source: git/Singular/LIB/numerAlg.lib @ 33b509

///////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Algebraic Geometry";
info="
LIBRARY:  NumerAlg.lib    Numerical Algebraic Algorithm
OVERVIEW:
        The library contains procedures to
        test the inclusion, the equality of two ideals defined by polynomial systems,
        compute the degree of a pure i-dimensional component of an algebraic variety
         defined by a polynomial system,
        compute the local dimension of an algebraic variety defined by a polynomial
         system at a point computed as an approximate value. The use of the library
         requires to install Bertini (http://www.nd.edu/~sommese/bertini).

AUTHOR: Shawki AlRashed, rashed@mathematik.uni-kl.de; sh.shawki@yahoo.de
PROCEDURES:

 Incl(ideal I, ideal J);   test if I containes J

 Equal(ideal I, ideal J);  test if I equals to J

 Degree(ideal I, int i);   computes the degree of a pure i-dimensional

 NumLocalDim(ideal I, p);  numerical local dimension at a point computed as
                                  an approximate value
";

LIB "numerDecom.lib";

///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc Degree(ideal I,int i)
"USAGE:  Degree(ideal I,int i); I ideal,  i positive integer
RETURN:  the degree of the pure i-dimensional component of the algebraic
          variety defined by I
EXAMPLE: example Degree; shows an example
"
{
 def S=basering;
 def W=WitSet(I);
 setring W;
 int j;
 if(size(W(i)[1])>1)
 {
  j=size(W(i));
 }
 else
 {
  j=-1; // no component of dimension i
 }
 "The Degree of Component";
 j;
 setring S;
 return (W);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   poly f1=(x2+y2+z2-6)*(x-y)*(x-1);
   poly f2=(x2+y2+z2-6)*(x-z)*(y-2);
   poly f3=(x2+y2+z2-6)*(x-y)*(x-z)*(z-3);
   ideal I=f1,f2,f3;
   def W=Degree(I,1);
        ==>
           The Degree of Component
           3
   def W=Degree(I,2);
        ==>
           The Degree of Component
           2
}
///////////////////////////////////////////////////////////////////////////////

