package body Generic_Integer_Linear_Solvers is

  use Common_Divisors;

-- SCALERS :

  function Divisors ( a : Matrix ) return Vectors.Vector is

  -- DESCRIPTION :
  --   Returns a vector containing the gcd of the elements of each row.

    v : Vectors.Vector(a'range(1));
   
  begin
    for i in a'range(1) loop
      v(i) := a(i,a'first(2));
      if not Equal(v(i),one)
       then for j in (a'first(2)+1)..a'last(2) loop
              v(i) := gcd(v(i),a(i,j));
              exit when Equal(v(i),one);
            end loop;
      end if;
    end loop;
    return v;
  end Divisors;

  function Scale ( a : Matrix ) return Matrix is

    v : Vectors.Vector(a'range(1)) := Divisors(a);
    b : Matrix(a'range(1),a'range(2));

  begin
    for i in b'range(1) loop
      if (Equal(v(i),one) or Equal(v(i),zero))
       then for j in b'range(2) loop
	      b(i,j) := a(i,j);
            end loop;
       else for j in b'range(2) loop
	      b(i,j) := a(i,j) / v(i);
            end loop;
      end if;
    end loop;
    return b;
  end Scale;

  procedure Scale ( a : in out Matrix; v : out Vectors.Vector ) is

    dv : Vectors.Vector(a'range(1)) := Divisors(a);

  begin
    for i in a'range(1) loop
      if (Equal(dv(i),one) or Equal(dv(i),zero))
       then null;
       else for j in a'range(2) loop
	      a(i,j) := a(i,j) / dv(i);
            end loop;
      end if;
    end loop;
    v := dv;
  end Scale;

  procedure Scale ( a : in out Matrix ) is

    v : Vectors.Vector(a'range(1)) := Divisors(a);

  begin
    for i in a'range(1) loop
      if (Equal(v(i),one) or Equal(v(i),zero))
       then null;
       else for j in a'range(2) loop
	      a(i,j) := a(i,j) / v(i);
            end loop;
      end if;
    end loop;
  end Scale;

  procedure Scale ( a : in out Matrix; row,col : in integer ) is

    g : number := a(row,col);

  begin
    if not Equal(g,one)
     then for l in (col+1)..a'last(2) loop
            g := gcd(g,a(row,l));
            exit when Equal(g,one);
          end loop;
    end if;
    if (not Equal(g,zero)) and (not Equal(g,one))
     then for l in row..a'last(2) loop
            a(row,l) := a(row,l)/g;
          end loop;
    end if;
  end Scale;

-- STATIC TRIANGULATORS :

  procedure Upper_Triangulate ( l : out Matrix; a : in out Matrix ) is

    row,pivot : integer;
    temp,aa,bb,ka,lb,d,a_rowk,a_ik : number;
    ll : Matrix(a'range(1),a'range(1));

  begin
    for i in ll'range(1) loop
      for j in ll'range(2) loop
        Copy(zero,ll(i,j));
      end loop;
      Copy(one,ll(i,i));
    end loop;
    row := a'first(1);
    for j in a'first(2)..a'last(2) loop
      pivot := row-1;                                         -- find pivot
      for i in row..a'last(1) loop
        if not Equal(a(i,j),zero)
         then pivot := i;
              exit;
        end if;
      end loop;
      if pivot >= row 
       then if pivot /= row                      -- interchange if necessary
             then for k in a'range(2) loop
                    Copy(a(row,k),temp);
                    Copy(a(pivot,k),a(row,k));
                    Copy(temp,a(pivot,k));
                  end loop;
                  for k in ll'range(2) loop
                    Copy(ll(row,k),temp);
                    Copy(ll(pivot,k),ll(row,k));
                    Copy(temp,ll(pivot,k));
                  end loop;
            end if;
            for i in (row+1)..a'last(1) loop                  -- make zeroes
              if not Equal(a(i,j),zero)
               then gcd(a(row,j),a(i,j),ka,lb,d);
                    aa := a(row,j)/d;     bb := a(i,j)/d;
                    if Equal(aa,bb) and then Equal(ka,zero)
                     then Copy(lb,ka); Copy(zero,lb);
                    end if;
                    if Equal(aa,-bb) and then Equal(ka,zero)
                     then ka := -lb; Copy(zero,lb);
                    end if;
                    for k in j..a'last(2) loop
                      Copy(a(row,k),a_rowk); Clear(a(row,k));
                         -- a_rowk := a(row,k);
                      Copy(a(i,k),a_ik); Clear(a(i,k)); -- a_ik := a(i,k);
                      a(row,k) :=  ka*a_rowk + lb*a_ik;
                      a(i,k)   := -bb*a_rowk + aa*a_ik;
                    end loop;
                    for k in ll'range(2) loop
                      a_rowk := ll(row,k);
                      a_ik := ll(i,k);
                      ll(row,k) :=  ka*a_rowk + lb*a_ik;
                      ll(i,k)   := -bb*a_rowk + aa*a_ik;
                    end loop;
              end if;
           end loop;
           row := row + 1;
      end if;
      exit when row > a'last(1);
    end loop;
    l := ll;
  end Upper_Triangulate;

