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rootpol.c

/* $Id: rootpol.c,v 1.21.2.4 2002/01/10 21:50:18 cvsadmin Exp $

Copyright (C) 2000  The PARI group.

This file is part of the PARI/GP package.

PARI/GP is free software; you can redistribute it and/or modify it under the
terms of the GNU General Public License as published by the Free Software
Foundation. It is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY WHATSOEVER.

Check the License for details. You should have received a copy of it, along
with the package; see the file 'COPYING'. If not, write to the Free Software
Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */

/*******************************************************************/
/*                                                                 */
/*                ROOTS OF COMPLEX POLYNOMIALS                     */
/*                (written by Xavier Gourdon)                      */
/*                                                                 */
/*******************************************************************/
#include "pari.h"

extern GEN polrecip_i(GEN x);
#define pariINFINITY 100000
#define NEWTON_MAX 10

static GEN globalu, *radius;
static long KARASQUARE_LIMIT, COOK_SQUARE_LIMIT, Lmax, step4;

/********************************************************************/
/**                                                                **/
/**                      ARITHMETIQUE RAPIDE                       **/
/**                                                                **/
/********************************************************************/

/* fast product of x,y which must be integer or complex of integer */
static GEN
quickmulcc(GEN x, GEN y)
{
  long tx=typ(x),ty=typ(y);
  GEN z;

  if (tx==t_INT)
  {
    if (ty==t_INT) return mulii(x,y);
    if (ty==t_COMPLEX)
    {
      z=cgetg(3,t_COMPLEX);
      z[1]=(long) mulii(x,(GEN) y[1]);
      z[2]=(long) mulii(x,(GEN) y[2]);
      return z;
    }
  }

  if (tx==t_COMPLEX)
  {
    if (ty==t_INT)
    {
      z=cgetg(3,t_COMPLEX);
      z[1]=(long) mulii((GEN)x[1],y);
      z[2]=(long) mulii((GEN)x[2],y);
      return z;
    }
    if (ty==t_COMPLEX)
    {
      long av,tetpil;
      GEN p1,p2;

      z=cgetg(3,t_COMPLEX); av=avma;
      p1=mulii((GEN)x[1],(GEN)y[1]); p2=mulii((GEN)x[2],(GEN)y[2]);
      x=addii((GEN)x[1],(GEN)x[2]); y=addii((GEN)y[1],(GEN)y[2]);
      y=mulii(x,y); x=addii(p1,p2);
      tetpil=avma; z[1]=lsubii(p1,p2); z[2]=lsubii(y,x);
      gerepilemanyvec(av,tetpil,z+1,2);
      return z;
    }
  }
  err(talker,"bug in quickmulcc");
  return NULL; /* not reached */
}

static void
set_karasquare_limit(long bitprec)
{
  if (bitprec<600) { KARASQUARE_LIMIT=8; COOK_SQUARE_LIMIT=400; return; }
  if (bitprec<2000) { KARASQUARE_LIMIT=4; COOK_SQUARE_LIMIT=200; return; }
  if (bitprec<3000) { KARASQUARE_LIMIT=4; COOK_SQUARE_LIMIT=125; return; }
  if (bitprec<5000) { KARASQUARE_LIMIT=2; COOK_SQUARE_LIMIT=75; return; }
  KARASQUARE_LIMIT=1; COOK_SQUARE_LIMIT=50;
}

/* the pari library does not have specific procedure for the square of
polynomials. This one is twice faster than gmul */
static GEN
mysquare(GEN p)
{
  GEN s,aux1,aux2;
  long i,j,n=lgef(p)-3,nn,ltop,lbot;

  if (n==-1) return gcopy(p);
  nn=n<<1; s=cgetg(nn+3,t_POL);
  s[1] = evalsigne(1) | evalvarn(varn(p)) | evallgef(nn+3);
  for (i=0; i<=n; i++)
  {
    aux1=gzero; ltop=avma;
    for (j=0; j<(i+1)>>1; j++)
    {
      aux2=quickmulcc((GEN) p[j+2],(GEN)p[i-j+2]);
      aux1=gadd(aux1,aux2);
    }
    if (i%2==1) { lbot=avma; s[i+2]=lpile(ltop,lbot,gshift(aux1,1)); }
    else
    {
      aux1=gshift(aux1,1);
      aux2=quickmulcc((GEN)p[2+(i>>1)],(GEN)p[2+(i>>1)]);
      lbot=avma; s[i+2]=lpile(ltop,lbot,gadd(aux1,aux2));
    }
  }
  for (i=n+1; i<=nn; i++)
  {
    aux1=gzero; ltop=avma;
    for (j=i-n; j<(i+1)>>1; j++)
    {
      aux2=quickmulcc((GEN)p[j+2],(GEN)p[i-j+2]);
      aux1=gadd(aux1,aux2);
    }
    if (i%2==1) { lbot=avma; s[i+2]=lpile(ltop,lbot,gshift(aux1,1)); }
    else
    {
      aux1=gshift(aux1,1);
      aux2=quickmulcc((GEN)p[2+(i>>1)],(GEN)p[2+(i>>1)]);
      lbot=avma; s[i+2]=lpile(ltop,lbot,gadd(aux1,aux2));
    }
  }
  return s;
}

static GEN
karasquare(GEN p)
{
  GEN p1,s0,s1,s2,aux;
  long n=lgef(p)-3,n0,n1,i,var,nn0,ltop,lbot;

  if (n<=KARASQUARE_LIMIT) return mysquare(p);
  ltop=avma;
  var=evalsigne(1)+evalvarn(varn(p)); n0=n>>1; n1=n-n0-1;
  setlgef(p,n0+3); /* hack to have the first half of p */
  s0=karasquare(p);
  p1=cgetg(n1+3,t_POL); p1[1]=var+evallgef(n1+3);
  for (i=2; i<=n1+2; i++) p1[i]=p[1+i+n0];
  s2=karasquare(p1);
  s1=karasquare(gadd(p,p1));
  s1=gsub(s1,gadd(s0,s2));
  nn0=n0<<1;
  aux=cgetg((n<<1)+3,t_POL); aux[1]=var+evallgef(2*n+3);
  var=lgef(s0);
  for (i=2; i<var; i++) aux[i]=s0[i];
  for (   ; i<=nn0+2; i++) aux[i]=zero;
  var=lgef(s2);
  for (i=2; i<var; i++) aux[2+i+nn0]=s2[i];
  for (i=var-2; i<=(n1<<1); i++) aux[4+i+nn0]=zero;
  aux[3+nn0]=zero;
  for (i=3; i<=lgef(s1); i++)
    aux[i+n0]=ladd((GEN) aux[i+n0],(GEN) s1[i-1]);
  setlgef(p,n+3); /* recover all the polynomial p */
  lbot=avma; return gerepile(ltop,lbot,gcopy(aux));
}

static GEN
cook_square(GEN p)
{
  GEN p0,p1,p2,p3,q,aux0,aux1,r,aux,plus,moins;
  long n=lgef(p)-3,n0,n3,i,j,ltop=avma,lbot,var;

  if (n<=COOK_SQUARE_LIMIT) return karasquare(p);

  n0=(n+1)/4; n3=n+1-3*n0;
  p0=cgetg(n0+2,t_POL); p1=cgetg(n0+2,t_POL); p2=cgetg(n0+2,t_POL);
  p3=cgetg(n3+2,t_POL);
  var=evalsigne(1)|evalvarn(varn(p));
  p0[1]=p1[1]=p2[1]=var|evallgef(n0+2); p3[1]=var|evallgef(n3+2);

  for (i=0; i<n0; i++)
  {
    p0[i+2]=p[i+2]; p1[i+2]=p[i+n0+2]; p2[i+2]=p[i+2*n0+2];
  }
  for (i=0; i<n3; i++) p3[i+2]=p[i+3*n0+2];