proc Incl(ideal I, ideal J)
"USAGE:  Incl(ideal I, ideal J); I, J ideals
RETURN:  t=1 if the algebraic variety defined by I contains the algebraic
           variety defined by J, otherwise t=0
EXAMPLE: example Incl; shows an example
"
{
 def S=basering;
 int n=nvars(basering);
 int i,j,ii,k,z,zi,dd;
 if(dim(std(I))==0)
 {
  def W=solve(I,"nodisplay");
  setring W;
  ideal J=imap(S,J);
  ideal I=imap(S,I);
  list w;
  poly tj;
  number al,ar,ai,ri,jj;
  zi=size(SOL);
  for(j=1;j<=zi;j++)
  {
   w=SOL[j];
   for(k=1;k<=size(J);k++)
   {
    tj=J[k];
    for(ii=1;ii<=n;ii++)
    {
     tj=subst(tj,var(ii),w[ii]);
    }
    al=leadcoef(tj);
    ar=repart(al);
    ai=impart(al);
    ri=ar^2+ai^2;
    if(ri>0.000000000000001)
    {
     jj=0;
     k=size(I)+1;
     j=zi+1;
    }
    else
    {
     jj=1;
     ri=0;
    }
   }
  }
 }
 else
 {
  def W=WitSupSet(I);
  setring W;
  ideal J=imap(S,J);
  ideal I=imap(S,I);
  list w;
  number al,ar,ai,ri,jj;
  poly tj;
  dd=size(L);
  for(i=0;i<=dd;i++)
  {
   z=size(W(i)[1]);
   zi=size(W(i));
   if(z>1)
   {
    for(j=1;j<=zi;j++)
    {
     w=W(i)[j];
     for(k=1;k<=size(J);k++)
     {
      tj=J[k];
      for(ii=1;ii<=n;ii++)
      {
       tj=subst(tj,var(ii),w[ii]);
      }
      al=leadcoef(tj);
      ar=repart(al);
      ai=impart(al);
      ri=ar^2+ai^2;
      if(ri>0.000000000000001)
      {
       jj=-1;
       k=size(J)+1;
       j=zi+1;
       z=0;
       i=dd+1;
      }
      else
      {
       jj=1;
       ri=0;
      }
     }
    }
   }
  }
 }
 if(ri>0.000000000000001)
 {
  jj=0;
 }
 else
 {
  jj=1;
 }
"================================================";
 "Inclusion:";
 jj;
"================================================";
 export(jj);
 export(J);
 export(I);
   system("sh","rm singular_solutions");
   system("sh","rm nonsingular_solutions");
   system("sh","rm real_solutions");
   system("sh","rm raw_solutions");
   system("sh","rm raw_data");
   system("sh","rm output");
   system("sh","rm midpath_data");
   system("sh","rm main_data");
   system("sh","rm input");
   system("sh","rm failed_paths");
 setring S;
 return (W);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   poly f1=(x2+y2+z2-6)*(x-y)*(x-1);
   poly f2=(x2+y2+z2-6)*(x-z)*(y-2);
   poly f3=(x2+y2+z2-6)*(x-y)*(x-z)*(z-3);
   ideal I=f1,f2,f3;
   poly g1=(x2+y2+z2-6)*(x-1);
   poly g2=(x2+y2+z2-6)*(y-2);
   poly g3=(x2+y2+z2-6)*(z-3);
   ideal J=g1,g2,g3;
   def W=Incl(I,J);
      ==>
         Inclusion:
         0
 def W=Incl(J,I);
      ==>
         Inclusion:
         1
}
///////////////////////////////////////////////////////////////////////////////
proc Equal(ideal I, ideal J)
"USAGE:  Equal(ideal I, ideal J); I, J ideals
RETURN:  t=1 if the algebraic variety defined by I equals to the algebraic
           variety defined by J, otherwise t=0
EXAMPLE: example Equal; shows an example
"
{
 def S=basering;
 int n=nvars(basering);
 def W1=Incl(J,I);
 setring W1;
 number j1=jj;
 execute("ring q=(real,0),("+varstr(S)+"),dp;");
 ideal I=imap(W1,I);
 ideal J=imap(W1,J);
 execute("ring qq=0,("+varstr(S)+"),dp;");
 ideal I=imap(S,I);
 ideal J=imap(S,J);
 def W2=Incl(I,J);
 setring W2;
 number j2=jj;
 number j;
 number j1=imap(W1,j1);
 if(j2==1)
 {
  if(j1==1)
  {
   j=1/1;
  }
  else
  {
   j=0/1;
  }
 }
 else
 {
  j=0/1;
 }
"================================================";
 "Equality:";
 j;
"================================================";
 setring S;
 return (W2);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   poly f1=(x2+y2+z2-6)*(x-y)*(x-1);
   poly f2=(x2+y2+z2-6)*(x-z)*(y-2);
   poly f3=(x2+y2+z2-6)*(x-y)*(x-z)*(z-3);
   ideal I=f1,f2,f3;
   poly g1=(x2+y2+z2-6)*(x-1);
   poly g2=(x2+y2+z2-6)*(y-2);
   poly g3=(x2+y2+z2-6)*(z-3);
   ideal J=g1,g2,g3;
   def W=Equal(I,J);
        ==>
           Equality:
           0


