pol2bisspecial.hpp

Go to the documentation of this file.

#ifndef ALREADY_pol2bisspecial
#define ALREADY_pol2bisspecial
#ifndef LEVELSELECT
#define LEVELSELECT 1e-6
#endif

using std::list;
using std::vector;
using std::cout; 
using std::endl;

void inverse(int*&, int);
template<typename mon, typename T,typename Base ,typename monomial_server>
mon choice_naif(const polbis<mon,T> & p, const Base &b, monomial_server &serv)
{
  list<mon> tmp;
  mon m; 
  int maxdeg=-1,k=0;
  //int maxtmpdeg=0;
  for(int i=0;i<p.sizenf;i++)
    {
      if (p.nfind[i])
    for(int j=0;j<8;j++)
      if(p.nfind[i]>>j&1)
      {
        mon tmpmon;
        serv.int2mon(8*i+j,tmpmon);
        if(tmpmon.GetDegree()>maxdeg)
          {
        tmp.erase(tmp.begin(),tmp.end());
        maxdeg=tmpmon.GetDegree();
          }
        if(tmpmon.GetDegree()==maxdeg)
          {
        tmpmon.SetCoeff(p.nf[k]);
        tmp.push_back(tmpmon);
          }
        k++;
      }
    }
  m=tmp.front();
  //  cout<<"monom choisi "<<m<<endl;
  return m;
}

template<typename mon, typename T,typename Base, typename monomial_server>
mon choice_num(const polbis<mon,T> & p, const Base &b, monomial_server &serv)
{
  list<mon> tmp;
  mon m; 
  int maxdeg=-1,k=0;
  //int maxtmpdeg=0;
  for(int i=0;i<p.sizenf;i++)
    {
      if (p.nfind[i])
    for(int j=0;j<8;j++)
      if(p.nfind[i]>>j&1)
      {
        mon tmpmon;
        serv.int2mon(8*i+j,tmpmon);
        if(tmpmon.GetDegree()>maxdeg)
          {
        tmp.erase(tmp.begin(),tmp.end());
        maxdeg=tmpmon.GetDegree();
          }
        if(tmpmon.GetDegree()==maxdeg)
          {
        tmpmon.SetCoeff(p.nf[k]);
        tmp.push_back(tmpmon);
          }
        k++;
      }
    }
  m=tmp.front();
  for(typename list<mon>::iterator itertmp=tmp.begin();itertmp!=tmp.end();itertmp++)
    {
      if(m.GetCoeff()<itertmp->GetCoeff()) {m=*itertmp;};
    }
  //  cout<<"monom choisi "<<m<<endl;
  return m;
}

template<typename mon, typename T,typename Base, typename monomial_server>
mon choice_grevlex(const polbis<mon,T> & p,const Base &b, 
           monomial_server &serv)
{
  list<mon> tmp;
  mon m; 
  int maxdeg=-1,k=0;
  //int maxtmpdeg=0;
  for(int i=0;i<p.sizenf;i++)
    {
      if (p.nfind[i])
    for(int j=0;j<8;j++)
      if(p.nfind[i]>>j&1)
        
    {
      mon tmpmon;
      serv.int2mon(8*i+j,tmpmon);
      if(tmpmon.GetDegree()>maxdeg)
        {
          tmp.erase(tmp.begin(),tmp.end());
          maxdeg=tmpmon.GetDegree();
        }
      if(tmpmon.GetDegree()==maxdeg)
        {
          tmpmon.SetCoeff(p.nf[k]);
          tmp.push_back(tmpmon);
        }
      k++;
    }
    }
  m=tmp.front();
  for(typename list<mon>::iterator itertmp=tmp.begin();itertmp!=tmp.end();itertmp++)
    {
      if(my_ord(*itertmp,m)<0) {m=*itertmp;};
    }
  // cout<<"monom choisi "<<m<<endl;
  return m;
}