  procedure Upper_Triangulate ( a : in out Matrix ) is

    row,pivot : integer;
    temp,aa,bb,ka,lb,d,a_rowk,a_ik : number;

  begin
    row := a'first(1);
    for j in a'first(2)..a'last(2) loop
      pivot := row-1;                                           -- find pivot
      for i in row..a'last(1) loop
        if not Equal(a(i,j),zero)
         then pivot := i;
              exit;
        end if;
      end loop;
      if pivot >= row 
       then if pivot /= row                       -- interchange if necessary 
            then for k in a'range(2) loop
	           Copy(a(row,k),temp);
	           Copy(a(pivot,k),a(row,k));
	           Copy(temp,a(pivot,k));
	         end loop;
           end if;
           for i in (row+1)..a'last(1) loop                    -- make zeroes
             if not Equal(a(i,j),zero)
              then gcd(a(row,j),a(i,j),ka,lb,d);
                   aa := a(row,j)/d;     bb := a(i,j)/d;
                   if Equal(aa,bb) and then Equal(ka,zero)
                    then Copy(lb,ka); Copy(zero,lb);
                   end if;
                   if Equal(aa,-bb) and then Equal(ka,zero)
                    then ka := -lb; Copy(zero,lb);
                   end if;
		   for k in j..a'last(2) loop
                     Copy(a(row,k),a_rowk); Clear(a(row,k));
                               --a_rowk := a(row,k);
                     Copy(a(i,k),a_ik); Clear(a(i,k)); --a_ik   := a(i,k);
                     a(row,k) :=  ka*a_rowk + lb*a_ik;
                     a(i,k)   := -bb*a_rowk + aa*a_ik;
                   end loop;
             end if;
           end loop;
	   row := row + 1;
       end if;
       exit when row > a'last(1);
    end loop;
  end Upper_Triangulate;

  procedure Upper_Triangulate 
                 ( a : in out Matrix;
                   ipvt : in out Standard_Integer_Vectors.Vector ) is

    row,pivot,tmppiv : integer;
    temp,aa,bb,ka,lb,d,a_rowk,a_ik : number;