  def W=Equal(J,J);
        ==>
           Equality:
           1
}
///////////////////////////////////////////////////////////////////////////////
proc NumLocalDim(ideal J, list w, int e)
"USAGE:  NumLocalDim(ideal J, list w, int e); J ideal,
           w list of an approximate value of a point v in the algebraic variety defined by J,
               e integer
RETURN: the local dimension of the algebraic variety defined by J at v
EXAMPLE: example NumLocalDim; shows an example
"
{
 def S=basering;
 int n=nvars(basering);
 int sI=size(J);
 int i,j,jj,t,tt,sz1,sz2,ii,ph,ci,k;
 poly p,pp;
 list rw,iw;
 for(i=1;i<=sI;i++)
 {
  p=J[i];
  for(j=1;j<=n;j++)
  {
   w[j]=w[j]+I*0;
   rw[j]=repart(w[j]);
   iw[j]=impart(w[j]);
   p=subst(p,var(j),w[j]);
  }
  pp=pp+p;
 }
 number u=leadcoef(pp);
 if((u^2)==0)
 {
  execute("ring A=(real,e-1),("+varstr(S)+",I),ds;");
  ideal II=imap(S,J);
  list rw=imap(S,rw);
  list iw=imap(S,iw);
  poly p(1..n);
  for(j=1;j<=n;j++)
  {
   p(j)=var(j)+rw[j]+I*iw[j];
  }
  map f=A,p(1..n);
  ideal T=f(II);
  tt=dim(std(T));
  t=tt-1;
 }
 else
 {
  int d=dim(std(J));
  execute("ring R=(complex,e-1,I),("+varstr(S)+"),ds;");
  list w=imap(S,w);
  ideal II=imap(S,J);
  ideal JJ;
  poly p, p(1..n);
  for(i=1;i<=sI;i++)
  {
   p=II[i];
   for(j=1;j<=n;j++)
   {
    p=subst(p,var(j),w[j]);
   }
   JJ[i]=II[i]-p;
  }
  for(j=1;j<=n;j++)
  {
   p(j)=var(j)+w[j];
  }
  map f=R,p(1..n);
  ideal T=f(JJ);
  tt=dim(std(T));
  if(tt==d)
  {
   execute("ring A=(complex,e,I),("+varstr(S)+"),dp;");
   t=tt;
  }
  else
  {
   execute("ring RR=(real,e-2),("+varstr(S)+",I),dp;");
   ideal II=imap(S,J);
   list rw=imap(S,rw);
   list iw=imap(S,iw);
   ideal L,LL,H,HH;
   poly l(1..d),ll(1..d);
   int c;
   for(i=1;i<=d;i++)
   {
    for(j=1;j<=n;j++)
    {
     c=random(1,100);
     l(i)=l(i)+c*(var(j));
     ll(i)=ll(i)+c*(var(j)-rw[j]-I*iw[j]);
    }
    l(i)=l(i)+random(101,200);
    L[i]=l(i);
    LL[i]=ll(i);
   }
   poly pi=I^2+1;
   H=L,II,pi;
   ideal JJ;
   poly p, p(1..n);
   for(i=1;i<=sI;i++)
   {
    p=II[i];
    for(j=1;j<=n;j++)
    {
     p=subst(p,var(j),rw[j]+I*iw[j]);
    }
    JJ[i]=II[i]-p;
   }
   HH=LL,JJ,pi;
   if(dim(std(H))==0)
   {
    def M=solve(H,100,"nodisplay");
    setring M;
    sz1=size(SOL);
    execute("ring RRRQ=(real,e-1),("+varstr(S)+",I),dp;");
    ideal HH=imap(RR,HH);
    if(dim(std(HH))==0)
    {
     def MM=solve(HH,100,"nodisplay");
     setring MM;
     sz2=size(SOL);
    }
   }
   else
   {
    sz1=1;
   }
   if(sz1==sz2)
   {
    execute("ring A=(complex,e,I),("+varstr(S)+"),dp;");
    t=d;
   }
   else
   {
    execute("ring RQ=(real,e-1),("+varstr(S)+"),dp;");
    ideal II=imap(S,J);
    def RW=WitSet(II);
    setring RW;
    list v;
    list w=imap(S,w);
    number nr,ni;
    if(tt<0)
    {
     tt=0;
    }
    for(ii=tt;ii<=d;ii++)
    {
     list W(ii)=imap(RW,W(ii));
     if(size(W(ii)[1])>1)
     {
      if(ii==0)
      {
       for(i=1;i<=size(W(0));i++)
       {
        v=W(ii)[i];