  q=cgetg(8,t_VEC); q=q+4;

  q[0]=(long) p0;
  aux0=gadd(p0,p2); aux1=gadd(p1,p3);
  q[-1]=lsub(aux0,aux1); q[1]=ladd(aux0,aux1);
  aux0=gadd(p0,gmulgs(p2,4)); aux1=gmulgs(gadd(p1,gmulgs(p3,4)),2);
  q[-2]=lsub(aux0,aux1); q[2]=ladd(aux0,aux1);
  aux0=gadd(p0,gmulgs(p2,9)); aux1=gmulgs(gadd(p1,gmulgs(p3,9)),3);
  q[-3]=lsub(aux0,aux1); q[3]=ladd(aux0,aux1);
  for (i=-3; i<=3; i++) q[i]=(long) cook_square((GEN)q[i]);
  r=new_chunk(7);
  plus=cgetg(4,t_VEC); moins=cgetg(4,t_VEC);
  for (i=1; i<=3; i++)
  {
    plus[i]=ladd((GEN)q[-i],(GEN)q[i]);
    moins[i]=lsub((GEN)q[-i],(GEN)q[i]);
  }
  r[0]=q[0];
  r[1]=ldivgs(
            gsub(
               gsub(gmulgs((GEN)moins[2],9),(GEN)moins[3]),
               gmulgs((GEN)moins[1],45)),
            60);
  r[2]=ldivgs(
            gadd(
               gadd(gmulgs((GEN)plus[1],270),gmulgs((GEN)q[0],-490)),
               gadd(gmulgs((GEN)plus[2],-27),gmulgs((GEN)plus[3],2))),
            360);
  r[3]=ldivgs(
            gadd(
               gadd(gmulgs((GEN)moins[1],13),gmulgs((GEN)moins[2],-8)),
               (GEN)moins[3]),
            48);
  r[4]=ldivgs(
            gadd(
               gadd(gmulgs((GEN)q[0],56),gmulgs((GEN)plus[1],-39)),
               gsub(gmulgs((GEN)plus[2],12),(GEN)plus[3])),
            144);
  r[5]=ldivgs(
            gsub(
               gadd(gmulgs((GEN)moins[1],-5),gmulgs((GEN)moins[2],4)),
               (GEN)moins[3]),
            240);
  r[6]=ldivgs(
            gadd(
               gadd(gmulgs((GEN)q[0],-20),gmulgs((GEN)plus[1],15)),
               gadd(gmulgs((GEN)plus[2],-6),(GEN)plus[3])),
            720);
  q=cgetg(2*n+3,t_POL); q[1]=var|evallgef(2*n+3);
  for (i=0; i<=2*n; i++) q[i+2]=zero;
  for (i=0; i<=6; i++)
  {
    aux=(GEN) r[i];
    for (j=0; j<=lgef(aux)-3; j++)
      q[n0*i+j+2]=ladd((GEN)q[n0*i+j+2],(GEN)aux[j+2]);
  }
  lbot=avma; return gerepile(ltop,lbot,gcopy(q));
}

static GEN
graeffe(GEN p)
{
  GEN p0,p1,s0,s1,ss1;
  long n=lgef(p)-3,n0,n1,i,auxi,ns1;

  if (n==0) return gcopy(p);
  n0=n>>1; n1=(n-1)>>1;
  auxi=evalsigne(1)|evalvarn(varn(p));
  p0=cgetg(n0+3,t_POL); p0[1]=auxi|evallgef(n0+3);
  p1=cgetg(n1+3,t_POL); p1[1]=auxi|evallgef(n1+3);
  for (i=0; i<=n0; i++) p0[i+2]=p[2+(i<<1)];
  for (i=0; i<=n1; i++) p1[i+2]=p[3+(i<<1)];

  s0=cook_square(p0);
  s1=cook_square(p1); ns1 = lgef(s1)-3;
  ss1 = cgetg(ns1+4, t_POL);
  ss1[1] = auxi | evallgef(ns1+4);
  ss1[2]=zero;
  for (i=0; i<=ns1; i++) ss1[3+i]=lneg((GEN)s1[2+i]);
  /* now ss1 contains -x * s1 */
  return gadd(s0,ss1);
}

/********************************************************************/
/**                                                                **/
/**        FACTORISATION SQUAREFREE AVEC LE GCD MODULAIRE          **/
/**                                                                **/
/********************************************************************/

/* return a n x 2 matrix:
 *   col 1 contains the i's such that A_i non constant
 *   col 2 the A_i's, s.t. pol = A_i1^i1.A_i2^i2...A_in^in.
 * if pol is constant return [;]
 */
GEN
square_free_factorization(GEN pol)
{
  long deg,i,j,m;
  GEN p1,x,t1,v1,t,v,A;

  if (typ(pol)!=t_POL) err(typeer,"square_free_factorization");
  deg=lgef(pol)-3; if (deg<1) return cgetg(1,t_MAT);
  p1 = content(pol); if (!gcmp1(p1)) pol = gdiv(pol,p1);

  x=cgetg(3,t_MAT);
  t1 = NULL; /* gcc -Wall */
  if (deg > 1)
  {
    t1 = modulargcd(pol,derivpol(pol));
    if (isscalar(t1)) deg = 1;
  }
  if (deg==1)
  {
    x[1]=lgetg(2,t_COL); p1=(GEN)x[1]; p1[1]=un;
    x[2]=lgetg(2,t_COL); p1=(GEN)x[2]; p1[1]=(long)pol; return x;
  }
  A=new_chunk(deg+1); v1=gdivexact(pol,t1); v=v1; i=0;
  while (lgef(v)>3)
  {
    v=modulargcd(t1,v1); i++;
    A[i]=(long)gdivexact(v1,v);
    t=gdivexact(t1,v); v1=v; t1=t;
  }
  m=1; x[1]=lgetg(deg+1,t_COL); x[2]=lgetg(deg+1,t_COL);
  for (j=1; j<=i; j++)
    if (isnonscalar(A[j]))
    {
      p1=(GEN)x[1]; p1[m] = lstoi(j);
      p1=(GEN)x[2]; p1[m] = A[j];
      m++;
    }
  setlg(x[1],m); setlg(x[2],m); return x;
}

/********************************************************************/
/**                                                                **/
/**                 CALCUL DU MODULE DES RACINES                   **/
/**                                                                **/
/********************************************************************/

static double
log2ir(GEN x)
{
  double l;

  if (signe(x)==0) return (double) -pariINFINITY;
  if (typ(x)==t_INT)
  {
    if (lgefint(x)==3) return (double) log2( (double)(ulong) x[2]);
    l=(double)(ulong) x[2]+
      (double)(ulong) x[3] / exp2((double) BITS_IN_LONG);
    return log2(l)+ (double) BITS_IN_LONG * (lgefint(x)-3.);
  }
  /* else x is real */
  return 1.+ (double) expo(x)+log2( (double)(ulong) x[2]) - (double) BITS_IN_LONG;
}

static double
mylog2(GEN z)
{
  double x,y;

  if (typ(z)!=t_COMPLEX) return log2ir(z);

  x = log2ir((GEN) z[1]);
  y = log2ir((GEN) z[2]);
  if (fabs(x-y)>10) return (x>y)? x: y;
  return x+0.5*log2( 1 + exp2(2*(y-x)));
}

/* find s such that  A_h <= 2^s <= 2 A_i  for one h and all i < n = deg(p),
 * with  A_i := (binom(n,i) lc(p) / p_i) ^ 1/(n-i), and  p = sum p_i X^i */
static long
findpower(GEN p)
{
  double x, logbinomial, mins = pariINFINITY;
  long n = lgef(p)-3,i;

  logbinomial = mylog2((GEN)p[n+2]); /* log2(lc * binom(n,i)) */
  for (i=n-1; i>=0; i--)
  { 
    logbinomial += log2((double) (i+1) / (double) (n-i));
    x = mylog2((GEN)p[i+2]);
    if (x != -pariINFINITY)
    {
      double s = (logbinomial - x) / (double) (n-i);
      if (s < mins) mins = s;
    }
  }
  i = (long)ceil(mins);
  if (i - mins > 1 - 1e-12) i--;
  return i;
}