  begin
    row := a'first(1);
    for j in a'first(2)..a'last(2) loop
      pivot := row-1;                                           -- find pivot
      for i in row..a'last(1) loop
        if not Equal(a(i,j),zero)
         then pivot := i;
              exit;
        end if;
      end loop;
      if pivot >= row 
       then if pivot /= row                       -- interchange if necessary
            then for k in a'range(2) loop
                   Copy(a(row,k),temp);
                   Copy(a(pivot,k),a(row,k));
                   Copy(temp,a(pivot,k));
	         end loop;
                 tmppiv := ipvt(row);
                 ipvt(row) := ipvt(pivot);
                 ipvt(pivot) := tmppiv;
           end if;
           for i in (row+1)..a'last(1) loop                   -- make zeroes
             if not Equal(a(i,j),zero)
              then gcd(a(row,j),a(i,j),ka,lb,d);
                   aa := a(row,j)/d;     bb := a(i,j)/d;
                   if Equal(aa,bb) and then Equal(ka,zero)
                    then Copy(lb,ka); Copy(zero,lb);
                   end if;
                   if Equal(aa,-bb) and then Equal(ka,zero)
                    then ka := -lb; Copy(zero,lb);
                   end if;
                   for k in j..a'last(2) loop
                     Copy(a(row,k),a_rowk); Clear(a(row,k));
                         -- a_rowk := a(row,k);
                     Copy(a(i,k),a_ik); Clear(a(i,k)); -- a_ik   := a(i,k);
                     a(row,k) :=  ka*a_rowk + lb*a_ik;
                     a(i,k)   := -bb*a_rowk + aa*a_ik;
                   end loop;
             end if;
           end loop;
	   row := row + 1;
       end if;
       exit when row > a'last(1);
    end loop;
  end Upper_Triangulate;

  procedure Lower_Triangulate ( a : in out Matrix; u : out Matrix ) is

    column,pivot : integer;
    temp,aa,bb,ka,lb,d,a_kcolumn,a_kj : number;
    uu : Matrix(a'range(2),a'range(2));

  begin

    for i in uu'range(1) loop
      for j in uu'range(2) loop
        Copy(zero,uu(i,j));
      end loop;
      Copy(one,uu(i,i));
    end loop;
    column := a'first(2);
    for i in a'first(1)..a'last(1) loop
      pivot := column-1;                                      -- find pivot
      for j in column..a'last(2) loop
        if not Equal(a(i,j),zero)
         then pivot := j;
              exit;
        end if;
      end loop;
      if pivot >= column
       then if pivot /= column                  -- interchange if necessary
             then for k in a'range(1) loop
                    Copy(a(k,column),temp);
                    Copy(a(k,pivot),a(k,column));
                    Copy(temp,a(k,pivot));
                  end loop;
                  for k in uu'range(1) loop
                    Copy(uu(k,column),temp);
                    Copy(uu(k,pivot),uu(k,column));
                    Copy(temp,uu(k,pivot));
                  end loop;
            end if;
            for j in (column+1)..a'last(2) loop               -- make zeroes
              if not Equal(a(i,j),zero)
               then gcd(a(i,column),a(i,j),ka,lb,d);
                    aa := a(i,column)/d;        bb := a(i,j)/d;
                    if Equal(aa,bb) and then Equal(ka,zero)
                     then Copy(lb,ka); Copy(zero,lb);
                    end if;
                    if Equal(aa,-bb) and then Equal(ka,zero)
                     then ka := -lb; Copy(zero,lb);
                    end if;
                    for k in i..a'last(1) loop
                      Copy(a(k,column),a_kcolumn); Clear(a(k,column));
                                              -- a_kcolumn := a(k,column);
                      Copy(a(k,j),a_kj); Clear(a(k,j)); -- a_kj := a(k,j);
                      a(k,column) := a_kcolumn*ka    + a_kj*lb;
                      a(k,j)      := a_kcolumn*(-bb) + a_kj*aa;
                    end loop;
                    for k in uu'range(1) loop
                      Copy(uu(k,column),a_kcolumn); Clear(uu(k,column));
                                               -- a_kcolumn := uu(k,column);
                      Copy(uu(k,j),a_kj); Clear(u(k,j)); -- a_kj := uu(k,j);
                      uu(k,column) := a_kcolumn*ka    + a_kj*lb;
                      uu(k,j)      := a_kcolumn*(-bb) + a_kj*aa;
                    end loop;
              end if;
            end loop;
            column := column + 1;
      end if;
      exit when column > a'last(2);
    end loop;
    u := uu;
  end Lower_Triangulate;

  procedure Lower_Triangulate ( a : in out Matrix ) is

    column,pivot : integer;
    temp,aa,bb,ka,lb,d,a_kcolumn,a_kj : number;