template<typename mon, typename T,typename Base, typename monomial_server>
mon choice_dlex(const polbis<mon,T> & p,const Base &b, monomial_server &serv)
{
  list<mon> tmp;
  mon m; 
  int maxdeg=-1,k=0;
  //int maxtmpdeg=0;
  for(int i=0;i<p.sizenf;i++)
    {
      if (p.nfind[i])
    for(int j=0;j<8;j++)
      if(p.nfind[i]>>j&1)
        {
          mon tmpmon;
          serv.int2mon(8*i+j,tmpmon);
          if(tmpmon.GetDegree()>maxdeg)
        {
          tmp.erase(tmp.begin(),tmp.end());
          maxdeg=tmpmon.GetDegree();
        }
          if(tmpmon.GetDegree()==maxdeg)
        {
          tmpmon.SetCoeff(p.nf[k]);
          tmp.push_back(tmpmon);
        }
          k++;
        }
    }
  m=tmp.front();
  for(typename list<mon>::iterator itertmp=tmp.begin();itertmp!=tmp.end();itertmp++)
    {
      if(my_ord(*itertmp,m)>0) {m=*itertmp;};
    }
  //  cout<<"monom choisi "<<m<<endl;
  return m;
}


template<typename mon, typename T,typename Base, typename monomial_server>
mon choice_mac(const polbis<mon,T> & p,const Base &b, monomial_server &serv)
{
  list<mon> tmp;
  mon m; 
  int maxdeg=-1,k=0;
  int maxtmpdeg=0;
  for(int i=0;i<p.sizenf;i++)
    {
      if (p.nfind[i])
    for(int j=0;j<8;j++)
      if(p.nfind[i]>>j&1)
        {
          mon tmpmon;
          serv.int2mon(8*i+j,tmpmon);
          if(tmpmon.GetDegree()>maxdeg)
        {
          tmp.erase(tmp.begin(),tmp.end());
          maxdeg=tmpmon.GetDegree();
        }
          if(tmpmon.GetDegree()==maxdeg)
        {
          tmpmon.SetCoeff(p.nf[k]);
          tmp.push_back(tmpmon);
        }
          k++;
        }
    }
  m=tmp.front();
  for(typename list<mon>::iterator itertmp=tmp.begin();itertmp!=tmp.end();itertmp++)
    {
      //cout<<"monomes regarde "<<*itertmp<<" Lavr  "<<lvar(itertmp->rep)<<endl;
      for(int i=0;i<=lvar(itertmp->rep);i++)
    {
      if(itertmp->GetDegree(i)>=maxtmpdeg)
        {
          maxtmpdeg=itertmp->GetDegree(i);
          m=*itertmp;
        }
    }
    }
  return m;
}


template<typename mon, typename T,typename Base, typename monomial_server>
mon choice_mpq(const polbis<mon,T> & p,const Base &b, monomial_server &serv)
{
  int i,degp=Degree(p),k=0;
  int mini=INT_MAX;
  mon tmpmon;
  mon stockmon;
  for(i=0;i<p.sizenf;i++)
    if(p.nfind[i])
      for(int j=0;j<8;j++)
    if(p.nfind[i]>>j&1)
      {
        serv.int2mon(8*i+j,tmpmon);
        if(tmpmon.GetDegree()==degp)
          {
        __mpz_struct  *numref,*denref;
        numref=mpq_numref(&(p.nf[k].rep()));
        denref=mpq_denref(&(p.nf[k].rep()));
        if(mpz_size(numref)+
           mpz_size(denref)<mini) 
          {
            mini=mpz_size(mpq_numref(&(p.nf[k].rep())))+
              mpz_size(mpq_denref(&(p.nf[k].rep())));
            //cout<<"newmax "<<max<<endl;
            //cout<<" les monomes "<<*iter<<endl;
            stockmon=tmpmon*p.nf[k];
          }
          }
        k++;
      }
  //cout<<"je retourne "<<stockmon<<endl;
  return stockmon;
}