/* returns the exponent for the procedure modulus, from the newton diagram */
static long
polygone_newton(GEN p, long k)
{
  double *logcoef,pente;
  long n=lgef(p)-3,i,j,h,l,*sommet,pentelong;

  logcoef=(double*) gpmalloc((n+1)*sizeof(double));
  sommet=(long*) gpmalloc((n+1)*sizeof(long));

  /* sommet[i]=1 si i est un sommet, =0 sinon */
  for (i=0; i<=n; i++) { logcoef[i]=mylog2((GEN)p[2+i]); sommet[i]=0; }
  sommet[0]=1; i=0;
  while (i<n)
  {
    pente=logcoef[i+1]-logcoef[i];
    h=i+1;
    for (j=i+1; j<=n; j++)
    {
      if (pente<(logcoef[j]-logcoef[i])/(double)(j-i))
      {
      h=j;
      pente=(logcoef[j]-logcoef[i])/(double)(j-i);
      }
    }
    i=h;
    sommet[h]=1;
  }
  h=k; while (!sommet[h]) h++;
  l=k-1; while (!sommet[l]) l--;
  pentelong=(long) floor((logcoef[h]-logcoef[l])/(double)(h-l)+0.5);
  free(logcoef); free(sommet); return pentelong;
}

/* change z into z*2^e, where z is real or complex of real */
static void
myshiftrc(GEN z, long e)
{
  if (typ(z)==t_COMPLEX)
  {
    if (signe(z[1])!=0) setexpo(z[1], expo(z[1])+e);
    if (signe(z[2])!=0) setexpo(z[2], expo(z[2])+e);
  }
  else
    if (signe(z)!=0) setexpo(z,expo(z)+e);
}

/* return z*2^e, where z is integer or complex of integer (destroy z) */
static GEN
myshiftic(GEN z, long e)
{
  if (typ(z)==t_COMPLEX)
  {
    z[1]=signe(z[1])? lmpshift((GEN) z[1],e): zero;
    z[2]=lmpshift((GEN) z[2],e);
    return z;
  }
  return signe(z)? mpshift(z,e): gzero;
}

/* as realun with precision in bits, not in words */
static GEN
myrealun(long bitprec)
{
  GEN x;
  if (bitprec < 0) bitprec = 0;
  x = cgetr(bitprec/BITS_IN_LONG+3);
  affsr(1,x); return x;
}

static GEN
mygprecrc(GEN x, long bitprec, long e)
{
  long tx=typ(x);
  GEN y;

  if (bitprec<0) bitprec=0; /* should rarely happen */
  switch(tx)
  {
    case t_REAL:
      y=cgetr(bitprec/BITS_IN_LONG+3); affrr(x,y);
      if (!signe(x)) setexpo(y,-bitprec+e);
      break;
    case t_COMPLEX:
      y=cgetg(3,t_COMPLEX);
      y[1]=(long) mygprecrc((GEN)x[1],bitprec,e);
      y[2]=(long) mygprecrc((GEN)x[2],bitprec,e);
      break;
    default: y=gcopy(x);
  }
  return y;
}

/* gprec behaves badly with the zero for polynomials.
The second parameter in mygprec is the precision in base 2 */
static GEN
mygprec(GEN x, long bitprec)
{
  long tx=typ(x),lx,i,e = gexpo(x);
  GEN y;

  switch(tx)
  {
    case t_POL:
      lx=lgef(x); y=cgetg(lx,tx); y[1]=x[1];
      for (i=2; i<lx; i++) y[i]=(long) mygprecrc((GEN)x[i],bitprec,e);
      break;

    default: y=mygprecrc(x,bitprec,e);
  }
  return y;
}

/* the round fonction has a bug in pari. Thus I create mygfloor, using gfloor
which has no bug (destroy z)*/
static GEN
mygfloor(GEN z)
{
  if (typ(z)!=t_COMPLEX) return gfloor(z);
  z[1]=lfloor((GEN)z[1]); z[2]=lfloor((GEN)z[2]); return z;
}

/* returns a polynomial q in (Z[i])[x] keeping bitprec bits of p */
static GEN
eval_rel_pol(GEN p,long bitprec)
{
  long e=gexpo(p),n=lgef(p),i,shift;
  GEN q = gprec(p,(long) ((double) bitprec * L2SL10)+2);

  shift=bitprec-e+1;
  for (i=2; i<n; i++)
    q[i]=(long) mygfloor(myshiftic((GEN)q[i],shift));
  return q;
}

/* normalize a polynomial p, that is change it with coefficients in Z[i],
after making product by 2^bitprec */
static void
pol_to_gaussint(GEN p, long bitprec)
{
  long i,n=lgef(p);
  for (i=2; i<n; i++)
  {
    myshiftrc((GEN) p[i],bitprec);
    p[i]=(long) mygfloor((GEN) p[i]);
  }
}

/* returns p(R*x)/R^n (in R or R[i]), n=deg(p), bitprec bits of precision */
static GEN
homothetie(GEN p, GEN R, long bitprec)
{
  GEN q,r,gR,aux;
  long n=lgef(p)-3,i;

  gR=mygprec(ginv(R),bitprec);
  q=mygprec(p,bitprec);
  r=cgetg(n+3,t_POL); r[1]=p[1];
  aux=gR; r[n+2]=q[n+2];
  for (i=n-1; i>=0; i--)
  {
    r[i+2] = lmul(aux,(GEN)q[i+2]);
    aux = mulrr(aux,gR);
  }
  return r;
}

/* change q in 2^(n*e) p(x*2^(-e)), n=deg(q) */
static void
homothetie2n(GEN p, long e)
{
  if (e)
  {
    long i,n=lgef(p)-1;
    for (i=2; i<=n; i++) myshiftrc((GEN) p[i],(n-i)*e);
  }
}

/* return 2^f * 2^(n*e) p(x*2^(-e)), n=deg(q) */
static void
homothetie_gauss(GEN p, long e,long f)
{
  if (e||f)
  {
    long i, n=lgef(p)-1;
    for (i=2; i<=n; i++) p[i]=(long) myshiftic((GEN) p[i],f+(n-i)*e);
  }
}

static long
valuation(GEN p)
{
  long j=0,n=lgef(p)-3;

  while ((j<=n) && isexactzero((GEN)p[j+2])) j++;
  return j;
}

/* provides usually a good lower bound on the largest modulus of the roots,
puts in k an upper bound of the number of roots near the largest root
at a distance eps */
static double
lower_bound(GEN p, long *k, double eps)
{
  long n=lgef(p)-3,i,j,ltop=avma;
  GEN a,s,icd;
  double r,*rho;

  if (n<4) { *k=n; return 0.; }
  a=cgetg(6,t_POL); s=cgetg(6,t_POL);
  rho=(double *) gpmalloc(4*sizeof(double));
  icd = gdiv(realun(DEFAULTPREC), (GEN) p[n+2]);
  for (i=1; i<=4; i++) a[i+1]=lmul(icd,(GEN)p[n+2-i]);
  for (i=1; i<=4; i++)
  {
    s[i+1]=lmulsg(i,(GEN)a[i+1]);
    for (j=1; j<i; j++)
      s[i+1]=ladd((GEN)s[i+1],gmul((GEN)s[j+1],(GEN)a[i+1-j]));
    s[i+1]=lneg((GEN)s[i+1]);
    r=gtodouble(gabs((GEN) s[i+1],3));
    if (r<=0.)  /* should not be strictly negative */
      rho[i-1]=0.;
    else
      rho[i-1]=exp(log(r/(double)n)/(double) i);
  }
  r=0.;
  for (i=0; i<4; i++) if (r<rho[i]) r=rho[i];
  if (r>0. && eps<1.2)
    *k=(long) floor((n*rho[0]/r+n)/(1+exp(-eps)*cos(eps)));
  else
    *k=n;
  free(rho); avma=ltop; return r;
}