  begin
    column := a'first(2);
    for i in a'first(1)..a'last(1) loop
      pivot := column-1;                                      -- find pivot
      for j in column..a'last(2) loop
        if not Equal(a(i,j),zero)
         then pivot := j;
              exit;
        end if;
      end loop;
      if pivot >= column
       then if pivot /= column                  -- interchange if necessary
             then for k in a'range(1) loop
                    Copy(a(k,column),temp);
                    Copy(a(k,pivot),a(k,column));
                    Copy(temp,a(k,pivot));
                  end loop;
            end if;
            for j in (column+1)..a'last(2) loop               -- make zeroes
             if not Equal(a(i,j),zero)
              then gcd(a(i,column),a(i,j),ka,lb,d);
                   aa := a(i,column)/d; bb := a(i,j)/d;
                   if Equal(aa,bb) and then Equal(ka,zero)
                    then Copy(lb,ka); Copy(zero,lb);
                   end if;
                   if Equal(aa,-bb) and then Equal(ka,zero)
                    then ka := -lb; Copy(zero,lb);
                   end if;
                   for k in i..a'last(1) loop
                     Copy(a(k,column),a_kcolumn); Clear(a(k,column));
                                         -- a_kcolumn := a(k,column);
                     Copy(a(k,j),a_kj); Clear(a(k,j)); -- a_kj := a(k,j);
                     a(k,column) := a_kcolumn*ka    + a_kj*lb;
                     a(k,j)      := a_kcolumn*(-bb) + a_kj*aa;
                   end loop;
             end if;
           end loop;
           column := column + 1;
      end if;
      exit when column > a'last(2);
    end loop;
  end Lower_Triangulate;

  procedure Lower_Triangulate
               ( a : in out Matrix;
                 ipvt : in out Standard_Integer_Vectors.Vector ) is

    column,pivot,tmppiv : integer;
    temp,aa,bb,ka,lb,d,a_kcolumn,a_kj : number;

  begin
    column := a'first(2);
    for i in a'first(1)..a'last(1) loop
      pivot := column-1;                                       -- find pivot
      for j in column..a'last(2) loop
        if not Equal(a(i,j),zero)
         then pivot := j;
              exit;
        end if;
      end loop;
      if pivot >= column
       then if pivot /= column                   -- interchange if necessary
             then for k in a'range(1) loop
                    Copy(a(k,column),temp);
                    Copy(a(k,pivot),a(k,column));
                    Copy(temp,a(k,pivot));
                  end loop;
                  tmppiv := ipvt(column);
                  ipvt(column) := ipvt(pivot);
                  ipvt(pivot) := tmppiv;
            end if;
            for j in (column+1)..a'last(2) loop               -- make zeroes
              if not Equal(a(i,j),zero)
               then gcd(a(i,column),a(i,j),ka,lb,d);
                    aa := a(i,column)/d; bb := a(i,j)/d;
                    if Equal(aa,bb) and then Equal(ka,zero)
                     then Copy(lb,ka); Copy(zero,lb);
                    end if;
                    if Equal(aa,-bb) and then Equal(ka,zero)
                     then ka := -lb; Copy(zero,lb);
                    end if;
                    for k in i..a'last(1) loop
                      Copy(a(k,column),a_kcolumn); Clear(a(k,column));
                                                -- a_kcolumn := a(k,column);
                      Copy(a(k,j),a_kj); Clear(a(k,j)); -- a_kj := a(k,j);
                      a(k,column) := a_kcolumn*ka    + a_kj*lb;
                      a(k,j)      := a_kcolumn*(-bb) + a_kj*aa;
                    end loop;
              end if;
            end loop;
            column := column + 1;
      end if;
      exit when column > a'last(2);
    end loop;
  end Lower_Triangulate;

-- SELECTORS :

  function Det ( a : Matrix ) return number is

  -- NOTE :
  --   The triangulation is implemented independently to keep track
  --   of row interchanges.