template<typename mon, typename T, typename Base, typename monomial_server>
mon choice_fall(const polbis<mon,T> & p, const Base &b, monomial_server &serv)
{
  int i,k=0;
  int maxi=-1;
  mon tmpmon;
  mon stockmon;
  mon res=choice_mac(p,b,serv);
  typename Base::const_iterator iterb;
  list<mon> tmplist;
  int degp=Degree(p,serv);
#ifdef DEB 
  cout<<"Degree(p) "<<degp<<endl;
#endif
  for(i=0;i<p.sizenf;i++)
    if(p.nfind[i])
      for(int j=0;j<8;j++)
    if((p.nfind[i]>>j)&1) 
      {
        serv.int2mon(8*i+j,tmpmon);
        if (tmpmon.GetDegree()==degp)// && (!Iszero(p.nf[k])))
          tmplist.push_back(tmpmon*p.nf[k]);
        k++;    
        //      cout<<"k "<<k<<" "<<tmpmon<<endl;
      }
  for(typename list<mon>::iterator iterlist=tmplist.begin()
    ;iterlist!=tmplist.end();iterlist++)
    {
      for(iterb=b.begin();iterb!=b.end();iterb++)
    {
      int isgood=0;
      for(int i=0;i<iterb->taille1;i++)
        if(isdivisible(iterb->accept[i],*iterlist,b))
          {isgood=1;break;}
      for(int i=0;i<iterb->taille2;i++)
        if(isdivisible(iterb->refuse[i],*iterlist,b))
          {isgood=0;break;}
      if(isgood)
        {

          for(int i=0;i<iterb->taille2;i++)
        {
          mon m=lcm(*iterlist,iterb->refuse[i]);
          int dist=0;
          for(int j=0;j<iterb->taille2;j++)
            if ((j!=i) && (isdivisible(iterb->refuse[j],m,b)))
              isgood=0;
          if(isgood)
            {
              for(int j=0;j<b.nbvar();j++)
            if(m.GetDegree(j)>iterb->refuse[i].GetDegree(j))
              dist+=m.GetDegree(j)-iterb->refuse[i].GetDegree(j);
              if(dist>maxi)
            {
              res=*iterlist;
              maxi=dist;
            }
            }
        }
        }
      
    }
    }
#ifdef BONRAJOUT  
  cout<<"le pol "<<invconv<Poly>(p)<<endl<<"le mon choisi "<<res<<endl;
#endif
  return res;
}


template<typename mon, typename T, int alpha,typename Base, typename monomial_server>
mon choice_fall_num(const polbis<mon,Z<alpha,T> >& pol, 
const Base &b, monomial_server &serv)
{
  return choice_fall(pol,b,serv);
}
template<typename mon, typename T, typename Base, typename monomial_server>
mon choice_fall_num(const polbis<mon,T> & p, const Base &b, monomial_server &serv)
{
  int i,k=0;
  int maxi=-1;
  mon tmpmon;
  mon stockmon;
  mon res=choice_mac(p,b,serv);
  //cout<<"monome choisi "<<res<<endl;
  typename Base::const_iterator iterb;
  list<mon> tmplist;
  int degp=Degree(p,serv);
  typename mon::coeff_t maxcoeff=0;
  for(i=0;i<p.sizenf;i++)
    if(p.nfind[i])
      for(int j=0;j<8;j++)
    if((p.nfind[i]>>j)&1) 
      {
        serv.int2mon(8*i+j,tmpmon);
        if (tmpmon.GetDegree()==degp)// && (!Iszero(p.nf[k])))
          tmplist.push_back(tmpmon*p.nf[k]);
        k++;    
        //      cout<<"k "<<k<<" "<<tmpmon<<endl;
      }
  for(typename list<mon>::iterator iterlist=tmplist.begin()
    ;iterlist!=tmplist.end();iterlist++)
    {
      if(maxcoeff<iterlist->GetCoeff())
    maxcoeff=iterlist->GetCoeff();
      if(maxcoeff<-1*iterlist->GetCoeff())
    maxcoeff=-1*iterlist->GetCoeff();
 
    }
  for(typename list<mon>::iterator iterlist=tmplist.begin()
    ;iterlist!=tmplist.end();iterlist++)
    {
      if((iterlist->GetCoeff()<0?-1*iterlist->GetCoeff():iterlist->GetCoeff())
     <LEVELSELECT*maxcoeff)
    continue;
      
      for(iterb=b.begin();iterb!=b.end();iterb++)
    {
      int isgood=0;
      for(int i=0;i<iterb->taille1;i++)
        if(isdivisible(iterb->accept[i],*iterlist,b))
          {isgood=1;break;}
      for(int i=0;i<iterb->taille2;i++)
        if(isdivisible(iterb->refuse[i],*iterlist,b))
          {isgood=0;break;}
      if(isgood)
        {

          for(int i=0;i<iterb->taille2;i++)
        {
          mon m=lcm(*iterlist,iterb->refuse[i]);
          int dist=0;
          for(int j=0;j<iterb->taille2;j++)
            if ((j!=i) && (isdivisible(iterb->refuse[j],m,b)))
              isgood=0;
          if(isgood)
            {
              for(int j=0;j<b.nbvar();j++)
            if(m.GetDegree(j)>iterb->refuse[i].GetDegree(j))
              dist+=m.GetDegree(j)-iterb->refuse[i].GetDegree(j);
              if(dist>maxi)
            {
              res=*iterlist;
              maxi=dist;
            }
            }
        }
        }
      