/* returns the maximum of the modulus of p with a relative error tau */
static GEN
max_modulus(GEN p, double tau)
{
  GEN q,r,aux,gunr;
  long i,j,k,valuat,n=lgef(p)-3,nn,av,ltop=avma,bitprec,M,e;
  double rho,tau2,eps;

  r = cgeti(BIGDEFAULTPREC);
  av = avma;

  if (tau>3.0) tau=3.0; /* fix PZ 7.2.98: ensures eps is positive */
  eps=1/log(4./tau); tau2=tau/6.;
  bitprec=(long) ((double) n*log2(1./tau2)+3*log2((double) n))+1;
  gunr=myrealun(bitprec+2*n);
  aux=gdiv(gunr,(GEN) p[2+n]);
  q=gmul(aux,p); q[2+n]=lcopy(gunr);
  k=nn=n;
  e=findpower(q); homothetie2n(q,e);
  affsi(e,r);
  q=mygprec(q,bitprec+(n<<1));
  pol_to_gaussint(q,bitprec);
  M=(long) ((log(log(4.*n))+log(3./tau))/log(2.))+2;
  for (i=0,e=0;;)
  {
    rho=lower_bound(q,&k,eps);
    if (rho>exp2(-(double) e)) e = (long) -floor(log2(rho));
    affii(shifti(addis(r,e), 1), r);
    if (++i == M) break;

    bitprec=(long) ((double) k* log2(1./tau2)+
                    (double) (nn-k)*log2(1./eps)+
                    3*log2((double) nn))+1;
    homothetie_gauss(q,e,bitprec-(long)floor(mylog2((GEN) q[2+nn])+0.5));
    valuat=valuation(q);
    if (valuat>0)
    {
      nn-=valuat; setlgef(q,nn+3);
      for (j=0; j<=nn; j++) q[2+j]=q[2+valuat+j];
    }
    set_karasquare_limit(gexpo(q));
    q = gerepileupto(av, graeffe(q));
    tau2=1.5*tau2; eps=1/log(1./tau2);
    e = findpower(q);
  }
  if (!signe(r)) { avma = ltop; return realun(DEFAULTPREC); }
  aux = cgetr(DEFAULTPREC);
  affir(r,aux); r = aux; setexpo(r, expo(r) - M);
  rho = rtodbl(r);
  /* rho = sum e_i 2^-i */
  avma = ltop;
  return mpexp(dbltor(-rho * LOG2)); /* 2^-r */
}

/* return the k-th modulus (in ascending order) of p, rel. error tau*/
static GEN
modulus(GEN p, long k, double tau)
{
  GEN q,gunr;
  long i,j,kk=k,imax,n=lgef(p)-3,nn,nnn,valuat,av,ltop=avma,bitprec,decprec,e;
  double tau2,r;

  tau2=tau/6; nn=n;
  bitprec= (long) ((double) n*(2.+log2(3.*(double) n)+log2(1./tau2)));
  decprec=(long) ((double) bitprec * L2SL10)+1;
  gunr=myrealun(bitprec);
  av = avma;
  q=gprec(p,decprec);
  q=gmul(gunr,q);
  e=polygone_newton(q,k);
  homothetie2n(q,e);
  r=(double) e;
  imax=(long) ((log2(3./tau)+log2(log(4.*(double) n)) ))+1;
  for (i=1; i<imax; i++)
  {
    q=eval_rel_pol(q,bitprec);

    nnn=lgef(q)-3; valuat=valuation(q);
    if (valuat>0)
    {
      kk-=valuat;
      for (j=0; j<=nnn-valuat; j++) q[2+j]=q[2+valuat+j];
      setlgef(q,nnn-valuat+3);
    }
    nn=nnn-valuat;

    set_karasquare_limit(bitprec);
    q = gerepileupto(av, graeffe(q));
    e=polygone_newton(q,kk);
    r += e / exp2((double)i);
    q=gmul(gunr,q);
    homothetie2n(q,e);

    tau2=1.5*tau2; if (tau2>1.) tau2=1.;
    bitprec= 1+(long) ((double) nn*(2.+log2(3.*(double) nn)+log2(1./tau2)));
  }
  avma=ltop; return mpexp(dbltor(-r * LOG2));
}

/* return the k-th modulus r_k of p, rel. error tau, knowing that
rmin < r_k < rmax. This helps because the information enable us to use
less precision... quicker ! */
static GEN
pre_modulus(GEN p, long k, double tau, GEN rmin, GEN rmax)
{
  GEN R, q, aux;
  long n=lgef(p)-3,i,imax,imax2,bitprec,ltop=avma, av;
  double tau2, aux2;

  tau2=tau/6.;
  aux = mulrr(mpsqrt(divrr(rmax,rmin)), dbltor(exp(4*tau2)));
  imax = (long) log2(log((double)n)/ rtodbl(mplog(aux)));
  if (imax<=0) return modulus(p,k,tau);

  R = mpsqrt(mulrr(rmin,rmax));
  av = avma;
  bitprec = (long) ((double) n*(2. + log2ir(aux) - log2(tau2)));
  q = homothetie(p,R,bitprec);
  imax2 = (long) ((log2(3./tau)+log2(log(4.*(double) n)) ))+1;
  if (imax>imax2) imax=imax2;

  for (i=0; i<imax; i++)
  {
    q = eval_rel_pol(q,bitprec);
    set_karasquare_limit(bitprec);
    q = gerepileupto(av, graeffe(q));
    affrr(mulrr(gsqr(aux), dbltor(exp(2*tau2))), aux);
    tau2 *= 1.5;
    bitprec= (long) ((double) n*(2. + log2ir(aux) - log2(1-exp(-tau2))));
    q = gmul(myrealun(bitprec),q);
  }

  aux2 = rtodbl(mplog(modulus(q,k,exp2((double)imax)*tau/3.)));
  R = mulrr(R, dbltor(exp(aux2*exp2(-(double)imax))));
  return gerepileupto(ltop, R);
}

/* returns the minimum of the modulus of p with a relative error tau */
static GEN
min_modulus(GEN p, double tau)
{
  long av=avma;
  GEN r;

  if (isexactzero((GEN)p[2])) return realzero(3);
  r = max_modulus(polrecip_i(p),tau);
  return gerepileupto(av, ginv(r));
}

/* returns k such that r_k e^(-tau) < R < r_{ k+1 } e^tau.
l is such that you know in advance that l<= k <= n-l */
static long
dual_modulus(GEN p, GEN R, double tau, long l)
{
  GEN q;
  long i,j,imax,k,delta_k=0,n=lgef(p)-3,nn,nnn,valuat,ltop=avma,bitprec,ll=l;
  double logmax,aux,tau2;

  tau2=7.*tau/8.;
  bitprec=6*n-5*l+(long) ((double) n*(log2(1/tau2)+8.*tau2/7.));
  q=homothetie(p,R,bitprec);
  nn=n;
  imax=(long)(log(log(2.*(double)n)/tau2)/log(7./4.)+1);

  for (i=0; i<imax; i++)
  {
    bitprec=6*nn-5*ll+(long) ((double) nn*(log2(1/tau2)+8.*tau2/7.));

    q=eval_rel_pol(q,bitprec);
    nnn=lgef(q)-3; valuat=valuation(q);
    if (valuat>0)
    {
      delta_k+=valuat;
      for (j=0; j<=nnn-valuat; j++) q[2+j]=q[2+valuat+j];
      setlgef(q,nnn-valuat+3);
    }
    ll= (-valuat<nnn-n)? ll-valuat: ll+nnn-n; /* min(ll-valuat,ll+nnn-n) */
    if (ll<0) ll=0;

    nn=nnn-valuat;
    if (nn==0) return delta_k;

    set_karasquare_limit(bitprec);
    q = gerepileupto(ltop, graeffe(q));
    tau2=tau2*7./4.;
  }
  k=-1; logmax=- (double) pariINFINITY;
  for (i=0; i<=lgef(q)-3; i++)
  {
    aux=mylog2((GEN)q[2+i]);
    if (aux>logmax) { logmax=aux; k=i; }
  }
  avma=ltop; return delta_k+k;
}