    res : number := one;
    m : matrix(a'range(1),a'range(2));
    row,pivot : integer;
    temp,aa,bb,ka,lb,d,m_rowk,m_ik : number;

  begin
    Copy(a,m);   -- triangulate m
    row := m'first(1);
    for j in m'first(2)..m'last(2) loop
      pivot := row-1;                                           -- find pivot
      for i in row..m'last(1) loop
        if not Equal(m(i,j),zero)
         then pivot := i;
              exit;
        end if;
      end loop;
      if pivot >= row
       then if pivot /= row                       -- interchange if necessary
             then for k in m'range(2) loop
                    Copy(m(row,k),temp);
                    Copy(m(pivot,k),m(row,k));
                    Copy(temp,m(pivot,k));
                  end loop;
                  Min(res);
            end if;
            for i in (row+1)..m'last(1) loop                   -- make zeroes
              if not Equal(m(i,j),zero)
               then gcd(m(row,j),m(i,j),ka,lb,d);
                    aa := m(row,j)/d; bb := m(i,j)/d;
                    if Equal(aa,bb) and then Equal(ka,zero)
                     then Copy(lb,ka); Copy(zero,lb);
                    end if;
                    if Equal(aa,-bb) and then Equal(ka,zero)
                     then ka := -lb; Copy(zero,lb);
                    end if;
                    for k in j..m'last(2) loop
                      Copy(m(row,k),m_rowk); Clear(m(row,k));
                                                         -- m_rowk := m(row,k);
                      Copy(m(i,k),m_ik); Clear(m(i,k));  -- m_ik   := m(i,k);
                      m(row,k) :=  ka*m_rowk + lb*m_ik;
                      m(i,k)   := -bb*m_rowk + aa*m_ik;
                    end loop;
              end if;
            end loop;
            row := row + 1;
       end if;
       exit when row > m'last(1);
    end loop;
    for k in m'range(1) loop
      Mul(res,m(k,k));
    end loop;
    Clear(m);
    return res;
  end Det;

  function Per ( i,n : natural; a : matrix;
                 kk : Standard_Integer_Vectors.Vector ) return number is
  begin
    if i = n+1
     then return one;
     else declare
            res,acc : number;
            kkk : Standard_Integer_Vectors.Vector(kk'range) := kk;
          begin
            Copy(zero,res);
            for j in kk'range loop
              if not Equal(a(i,j),zero) and then (kk(j) /= 0)
               then kkk(j) := kkk(j) - 1;
                    acc := Per(i+1,n,a,kkk);
                    Add(res,acc);
                    Clear(acc);
                    kkk(j) := kkk(j) + 1;
              end if;
            end loop;
            return res;
          end;
    end if;
  end Per;

  function Per ( i,n : natural; a : matrix;
                 kk : Standard_Integer_Vectors.Vector; max : number )
               return number is
  begin
    if i = n+1
     then return one;
     else declare
            res,acc : number;
            kkk : Standard_Integer_Vectors.Vector(kk'range) := kk;
          begin
            Copy(zero,res);
            for j in kk'range loop
              if not Equal(a(i,j),zero) and then (kk(j) /= 0)
               then kkk(j) := kkk(j) - 1;
                    acc := a(i,j)*Per(i+1,n,a,kkk,max);
                    Add(res,acc);
                    Clear(acc);
                    kkk(j) := kkk(j) + 1;
              end if;
              exit when (res > max) or Equal(res,max);
            end loop;
            return res;
          end;
    end if;
  end Per;

  function Per ( a : matrix; k : Standard_Integer_Vectors.Vector )
               return number is

  -- ALGORITHM :
  --   Row expansion without memory, as developed by C.W. Wampler,
  --   see `Bezout Number Calculations for Multi-Homogeneous Polynomial
  --        Systems', Appl. Math. Comput. 51:(2-3), 143-157, 1992.

  begin
    return Per(1,a'last(1),a,k);
  end Per;

  function Per ( a : matrix; k : Standard_Integer_Vectors.Vector;
                 max : number ) return number is

  -- ALGORITHM :
  --   Row expansion without memory, as developed by C.W. Wampler,
  --   see `Bezout Number Calculations for Multi-Homogeneous Polynomial
  --        Systems', Appl. Math. Comput. 51:(2-3), 143-157, 1992.