    }
    }
 // cout<<"pol especial"<<invconv<Poly>(p,serv)<<endl;
  //   cout<<"le mon choisi "<<res<<endl;
  return res;
}

template <typename T,typename mono, typename workspace>
list<polbis<mono,T> >& SolveUold(const harewell<T> & u ,list<polbis<mono,T> > &Pk
                  ,workspace &w)
{
  typedef list<polbis<mono,T> > typPk;
  typedef typename typPk::value_type elem_typ;
  typedef typename elem_typ::monom_t mon;
  typedef typename elem_typ::coeff_t coeff; 
  int t;
  vector<polbis<mono,T> > tmpPk;
  for(typename typPk::iterator iter=Pk.begin();iter!=Pk.end();iter++)
    tmpPk.push_back(*iter);
  Pk.erase(Pk.begin(),Pk.end());
  
  for(int i=u.ncol-1;i>=0;i--)
    //    if (!Iszero(u(i,i)))
      {
    T mult_coeff=(T)1;
    for (int j=(t=(i==(u.ncol-1))?(u.nnz-1):(u.colptr[i+1]-1))
           ;(j>=(u.colptr)[i]);j--)
      if (i==u.rowind[j]) 
        {
        mult_coeff=((T*)u.nzval)[j];
        //  cout<<"mult_coeff "<<i<<" "<<mult_coeff<<endl;
        }
    if(Iszero(mult_coeff)) continue;
    mult_coeff=((coeff)1)/mult_coeff;
    for (int j=(t=(i==(u.ncol-1))?(u.nnz-1):(u.colptr[i+1]-1))
           ;(j>=(u.colptr)[i]);j--)
      {
        //cout<<"j "<<j<<" i "<<i<<" rowind[j] "<<u.rowind[j]<<endl;
        if (i==u.rowind[j])//t==j 
          {
        if (Iszero(((T*)u.nzval)[j])) 
          //le system est il "degenere"?
          {//elemwrite(Pk,u.rowind[j],(elem_typ)(0));continue;}
            //cout<<"M#rde ici ligne "<<i<<endl;
            //exit(-1);
            break;
          }
        //oui on met 0 et on a 
        //plus rien a faire pour cette colonne
        else//non on calcule
              {
            //cout<<"Dans solveU le coeff vaut"
            //    <<((coeff*)u.nzval)[j]<<endl;
            mult_coeff=coeff(1);
            //elemwrite(Pk,u.rowind[j],
            //elemread(Pk,u.rowind[j])=
            
            // elemread(Pk,u.rowind[j])*
            //cout<<"rowind[i] "<<u.rowind[i]<<endl;
            tmpPk[u.rowind[j]]*
              (coeff)(((coeff)(1)/((coeff*)u.nzval)[j]));
              }
          }
        else {
          if(((T*)u.nzval)[j]!=(coeff)0)
        {
           //cout<<"((coeff*)u.nzval)[j]"<<((coeff*)u.nzval)[j]<<endl;
           //elemread(Pk,i)=
        //elemread(Pk,i)*mon((coeff)(((coeff*)u.nzval)[j]/mult_coeff));
          my_plus(tmpPk[u.rowind[j]],
              tmpPk[i],((coeff)(((coeff*)u.nzval)[j]*mult_coeff))
              ,&w);
          //elemread(Pk,i)=
     //     elemread(Pk,i)*((coeff)(mult_coeff/(((coeff*)u.nzval)[j])));
        }
       }
      }
      }
  typename list<polbis<mono, T> >::iterator iterPk=Pk.begin();
  for(typename vector<polbis<mono , T> >::iterator iter=tmpPk.begin()
    ;iter!=tmpPk.end();iter++)
        Pk.push_back(*iter);
    