/********************************************************************/
/**                                                                **/
/**       CALCUL D'UN FACTEUR PAR INTEGRATION SUR LE CERCLE        **/
/**                                                                **/
/********************************************************************/

static GEN
gmulbyi(GEN z)
{
  GEN aux = cgetg(3,t_COMPLEX);

  if (typ(z)!=t_COMPLEX)
  {
    aux[1]=zero;
    aux[2]=(long) z;
  }
  else
  {
    aux[1]=lneg((GEN)z[2]);
    aux[2]=z[1];
  }
  return aux;
}

static void
fft(GEN Omega, GEN p, GEN f, long Step, long l)
{
  long i,l1,l2,l3,rap=Lmax/l,rapi,Step4,ltop,lbot;
  GEN f1,f2,f3,f02,f13,g02,g13,ff;

  if (l==2)
  {
    f[0]=ladd((GEN)p[0],(GEN)p[Step]);
    f[1]=lsub((GEN)p[0],(GEN)p[Step]); return;
  }
  if (l==4)
  {
    f1=gadd((GEN)p[0],(GEN)p[(Step<<1)]);
    f3=gadd((GEN)p[Step],(GEN)p[3*Step]);
    f[0]=ladd(f1,f3);
    f[2]=lsub(f1,f3);

    f2=gsub((GEN)p[0],(GEN)p[(Step<<1)]);
    f02=gsub((GEN)p[Step],(GEN)p[3*Step]);
    f02=gmulbyi(f02);
    f[1]=ladd(f2,f02);
    f[3]=lsub(f2,f02);
    return;
  }

  l1=(l>>2); l2=(l1<<1); l3=l1+l2; Step4=(Step<<2);

  ltop=avma;
  fft(Omega,p,f,Step4,l1);
  fft(Omega,p+Step,f+l1,Step4,l1);
  fft(Omega,p+(Step<<1),f+l2,Step4,l1);
  fft(Omega,p+3*Step,f+l3,Step4,l1);

  ff=cgetg(l+1,t_VEC);
  for (i=0; i<l1; i++)
  {
    rapi=rap*i;
    f1=gmul((GEN)Omega[rapi],(GEN) f[i+l1]);
    f2=gmul((GEN)Omega[(rapi<<1)],(GEN) f[i+l2]);
    f3=gmul((GEN)Omega[3*rapi],(GEN) f[i+l3]);

    f02=gadd((GEN)f[i],f2);
    g02=gsub((GEN)f[i],f2);
    f13=gadd(f1,f3);
    g13=gmulbyi(gsub(f1,f3));

    ff[i+1]=ladd(f02,f13);
    ff[i+l1+1]=ladd(g02,g13);
    ff[i+l2+1]=lsub(f02,f13);
    ff[i+l3+1]=lsub(g02,g13);
  }
  lbot=avma; ff=gerepile(ltop,lbot,gcopy(ff));
  for (i=0; i<l; i++) f[i]=ff[i+1];
}

extern void mpsincos(GEN x, GEN *s, GEN *c);

/* return exp(ix), x a t_REAL */
static GEN
exp_i(GEN x)
{
  GEN v;
  
  if (!signe(x)) return realun(lg(x)); /* should not happen */
  v = cgetg(3,t_COMPLEX);
  mpsincos(x, (GEN*)(v+2), (GEN*)(v+1));
  return v;
}

/* returns a vector RU which contains exp(2*i*k*Pi/NN), k=0..NN-1 */
static GEN
initRU(long NN, long bitprec)
{
  GEN prim,aux,*RU,mygpi;
  long i,N2=(NN>>1),N4=(NN>>2),N8=(NN>>3);

  RU = (GEN*)cgetg(NN+1,t_VEC); RU++;
  mygpi = mppi(bitprec/BITS_IN_LONG+3);
  aux = gdivgs(mygpi,NN/2); /* 2 Pi/NN */
  prim = exp_i(aux);
  aux = gmulbyi(aux);

  RU[0] = myrealun(bitprec);
  for (i=1; i<=N8; i++) RU[i] = gmul(prim, RU[i-1]);
  for (i=1; i<N8; i++)
  {
    aux=cgetg(3,t_COMPLEX);
    aux[1]=RU[i][2];
    aux[2]=RU[i][1]; RU[N4-i]=aux;
  }
  for (i=0; i<N4; i++) RU[i+N4]=gmulbyi(RU[i]);
  for (i=0; i<N2; i++) RU[i+N2]=gneg(RU[i]);
  return (GEN)RU;
}

/* returns 1 if p has only real coefficients, 0 else */
static long
isreal(GEN p)
{
  long n=lgef(p)-3,i=0;

  while (i<=n && typ(p[i+2])!=t_COMPLEX) i++;
  return (i>n);
}

static void
parameters(GEN p, double *mu, double *gamma,
           long polreal, double param, double param2)
{
  GEN q,pc,Omega,coef,RU,prim,aux,aux0,ggamma,gx,mygpi;
  long ltop=avma,limite=stack_lim(ltop,1),n=lgef(p)-3,bitprec,NN,K,i,j,ltop2;
  double lx;

  bitprec=gexpo(p)+(long)param2+8;
  NN=(long) (param*3.14)+1; if (NN<Lmax) NN=Lmax;
  K=NN/Lmax; if (K%2==1) K++; NN=Lmax*K;
  mygpi=mppi(bitprec/BITS_IN_LONG+3);
  aux = gdivgs(mygpi,NN/2); /* 2 Pi/NN */
  prim = exp_i(aux);
  aux = gmulbyi(aux);
  RU = myrealun(bitprec);

  Omega=initRU(Lmax,bitprec);

  q=mygprec(p,bitprec);
  pc=cgetg(Lmax+1,t_VEC); pc++;
  for (i=n+1; i<Lmax; i++) pc[i]=zero;

  coef=cgetg(Lmax+1,t_VEC); coef++;
  *mu=(double)pariINFINITY; *gamma=0.;
  ggamma = gzero;
  aux0 = myrealun(bitprec);
  if (polreal) K=K/2+1;
  ltop2=avma;
  for (i=0; i<K; i++)
  {
    aux = aux0;
    for (j=0; j<=n; j++)
    {
      pc[j]=lmul((GEN)q[j+2],aux);
      aux=gmul(aux,RU); /* RU = prim^i, aux=prim^(ij) */
    }

    fft(Omega,pc,coef,1,Lmax);
    for (j=0; j<Lmax; j++)
    {
      aux=gprec((GEN)coef[j],DEFAULTPREC);
      gx=gabs(aux,DEFAULTPREC);
      lx=gtodouble(mplog(gx));
      if (lx<*mu) *mu=lx;
      if (polreal && (i>0 && i<K-1))
      {
      gx=gdiv(gdeux,gx);
      ggamma=gadd(ggamma,gx);
      }
      else
      ggamma=gadd(ggamma,ginv(gx));
    }
    RU=gmul(RU,prim);
    if (low_stack(limite, stack_lim(ltop,1)))
    {
      GEN *gptr[2];
      if(DEBUGMEM>1) err(warnmem,"parameters");
      gptr[0]=&ggamma; gptr[1]=&RU; gerepilemany(ltop2,gptr,2);
    }
  }
  ggamma=gdivgs(ggamma,NN);
  *gamma=gtodouble(glog(ggamma,DEFAULTPREC))/log(2.);
  avma=ltop;
}