  begin
    return Per(1,a'last(1),a,k,max);
  end Per;

  function Rank ( a : Matrix ) return natural is

    res : natural := 0; 
    m : Matrix(a'range(1),a'range(2));
    column : integer;

  begin
    Copy(a,m);
    Upper_Triangulate(m);
   -- compute the length of chain of nonzero elements in m :
   -- search first nonzero element in first row of m :
    column := m'first(2)-1;
    for k in m'range(2) loop
      if not Equal(m(m'first(1),k),zero)
       then column := k;
      end if;
      exit when (column = k);
    end loop;
    if column < m'first(2)
     then return 0;   -- all elements of m are zero
     else for k in m'range(1) loop
            exit when column > m'last(2);
            if not Equal(m(k,column),zero)
             then res := res + 1;
             else -- search for next nonzero element on row k :
                  for l in column+1..m'last(2) loop
                    if not Equal(m(k,l),zero)
                     then column := l;
                          res := res + 1;
                    end if;
                    exit when (column = l);
                  end loop;
            end if;
            column := column + 1;
          end loop;
    end if;
    Clear(m);
    return res;
  end Rank;

-- DYNAMIC TRIANGULATOR :

  procedure Triangulate ( l : in integer; m : in out matrix;
                          ipvt : in out Standard_Integer_Vectors.vector;
                          piv : out integer ) is

  -- DESCRIPTION :
  --   Updates lth row of m such that m remains upper triangular.

    tmp,a,b,lcmab,faca,facb : number;
    pivot,index,tmppiv : integer;

  begin
    Switch(ipvt,l,m);                      -- pivoting for previous unknowns
    index := 1;                         -- update : make l-1 zeroes in row l
    for k in 1..(l-1) loop
      if (not Equal(m(l,index),zero))
         and then (not Equal(m(k,index),zero))    -- make m(l,index) zero
       then a := m(k,index); b := m(l,index);
            lcmab := lcm(a,b);
            if lcmab < zero then lcmab := -lcmab; end if;
            facb := lcmab/b; faca := lcmab/a;
            if facb > zero
             then for i in index..m'last(2) loop
                    m(l,i) :=  facb*m(l,i) - faca*m(k,i);
                  end loop;
             else for i in index..m'last(2) loop
                    m(l,i) := -facb*m(l,i) + faca*m(k,i);
                  end loop;
            end if;
      end if;
      if not Equal(m(k,index),zero)
       then index := index + 1;
      end if;
    end loop;
    pivot := 0;                                              -- search pivot
    for k in l..m'last(2)-1 loop
      if not Equal(m(l,k),zero)
       then pivot := k;
      end if;
      exit when pivot /= 0;
    end loop;
    if pivot > l
     then for k in 1..l loop              -- interchange columns l and pivot
            Copy(m(k,l),tmp);
            Copy(m(k,pivot),m(k,l));
            Copy(tmp,m(k,pivot));
          end loop;
          tmppiv := ipvt(l);
          ipvt(l) := ipvt(pivot);
          ipvt(pivot) := tmppiv;
    end if;
    piv := pivot;
  end Triangulate;

  procedure Switch ( ipvt : in Standard_Integer_Vectors.vector;
                     index : in integer; m : in out matrix ) is

    tmpv : Vectors.Vector(m'range(2));

  begin
    for k in tmpv'range loop
      Copy(m(index,k),tmpv(k));
    end loop;
    for k in tmpv'range loop
      Copy(tmpv(ipvt(k)),m(index,k));
    end loop;
    Vectors.Clear(tmpv);
  end Switch;

  procedure Switch ( ipvt : in Standard_Integer_Vectors.vector;
                     first,last : in integer; m : in out matrix) is

    tmpv : Vectors.Vector(m'range(2));

  begin
    for index in first..last loop
      for k in tmpv'range loop
        Copy(m(index,k),tmpv(k));
      end loop;
      for k in tmpv'range loop
        Copy(tmpv(ipvt(k)),m(index,k));
      end loop;
      Vectors.Clear(tmpv);
    end loop;
  end Switch;

  procedure Switch ( l,pivot,index : in integer; m : in out matrix ) is

    tmp : number;

  begin
    if l /= pivot
     then Copy(m(index,l),tmp);
          Copy(m(index,pivot),m(index,l));
          Copy(tmp,m(index,pivot));
    end if;
  end Switch;

  procedure Switch ( l,pivot : in integer;
                     first,last : in integer; m : in out matrix ) is

    tmp : number;

  begin
    if l /= pivot
     then for index in first..last loop
            Copy(m(index,l),tmp);
            Copy(m(index,pivot),m(index,l));
            Copy(tmp,m(index,pivot));
          end loop;
    end if;
  end Switch;

