=n; j<n+p; j++) {
         aa[i,j] = z; /* zero */
       }
       for (j=0; j<n; j++) {
         aa[i,j] = mat[i,j];
       }
     }
     return(aa);
   }
   
   def test23() {
     w = ["Dx1",1,"Dx2",1,"Dx3",1,"x1",-1,"x2",-1,"x3",-1];
     Sweyl("x1,x2,x3",[w]);
     d2 = [[Dx1^2-Dx2*h] , [-Dx1*Dx2+Dx3*h] , [Dx2^2-Dx1*Dx3] ];
     d1 = [[-Dx2, -Dx1, -h],[Dx3,Dx2,Dx1]];
     LL = x1*Dx1 + 2*x2*Dx2+3*x3*Dx3;
     /* It is exact for LL = Dx1 + 2*Dx2+3*Dx3;  */
     u1 = [[LL+4*h^2,Poly("0")],[Poly("0"),LL+5*h^2]];
     u2 = [[LL+2*h^2,Poly("0"),Poly("0")],
           [Poly("0"),LL+3*h^2,Poly("0")],
           [Poly("0"),Poly("0"),LL+4*h^2]];
     u3 = [[LL]];
     Println("Checking if it is a double complex. ");
     Println("u^2 d^2 - d^2 u^3");
     sm1_pmat(u2*d2 - d2*u3);
     Println("u^1 d^1 - d^1 u^2");
     sm1_pmat(u1*d1 - d1*u2);
     aa = [
            Join(u3,d2),
            Join(FillFromLeft(u2,1,Poly("0"))-FillFromRight(d2,3,Poly("0")),
                 FillFromLeft(d1,1,Poly("0"))),
            FillFromLeft(u1,3,Poly("0"))-FillFromRight(d1,2,Poly("0"))
          ];
     Println([ aa[1]*aa[0], aa[2]*aa[1] ]);
     r= IsExact_h(aa,[x1,x2,x3]);
     Println(r);
     /* sm1_pmat(aa); */
     return(aa);
   }
   
   
   
   
   
   
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