/* NN is a multiple of Lmax */
static void
dft(GEN p, long k, long NN, long bitprec, GEN F, GEN H, long polreal)
{
  GEN Omega,q,qd,pc,pdc,alpha,beta,gamma,RU,aux,U,W,mygpi,prim,prim2;
  long limite,n=lgef(p)-3,i,j,K,ltop;

  mygpi=mppi(bitprec/BITS_IN_LONG+3);
  aux = gdivgs(mygpi,NN/2); /* 2 Pi/NN */
  prim = exp_i(aux);
  aux = gmulbyi(aux);
  prim2 = myrealun(bitprec);
  RU=cgetg(n+2,t_VEC); RU++;

  Omega=initRU(Lmax,bitprec);
  K=NN/Lmax; q=mygprec(p,bitprec);
  qd=derivpol(q);

  pc=cgetg(Lmax+1,t_VEC); pc++;
  for (i=n+1; i<Lmax; i++) pc[i]=zero;
  pdc=cgetg(Lmax+1,t_VEC); pdc++;
  for (i=n; i<Lmax; i++) pdc[i]=zero;

  alpha=cgetg(Lmax+1,t_VEC); alpha++;
  beta=cgetg(Lmax+1,t_VEC); beta++;
  gamma=cgetg(Lmax+1,t_VEC); gamma++;

  if (polreal) K=K/2+1;

  ltop=avma; limite = stack_lim(ltop,1);
  W=cgetg(k+1,t_VEC); U=cgetg(k+1,t_VEC);
  for (i=1; i<=k; i++) W[i]=U[i]=zero;

  for (i=0; i<K; i++)
  {
    RU[0]=(long) gun;
    for (j=1; j<=n; j++) RU[j]=lmul((GEN)RU[j-1],prim2);
    /* RU[j]=prim^{ ij }=prim2^j */

    for (j=0; j<n; j++) pdc[j]=lmul((GEN)qd[j+2],(GEN)RU[j]);
    fft(Omega,pdc,alpha,1,Lmax);
    for (j=0; j<=n; j++) pc[j]=lmul((GEN)q[j+2],(GEN)RU[j]);
    fft(Omega,pc,beta,1,Lmax);
    for (j=0; j<Lmax; j++) gamma[j]=linv((GEN)beta[j]);
    for (j=0; j<Lmax; j++) beta[j]=lmul((GEN)alpha[j],(GEN)gamma[j]);
    fft(Omega,beta,alpha,1,Lmax);
    fft(Omega,gamma,beta,1,Lmax);

    if (polreal) /* p has real coefficients */
    {
      if (i>0 && i<K-1)
      {
      for (j=1; j<=k; j++)
      {
        aux=gmul((GEN)alpha[j+1],(GEN)RU[j+1]);
        W[j]=ladd((GEN)W[j],gshift(greal(aux),1));
        aux=gmul((GEN)beta[j],(GEN)RU[j]);
        U[j]=ladd((GEN)U[j],gshift(greal(aux),1));
      }
      }
      else
      {
      for (j=1; j<=k; j++)
      {
        aux=gmul((GEN)alpha[j+1],(GEN)RU[j+1]);
        W[j]=ladd((GEN)W[j],greal(aux));
        aux=gmul((GEN)beta[j],(GEN)RU[j]);
        U[j]=ladd((GEN)U[j],greal(aux));
      }
      }
    }
    else
    {
      for (j=1; j<=k; j++)
      {
      W[j]=ladd((GEN)W[j],gmul((GEN)alpha[j+1],(GEN)RU[j+1]));
      U[j]=ladd((GEN)U[j],gmul((GEN)beta[j],(GEN)RU[j]));
      }
    }
    prim2=gmul(prim2,prim);
    if (low_stack(limite, stack_lim(ltop,1)))
    {
      GEN *gptr[3];
      if(DEBUGMEM>1) err(warnmem,"dft");
      gptr[0]=&W; gptr[1]=&U; gptr[2]=&prim2;
      gerepilemany(ltop,gptr,3);
    }
  }

  for (i=1; i<=k; i++)
  {
    aux=(GEN)W[i];
    for (j=1; j<i; j++) aux=gadd(aux,gmul((GEN)W[i-j],(GEN)F[k+2-j]));
    F[k+2-i] = ldivgs(aux,-i*NN);
  }
  for (i=0; i<k; i++)
  {
    aux=(GEN)U[k-i];
    for (j=1+i; j<k; j++) aux=gadd(aux,gmul((GEN)F[2+j],(GEN)U[j-i]));
    H[i+2] = ldivgs(aux,NN);
  }
}

static GEN
refine_H(GEN F, GEN G, GEN HH, long bitprec, long shiftbitprec)
{
  GEN H=HH,D,aux;
  long ltop=avma, limite=stack_lim(ltop,1),error=0,i,bitprec1,bitprec2,lbot;

  D=gsub(gun,gres(gmul(HH,G),F)); error=gexpo(D);
  bitprec2=bitprec+shiftbitprec;

  for (i=0; (error>-bitprec && i<NEWTON_MAX) && error<=0; i++)
  {
    if (low_stack(limite, stack_lim(ltop,1)))
    {
      GEN *gptr[2];
      if(DEBUGMEM>1) err(warnmem,"refine_H");
      gptr[0]=&D; gptr[1]=&H; gerepilemany(ltop,gptr,2);
    }
    bitprec1=-error+shiftbitprec;
    aux=gmul(mygprec(H,bitprec1),mygprec(D,bitprec1));
    aux=mygprec(aux,bitprec1);
    aux=gres(aux,mygprec(F,bitprec1));

    bitprec1=-error*2+shiftbitprec;
    if (bitprec1>bitprec2) bitprec1=bitprec2;
    H=gadd(mygprec(H,bitprec1),aux);
    D=gsub(gun,gres(gmul(H,G),F));
    error=gexpo(D); if (error<-bitprec1) error=-bitprec1;
  }
  if (error<=-bitprec/2) { lbot=avma; return gerepile(ltop,lbot,gcopy(H)); }
  avma=ltop; return gzero; /* procedure failed */
}

/* return 0 if fails, 1 else */
static long
refine_F(GEN p, GEN *F, GEN *G, GEN H, long bitprec, double gamma)
{
  GEN pp,FF,GG,r,HH,f0;
  long error,i,bitprec1=0,bitprec2,ltop=avma,shiftbitprec;
  long shiftbitprec2,n=lgef(p)-3,enh,normF,normG,limite=stack_lim(ltop,1);