//cout<<"exit SolveU"<<endl;
  return Pk;
}
template <typename T,typename mono, typename workspace>
list<polbis<mono,T> >& SolveU(const harewell<T> & u ,list<polbis<mono,T> > &Pk
                  ,workspace &w)
{
  typedef list<polbis<mono,T> > typPk;
  typedef typename typPk::value_type elem_typ;
  typedef typename elem_typ::monom_t mon;
  typedef typename elem_typ::coeff_t coeff; 
  int t;
  vector<polbis<mono,T> > tmpPk;
  for(typename typPk::iterator iter=Pk.begin();iter!=Pk.end();iter++)
    tmpPk.push_back(*iter);
  Pk.erase(Pk.begin(),Pk.end());
  
  for(int i=u.ncol-1;i>=0;i--)
      {
    T mult_coeff=(T)1;
    for (int j=(t=(i==(u.ncol-1))?(u.nnz-1):(u.colptr[i+1]-1))
           ;(j>=(u.colptr)[i]);j--)
      if (i==u.rowind[j]) 
        {
        mult_coeff=((T*)u.nzval)[j];
        }
    if(Iszero(mult_coeff)) continue;
    mult_coeff=((coeff)1)/mult_coeff;
    tmpPk[i]*mult_coeff;
    for (int j=(t=(i==(u.ncol-1))?(u.nnz-1):(u.colptr[i+1]-1))
           ;(j>=(u.colptr)[i]);j--)
      {
        if (i!=u.rowind[j])//t==j 
          if(((T*)u.nzval)[j]!=(coeff)0)
        {
          my_plus(tmpPk[u.rowind[j]],
              tmpPk[i],((coeff*)u.nzval)[j]
              ,&w);
        }
      }
      }
  typename list<polbis<mono, T> >::iterator iterPk=Pk.begin();
  for(typename vector<polbis<mono , T> >::iterator iter=tmpPk.begin()
    ;iter!=tmpPk.end();iter++)
        Pk.push_back(*iter);
  return Pk;
}

template <typename T,typename mono, typename workspace>
list<polbis<mono,T> >& SolveL(const harewell<T> &l,list<polbis<mono,T> > &Pk
                  ,workspace &w)
{
  typedef list<polbis<mono,T> > typPk;
  typedef typename typPk::value_type elem_typ;
  typedef typename elem_typ::monom_t mon;
  typedef typename elem_typ::coeff_t coeff; 
  //  elem_typ tmp;
  //cout<<"l.nnz "<<l.nnz<<endl;
  //cout<<"l.nzval "<<l.nzval<<endl;
  //cout<<"l.rowind "<<l.rowind<<endl;
  // affdebug(l);
  vector<polbis<mono,T> > tmpPk;
  for(typename typPk::iterator iter=Pk.begin();iter!=Pk.end();iter++)
    tmpPk.push_back(*iter);
    Pk.erase(Pk.begin(),Pk.end());
 