-- SOLVERS :

  function Check0 ( a : Matrix; x : Vectors.Vector ) return boolean is
   
   -- DESCRIPTION :
   --   Returns true if x is a solution of the system a*x = 0.

    tmp : Vectors.Vector(a'range(1));

  begin
    tmp := a*x;
    for i in tmp'range loop
      if not Equal(tmp(i),zero)
       then Vectors.Clear(tmp); return false;
      end if;
    end loop;
    Vectors.Clear(tmp);
    return true;
  end Check0;

  procedure Solve0 ( a : in Matrix; x : in out Vectors.Vector ) is

  -- ALGORITHM :
  --   An intermediate, generating matrix tmp will be constructed,
  --   such that 
  --      1) the solution x to tmp*x = 0 is the same of a*x = 0;
  --      2) tmp(i,i) and tmp(i,ind) are the only nonzero entries.
  --   Before this construction, it will be checked whether there
  --   exists a zero column and the index ind must be determined.
  --   After the definition of tmp, the back substitution process
  --   yields a solution.

    piv,ind : integer;
    tmp : Matrix(a'first(1)..(a'last(1)+1),a'range(2));
    res : Vectors.Vector(x'range);
    pivots : Standard_Integer_Vectors.Vector(x'range);
    zero_column : boolean;

  begin
   -- initialization of tmp,ind and pivots :
    for i in tmp'range(1) loop
      for j in tmp'range(2) loop
	Copy(zero,tmp(i,j));
      end loop;
    end loop;
    for i in pivots'range loop
      pivots(i) := i;
    end loop;
    ind := x'first(1)-1;
    for i in a'range(1) loop
      piv := pivots'first-1;
      for j in a'range(2) loop
	if not Equal(a(i,j),zero)
         then piv := pivots(j);
	      pivots(j) := pivots(i);
	      pivots(i) := piv;
	      exit;
        end if;
      end loop;
      zero_column := true;
      for j in a'first(1)..i loop
        Copy(a(j,pivots(i)),tmp(j,i));
	if zero_column and then not Equal(tmp(j,i),zero)
	 then zero_column := false;
        end if;
      end loop;
      if piv < pivots'first or else zero_column or else Equal(tmp(i,i),zero)
       then ind := i; exit;
      end if;
    end loop;
    if zero_column
     then for i in x'range loop
            Copy(zero,x(i));
          end loop;
	  Copy(one,x(ind));
     elsif (ind < x'first(1)) and (a'last(1) >= a'last(2))
         then for i in x'range loop
                Copy(zero,x(i));
              end loop;
         else
           if ind < x'first(1)
	    then ind := a'last(1)+1;
	         for j in tmp'range(2) loop
	           Copy(zero,tmp(ind,j));
                 end loop;
	         zero_column := true;
	         for j in a'range(1) loop
	           Copy(a(j,pivots(ind)),tmp(j,ind));
	           if zero_column and then not Equal(tmp(j,ind),zero)
	            then zero_column := false;
                   end if;
                 end loop;
           end if;
           if zero_column
	    then for i in x'range loop
                   Copy(zero,x(i));
                 end loop;
	         Copy(one,x(ind));
            else
             -- construct generating matrix :
              for i in reverse (tmp'first(2)+1)..(ind-1) loop  -- i = column
                for k in tmp'first(1)..(i-1) loop
                  if not Equal(tmp(k,i),zero)  -- make tmp(k,i) zero
                   then declare 
                          aa,bb,d : number;
                        begin
                          d := gcd(tmp(i,i),tmp(k,i));
                          aa := tmp(i,i)/d;     bb := tmp(k,i)/d;
                          for l in k..(i-1) loop
                            Mul(tmp(k,l),aa);
                          end loop;
                          Copy(zero,tmp(k,i)); --aa*tmp(k,i) - bb*tmp(i,i);
                          tmp(k,ind) := aa*tmp(k,ind) - bb*tmp(i,ind);
                        end;
                        Scale(tmp,k,k);  -- to avoid numeric_error
                  end if;
                end loop; -- upper half of ith colum consists of zero entries
              end loop;
             -- generate x by back substitution :
              x(ind) := tmp(x'first,x'first);
              for i in (x'first+1)..(ind-1) loop
	        if not Equal(tmp(i,i),zero)
                 then x(ind) := lcm(tmp(i,i),x(ind));
                end if;
              end loop;
              for i in x'first..(ind-1) loop
	        if Equal(tmp(i,i),zero)
	         then Copy(zero,x(i));
                 else x(i) := -(tmp(i,ind)*x(ind))/tmp(i,i);
                end if;
              end loop;
           end if;
    end if;
    for i in res'range loop                      -- take pivots into account
      Copy(zero,res(i));
    end loop;
    for i in x'first..ind loop
      Copy(x(i),res(pivots(i)));
    end loop;
    Vectors.Copy(res,x);
  end Solve0;