  FF=*F; HH=H;
  GG=poldivres(p,*F,&r);
  normF=gexpo(FF);
  normG=gexpo(GG);
  enh=gexpo(H); if (enh<0) enh=0;
  shiftbitprec=normF+2*normG+enh+(long) (4.*log2((double)n)+gamma) +1;
  shiftbitprec2=enh+2*(normF+normG)+(long) (2.*gamma+5.*log2((double)n))+1;
  bitprec2=bitprec+shiftbitprec;
  error=gexpo(r);
  if (error<-bitprec) error=1-bitprec;
  for (i=0; (error>-bitprec && i<NEWTON_MAX) && error<=0; i++)
  {
    if ((bitprec1==bitprec2) && (i>=2))
    {
      shiftbitprec+=n; shiftbitprec2+=n; bitprec2+=n;
    }
    if (low_stack(limite, stack_lim(ltop,1)))
    {
      GEN *gptr[4];
      if(DEBUGMEM>1) err(warnmem,"refine_F");
      gptr[0]=&FF; gptr[1]=&GG; gptr[2]=&r; gptr[3]=&HH;
      gerepilemany(ltop,gptr,4);
    }

    bitprec1=-error+shiftbitprec2;
    HH=refine_H(mygprec(FF,bitprec1),mygprec(GG,bitprec1),
            mygprec(HH,bitprec1),1-error,shiftbitprec2);
    if (HH==gzero) return 0; /* procedure failed */

    bitprec1=-error+shiftbitprec;
    r=gmul(mygprec(HH,bitprec1),mygprec(r,bitprec1));
    r=mygprec(r,bitprec1);
    f0=gres(r,mygprec(FF,bitprec1));

    bitprec1=-2*error+shiftbitprec;
    if (bitprec1>bitprec2) bitprec1=bitprec2;
    FF=gadd(mygprec(FF,bitprec1),f0);

    bitprec1=-3*error+shiftbitprec;
    if (bitprec1>bitprec2) bitprec1=bitprec2;
    pp=mygprec(p,bitprec1);
    GG=poldivres(pp,mygprec(FF,bitprec1),&r);
    error=gexpo(r); if (error<-bitprec1) error=-bitprec1;
  }
  if (error<=-bitprec)
  {
    *F=FF; *G=GG;
    return 1; /* procedure succeeds */
  }
  return 0; /* procedure failed */
}

/* returns F and G from the unit circle U such that |p-FG|<2^(-bitprec) |cd|,
where cd is the leading coefficient of p */
static void
split_fromU(GEN p, long k, double delta, long bitprec,
            GEN *F, GEN *G, double param, double param2)
{
  GEN pp,FF,GG,H;
  long n=lgef(p)-3,NN,bitprec2,
  ltop=avma,polreal=isreal(p);
  double mu,gamma;

  pp=gdiv(p,(GEN)p[2+n]);
  Lmax=4; while (Lmax<=n) Lmax=(Lmax<<1);
  parameters(pp,&mu,&gamma,polreal,param,param2);

  H =cgetg(k+2,t_POL); H[1] =evalsigne(1) | evalvarn(varn(p)) | evallgef(k+2);
  FF=cgetg(k+3,t_POL); FF[1]=evalsigne(1) | evalvarn(varn(p)) | evallgef(k+3);
  FF[k+2]=un;

  NN=(long) (0.5/delta); NN+=(NN%2); if (NN<2) NN=2;
  NN=NN*Lmax; ltop=avma;
  for(;;)
  {
    bitprec2=(long) (((double) NN*delta-mu)/log(2.))+gexpo(pp)+8;
    dft(pp,k,NN,bitprec2,FF,H,polreal);
    if (refine_F(pp,&FF,&GG,H,bitprec,gamma)) break;
    NN=(NN<<1); avma=ltop;
  }
  *G=gmul(GG,(GEN)p[2+n]); *F=FF;
}

static void
optimize_split(GEN p, long k, double delta, long bitprec,
            GEN *F, GEN *G, double param, double param2)
{
  long n=lgef(p)-3;
  GEN FF,GG;

  if (k<=n/2)
    split_fromU(p,k,delta,bitprec,F,G,param,param2);
  else
  { /* start from the reciprocal of p */
    split_fromU(polrecip_i(p),n-k,delta,bitprec,&FF,&GG,param,param2);
    *F=polrecip(GG); *G=polrecip(FF);
  }
}

/********************************************************************/
/**                                                                **/
/**             RECHERCHE DU CERCLE DE SEPARATION                  **/
/**                                                                **/
/********************************************************************/

/* return p(2^e*x) *2^(-n*e) */
static void
scalepol2n(GEN p, long e)
{
  long i,n=lgef(p)-1;
  for (i=2; i<=n; i++) p[i]=lmul2n((GEN)p[i],(i-n)*e);
}

/* returns p(x/R)*R^n */
static GEN
scalepol(GEN p, GEN R, long bitprec)
{
  GEN q,aux,gR;
  long i;

  aux = gR = mygprec(R,bitprec); q = mygprec(p,bitprec);
  for (i=lgef(p)-2; i>=2; i--)
  {
    q[i]=lmul(aux,(GEN)q[i]);
    aux = gmul(aux,gR);
  }
  return q;
}

/* returns q(x)=p(b), where b is (usually) a polynomial  */
static GEN
shiftpol(GEN p, GEN b)
{
  long av = avma,i, limit = stack_lim(av,1);
  GEN q = gzero;

  for (i=lgef(p)-1; i>=2; i--)
  {
    q = gadd((GEN)p[i], gmul(q,b));
    if (low_stack(limit, stack_lim(av,1))) q = gerepileupto(av,q);
  }
  return gerepileupto(av,q);
}

/* return (aX-1)^n * p[ (X-a) / (aX-1) ] */
static GEN
conformal_pol(GEN p, GEN a, long bitprec)
{
  GEN r,pui,num,aux, unr = myrealun(bitprec);
  long n=lgef(p)-3, i;

  aux = pui = cgetg(4,t_POL);
  pui[1] = evalsigne(1) | evalvarn(varn(p)) | evallgef(4);
  pui[2] = (long)negr(unr);
  pui[3] = lconj(a);
  num = cgetg(4,t_POL);
  num[1] = pui[1];
  num[2] = lneg(a);
  num[3] = (long)unr;
  r=(GEN)p[2+n];
  for (i=n-1; ; i--)
  {
    r=gadd(gmul(r,num),gmul(aux,(GEN) p[2+i]));
    if (i==0) return r;
    aux=gmul(pui,aux);
  }
} 

static GEN
compute_radius(GEN* radii, GEN p, long k, double aux, double *delta)
{   
  long i, n = lgef(p)-3;
  GEN rmin,rmax,p1;
  if (k>1)
  {
    i=k-1; while (i>0 && !signe(radii[i])) i--;
    rmin = pre_modulus(p,k,aux, radii[i], radii[k]);
  } 
  else /* k=1 */
    rmin = min_modulus(p,aux);
  affrr(rmin, radii[k]);
 
  if (k+1<n) 
  {   
    i=k+2; while (i<=n && !signe(radii[i])) i++;
    rmax = pre_modulus(p,k+1,aux, radii[k+1], radii[i]);
  }   
  else /* k+1=n */
    rmax = max_modulus(p,aux);
  affrr(rmax, radii[k+1]);
  
  p1 = radii[k];
  for (i=k-1; i>=1; i--)
  {
    if (!signe(radii[i]) || cmprr(radii[i], p1) > 0)
      affrr(p1, radii[i]);
    else
      p1 = radii[i];
  }
  p1 = radii[k+1];
  for (i=k+1; i<=n; i++)
  {
    if (!signe(radii[i]) || cmprr(radii[i], p1) < 0)
      affrr(p1, radii[i]);
    else      
      p1 = radii[i];
  }
  *delta = rtodbl(gmul2n(mplog(divrr(rmax,rmin)), -1));
  if (*delta > 1.) *delta = 1.;
  return mpsqrt(mulrr(rmin,rmax));
}

static GEN
update_radius(GEN *radii, GEN rho, double *par, double *par2)
{
  GEN p1, invrho = ginv(rho);
  long i, n = lg(radii);
  double t, param = 0., param2 = 0.;
  for (i=1; i<n; i++)
  {
    affrr(mulrr(radii[i], invrho), radii[i]);
    p1 = ginv(subsr(1, radii[i]));
    t = fabs(rtodbl(p1));
    param += t; if (t > 1.) param2 += log2(t);
  }
  *par = param; *par2 = param2; return invrho;
}