  for(int i=0;i<l.ncol;i++)
    {
      int j=l.colptr[i];
#if 0
      for (;(j<l.nnz-1)&&(j<(l.colptr)[i+1]-1);j+=2)
    {
      if (i==l.rowind[j]) 
        {
          my_plus(tmpPk[l.rowind[j+1]]
              ,tmpPk[i],(((coeff*)l.nzval)[j+1]));
        }
      else
        if (i==l.rowind[j+1]) 
          {
        my_plus(tmpPk[l.rowind[j]]
            ,tmpPk[i],(((coeff*)l.nzval)[j]));
          }
        else
          {
        //if((((coeff*)l.nzval)[j])!=(coeff)0)
        {
          my_plus2(tmpPk[l.rowind[j]],tmpPk[l.rowind[j+1]]
               ,tmpPk[i],(((coeff*)l.nzval)[j])
               ,(((coeff*)l.nzval)[j+1]));
          //cout<<"pointeurs apres "<<elemread(Pk,l.rowind[j]).nfind<<endl;
          //elemread(Pk,i)*(coeff)((coeff)1/((coeff*)l.nzval)[j]);
        }
          } 
    }
#endif 
      for (;(j<l.nnz)&&(j<(l.colptr)[i+1]);j++)
      {
    if (i==l.rowind[j]) 
      {
        //cout<<"j vaut "<<j<<endl;
        //cout<<"l.rowind["<<j<<"] "<<l.rowind[j]<<endl;
        //cout<<"l.nzval[j] "<<(((coeff *)l.nzval))[j]<<endl;
        //tmp=elemread(Pk,l.rowind[j]);
        // cout<<"Dans SolveL le coeff vaut "<< (1)/((((coeff *)l.nzval))[j])
        // <<endl;
        //elemwrite(Pk,l.rowind[j],
        //      tmp=tmp*(coeff)((coeff)(1)/((((coeff *)l.nzval))[j]));
      }
    else 
      {
        if((((coeff*)l.nzval)[j])!=(coeff)0)
          {
        //      cout<<"I vaut "<<i<<endl;
        //cout<<"j vaut "<<j<<endl;
        //  cout<<"l.rowind["<<j<<"] "<<l.rowind[j]<<endl;
        //cout<<"l.nzval[j]"<<((coeff*)l.nzval)[j]<<endl;
//      cout<<"Q.size vaut "<<(elemread(Pk,i).size)<<endl;
        //cout<<"pointeurs avant "<<elemread(Pk,l.rowind[j]).nfind<<endl;
        my_plus(tmpPk[l.rowind[j]]
            ,tmpPk[i],(((coeff*)l.nzval)[j]),&w);
        //cout<<"pointeurs apres "<<elemread(Pk,l.rowind[j]).nfind<<endl;
        //elemread(Pk,i)*(coeff)((coeff)1/((coeff*)l.nzval)[j]);
          }
      } 
      }
    }
  typename list<polbis<mono, T> >::iterator iterPk=Pk.begin();
  for(typename vector<polbis<mono , T> >::iterator iter=tmpPk.begin()
    ;iter!=tmpPk.end();iter++,iterPk++)
    Pk.push_back(*iter);
    //*iterPk=*iter;
  return Pk;
}


template<typename typmat ,typename T, typename mono, typename workspace> 
list<polbis<mono,T> > &
Solve(typmat &S, std::list< polbis<mono,T> > &TildePk,int **pr,int **pc
      ,workspace &w)
{
 typmat L,U;
 typedef list< polbis<mono,T> > typPk;
 int *Pr,*Pc;
 LUdecomp(S,&L,&U,&Pr,&Pc);
 inverse(Pr,L.nrow);
 Dopermu(TildePk,Pr);
 SolveL(L,TildePk,w);
 SolveU(S,TildePk,w);//SolveU(S,TildePk);
 *pc=Pc;*pr=Pr;
 L.destroystore();
 return TildePk;
}

template<typename typmat ,typename T, typename mono, typename workspace> 
list<polbis<mono,T> > &
Solve(typmat &S, typmat &L,
      std::list< polbis<mono,T> > &TildePk,int **pr,int **pc
      ,workspace &w)
{
 typmat U;
 typedef list< polbis<mono,T> > typPk;
 int *Pr,*Pc;//,maxsize=0;
 std::vector< polbis<mono,T> > localPk(TildePk.size());
 copy(TildePk.begin(),TildePk.end(),localPk.begin());
 //cout<<"M en entree"<<endl;
 //affdebug(S);
 //cout<<"apres aff"<<endl;
 LUdecomp(S,&L,&U,&Pr,&Pc);
// cout<<"perm row"<<endl;
// for(int i=0;i<L.nrow;i++)
//   cout<<Pr[i]<<" ";
// cout<<endl;
// cout<<"perm col"<<endl;
// for(int i=0;i<S.ncol;i++)
//   cout<<Pc[i]<<" ";
// cout<<endl;
 
 inverse(Pr,L.nrow);
 //cout<<"error inerse"<<endl;
 // Dopermu(TildePk,Pr);
 Dopermu(localPk,Pr);
 //cout<<"error perm"<<endl;
 //SolveL(L,TildePk,w);
 SolveL(L,localPk,w);
 //cout<<"error solveL"<<endl;
 SolveU(S,localPk,w);//SolveU(S,TildePk);
 // SolveU(S,TildePk,w);//SolveU(S,TildePk);
 // cout<<"L"<<endl;
 //affdebug(L);
 //cout<<"S"<<endl;
 //affdebug(S);
 *pc=Pc;*pr=Pr;
 copy(localPk.begin(),localPk.end(),TildePk.begin());
 return TildePk;
}



#endif //ALREADY_pol2bisspecial