  procedure Solve1 ( a : in Matrix; x : in out Vectors.Vector;
                     b : in Vectors.Vector; fail : out boolean ) is

    acc : number;

  begin
    fail := false;
    for i in reverse x'range loop
      Copy(b(i),x(i));
      for j in (i+1)..x'last loop
        acc := a(i,j)*x(j);
        Sub(x(i),acc);
        Clear(acc);
      end loop;
      if not Equal(x(i),zero) and then not Equal(a(i,i),zero)
       then acc := Rmd(x(i),a(i,i));
            if Equal(acc,zero)
             then Div(x(i),a(i,i));
             else fail := true;
            end if;
            Clear(acc);
            if fail
             then Vectors.Clear(x); return;
            end if;
      end if;
    end loop;
  end Solve1;

  procedure Solve1 ( a : in Matrix; b : in out Vectors.Vector;
                     fail : out boolean ) is

    acc : number;

  begin
    fail := false;
    for i in reverse b'range loop
      for j in (i+1)..b'last loop
        acc := a(i,j)*b(j);
        Sub(b(i),acc);
        Clear(acc);
      end loop;
      if not Equal(b(i),zero) and then not Equal(a(i,i),zero)
       then acc := Rmd(b(i),a(i,i));
            if Equal(acc,zero)
             then Div(b(i),a(i,i));
             else fail := true; 
            end if;
            Clear(acc);
            if fail
             then Vectors.Clear(b); return;
            end if;
      end if;
    end loop;
  end Solve1;

  function Solve ( m : Matrix; ipvt : Standard_Integer_Vectors.Vector )
                 return Vectors.Vector is

    x,res : Vectors.Vector(ipvt'range);
    a : matrix(m'first(1)..m'last(1)-1,m'range(2));
    ind : integer := a'first(1);       -- index for the current row number
    cnt0 : natural := 0;                 -- counts the number of zero rows

  begin
    for k in a'range(1) loop
      if not Equal(m(k,k),zero)               -- otherwise : skip zero row !
       then for l in a'range(2) loop
              Copy(m(k,l),a(ind,l));
            end loop;
            ind := ind + 1;
       else for l in a'range(2) loop
              Copy(m(k,l),a(a'last(1) - cnt0,l));
            end loop;
            cnt0 := cnt0 + 1;
      end if;
    end loop;
    for i in x'range loop
      Copy(zero,x(i));
    end loop;
    Solve0(a,x);
    for k in res'range loop
      res(ipvt(k)) := x(k);
    end loop;
    if res(res'last) < zero
     then Vectors.Min(res);
    end if;
    Clear(a);
    return res;
  end Solve;

end Generic_Integer_Linear_Solvers;