        nr=0;
        ni=0;
        for(j=1;j<=n;j++)
        {
         nr=nr+(repart(v[j])-repart(w[j]))^2;
         ni=ni+(impart(v[j])-impart(w[j]))^2;
        }
        if((ni+nr)<1/10^(2*e-3))
        {
         execute("ring A=(complex,e,I),("+varstr(S)+"),dp;");
         list W(ii)=imap(RW,W(ii));
         t=0;
         i=size(W(ii))+1;
         ii=d+1;
        }
       }
      }
      else
      {
       def SS=Singular2bertini(W(ii));
       execute("ring D=(complex,e,I),("+varstr(S)+",s,gamma),dp;");
       string nonsin;
       ideal H,L;
       ideal J=imap(RW,N(0));
       ideal LL=imap(RW,L);
       list w=imap(S,w);
       poly p;
       for(j=1;j<=ii;j++)
       {
        p=0;
        for(jj=1;jj<=n;jj++)
        {
         p=p+random(1,100)*(var(jj)-w[jj]);
        }
        L[j]=p;
       }
       for(jj=1;jj<=size(J);jj++)
       {
        H[jj]=s*gamma*J[jj]+(1-s)*J[jj];
       }
       for(jj=1;jj<=ii;jj++)
       {
        H[size(J)+jj]=s*gamma*LL[jj]+(1-s)*L[jj];
       }
       string sv=varstr(S);
       def Q(ii)=UseBertini(H,sv);
       system("sh","rm start");
       nonsin=read("nonsingular_solutions");
       if(size(nonsin)>=52)
       {
        def T(ii)=bertini2Singular("nonsingular_solutions",nvars(basering)-2);
        setring T(ii);
        list C=re;
        ci=size(C);
        number tr;
        list w=imap(S,w);
        for(jj=1;jj<=ci;jj++)
        {
         tr=0;
         for(k=1;k<=n;k++)
         {
          tr=tr+(repart(w[k])-repart(C[jj][k]))^2+(impart(w[k])-impart(C[jj][k]))^2;
         }
         if(tr<=1/10^(2*e-3))
         {
          execute("ring A=(complex,e,I),("+varstr(S)+"),dp;");
          t=ii;
          ii=d+1;
          jj=ci+1;
         }
        }
       }
      }
     }
    }
    system("sh","rm singular_solutions");
    system("sh","rm nonsingular_solutions");
    system("sh","rm real_solutions");
    system("sh","rm raw_solutions");
    system("sh","rm raw_data");
    system("sh","rm output");
    system("sh","rm midpath_data");
    system("sh","rm main_data");
    system("sh","rm input");
    system("sh","rm failed_paths");
   }
  }
 }
 "=============================================";
 "The Local Dimension:";
 t;
 setring S;
 return(A);
}
example
{ "EXAMPLE:"; echo = 2;
   int e=14;
   ring r=(complex,e,I),(x,y,z),dp;
   poly f1=(x2+y2+z2-6)*(x-y)*(x-1);
   poly f2=(x2+y2+z2-6)*(x-z)*(y-2);
   poly f3=(x2+y2+z2-6)*(x-y)*(x-z)*(z-3);
   ideal J=f1,f2,f3;
   list p0=0.99999999999999+I*0.00000000000001,2,3+I*0.00000000000001;
   list p2=1,0.99999999999998,2;
   list p1=5+I,4.999999999999998+I,5+I;
   def D=NumLocalDim(J,p0,e);
             ==>
               The Local Dimension:
                0
   def D=NumLocalDim(J,p1,e);
             ==>
               The Local Dimension:
                1
   def D=NumLocalDim(J,p2,e);
             ==>
               The Local Dimension:
                2
}

///////////////////////////////////////////////////////////////////////////////