/* apply the conformal mapping then split from U */
static void
conformal_mapping(GEN p, long k, long bitprec, double aux,GEN *F,GEN *G)
{
  long bitprec2,n=lgef(p)-3,decprec,i,ltop = avma, av;
  GEN q,FF,GG,a,R, *gptr[3];
  GEN rho,invrho;
  double delta,param,param2;

  bitprec2=bitprec+(long) (n*(2.*log2(2.732)+log2(1.5)))+1;
  a=gsqrt(stoi(3), 2*MEDDEFAULTPREC - 2);
  a=gmul(mygprec(a,bitprec2),mygprec(globalu,bitprec2));
  a=gdivgs(a,-6); /* a=-globalu/2/sqrt(3) */

  av = avma; q=mygprec(p,bitprec2);
  q = conformal_pol(q,a,bitprec2);
  for (i=1; i<=n; i++)
    if (signe(radius[i])) /* updating array radius */
    {
      long a = avma;
      GEN p1 = gsqr(radius[i]);
      p1 = divrr(gmul2n((subrs(p1,1)),1),
                   subrr(p1, mulsr(3,subrs(radius[i],1))));
      affrr(mpsqrt(addsr(1,p1)), radius[i]);
      avma = a;
    }

  rho = compute_radius(radius, q,k,aux/10., &delta);
  invrho = update_radius(radius, rho, &param, &param2);

  bitprec2 += (long) (((double)n) * fabs(log2ir(rho)) + 1.);
  R=mygprec(invrho,bitprec2);
  q=scalepol(q,R,bitprec2);
  gptr[0]=&q; gptr[1]=&R; gptr[2]=&invrho;
  gerepilemany(av,gptr,3);

  optimize_split(q,k,delta,bitprec2,&FF,&GG,param,param2);
  bitprec2 += n; R = ginv(R);
  FF=scalepol(FF,R,bitprec2);
  GG=scalepol(GG,R,bitprec2);

  a=mygprec(a,bitprec2);
  FF=conformal_pol(FF,a,bitprec2);
  GG=conformal_pol(GG,a,bitprec2);
  a=ginv(gsub(gun,gmul(a,gconj(a))));
  a=glog(a,(long) (bitprec2 * L2SL10)+1);

  decprec=(long) ((bitprec+n) * L2SL10)+1;
  FF=gmul(FF,gexp(gmulgs(a,k),decprec));
  GG=gmul(GG,gexp(gmulgs(a,n-k),decprec));

  *F=mygprec(FF,bitprec+n);
  *G=mygprec(GG,bitprec+n);
  gptr[0]=F; gptr[1]=G; gerepilemany(ltop,gptr,2);
}

/* split p, this time with no scaling. returns in F and G two polynomials
such that |p-FG|< 2^(-bitprec)|p| */
static void
split_2(GEN p, long bitprec, double thickness, GEN *F, GEN *G)
{
  GEN rmin,rmax,rho,invrho;
  double kappa,aux,delta,param,param2;
  long n=lgef(p)-3,i,j,k,bitprec2;
  GEN q,FF,GG,R;

  radius = (GEN*) cgetg(n+1, t_VEC);
  for (i=1; i<n; i++) radius[i]=realzero(3);
  aux = thickness/(double) n/4.;
  radius[1] = rmin = min_modulus(p, aux);
  radius[n] = rmax = max_modulus(p, aux);
  i=1; j=n;
  rho = mpsqrt(mulrr(rmin,rmax));
  k = dual_modulus(p,rho,aux,1);
  if (k<n/5. || (n/2.<k && k<(4*n)/5.))
    { rmax=rho; j=k+1; affrr(rho, radius[j]); }
  else
    { rmin=rho; i=k; affrr(rho, radius[i]); }
  while (j>i+1)
  {
    if (i+j==n+1)
      rho = mpsqrt(mulrr(rmin,rmax));
    else
    {
      kappa = 1. - log(1.+(double)min(i,n-j)) / log(1.+(double)min(j,n-i));
      if (i+j<n+1)
        rho = addrr(mulrr(mplog(rmax),dbltor(1+kappa)), mplog(rmin));
      else
        rho = addrr(mulrr(mplog(rmin),dbltor(1+kappa)), mplog(rmax));
      rho = mpexp(divrr(rho, dbltor(2+kappa)));
    }
    aux = rtodbl(mplog(divrr(rmax,rmin))) / (j-i) / 4.;
    k = dual_modulus(p,rho,aux, min(i,n+1-j));
    if (k-i < j-k-1 || (k-i == j-k-1 && 2*k > n))
      { rmax=rho; j=k+1; affrr(mulrr(rho, dbltor(exp(-aux))), radius[j]); }
    else
      { rmin=rho; i=k; affrr(mulrr(rho, dbltor(exp(aux))), radius[i]); }
  }
  aux = rtodbl(mplog(divrr(rmax, rmin)));

  if (step4==0)
  {
    rho = compute_radius(radius, p, k, aux/10., &delta);
    invrho = update_radius(radius, rho, &param, &param2);

    bitprec2 = bitprec + (long) ((double)n * fabs(log2ir(rho)) + 1.);
    R = mygprec(invrho,bitprec2);
    q = scalepol(p,R,bitprec2);

    optimize_split(q,k,delta,bitprec2,&FF,&GG,param,param2);
  }
  else
  {
    rho = mpsqrt(mulrr(rmax,rmin));
    invrho = ginv(rho);
    for (i=1; i<=n; i++)
      if (signe(radius[i])) affrr(mulrr(radius[i],invrho), radius[i]);

    bitprec2 = bitprec + (long) ((double)n * fabs(log2ir(rho)) + 1.);
    R = mygprec(invrho,bitprec2);
    q = scalepol(p,R,bitprec2);

    conformal_mapping(q, k, bitprec2, aux, &FF, &GG);
  }
  bitprec  += n;
  bitprec2 += n; R = ginv(mygprec(R,bitprec2));
  *F = mygprec(scalepol(FF,R,bitprec2), bitprec);
  *G = mygprec(scalepol(GG,R,bitprec2), bitprec);
}

/* procedure corresponding to steps 5,6,.. page 44 in the RR n. 1852 */
/* put in F and G two polynomial such that |p-FG|<2^(-bitprec)|p|
where the maximum modulus of the roots of p is <=1 and the sum of roots
is zero */

static void
split_1(GEN p, long bitprec, GEN *F, GEN *G)
{
  long bitprec2,bitprec3,i,imax,n=lgef(p)-3, polreal = isreal(p);
  GEN rmax,rmin,thickness,quo;
  GEN q,qq,FF,GG,v,gr,r, newq = NULL; /* gcc -Wall */

  r=max_modulus(p,0.01);
  bitprec2=bitprec+n;
  gr=mygprec(ginv(r),bitprec2);
  q=scalepol(p,gr,bitprec2);

  bitprec2=bitprec+gexpo(q)-gexpo(p);

  bitprec3=bitprec2+(long)((double)n*2.*log2(3.)+1);
  v=cgetg(5,t_VEC); v++;
  v[0]=lmul2n(myrealun(bitprec3), 1);
  v[1]=lneg((GEN)v[0]);
  v[2]=lmul((GEN)v[0],gi);
  v[3]=lneg((GEN)v[2]);
  imax = polreal? 3: 4;
  q=mygprec(q,bitprec3); thickness=realun(3);
  for (i=0; i<imax; i++)
  {
    qq = shiftpol(q,gadd(polx[varn(p)],(GEN)v[i]));
    rmin = min_modulus(qq,0.05);
    if (cmpsr(3, mulrr(rmin, thickness)) > 0)
    {
      rmax = max_modulus(qq,0.05);
      quo = divrr(rmax,rmin);
      if (cmprr(quo, thickness) > 0)
      {
      thickness=quo; newq=qq; globalu=(GEN)v[i];
      }
    }
    if (cmpsr(2 , thickness) < 0) break;
    if (polreal && i==1 && cmprr(thickness, dbltor(1.5)) > 0) break;
  }
  bitprec3=bitprec2+(long)((double) n*log2(3.)+1)+gexpo(newq)-gexpo(q);
  split_2(newq,bitprec3,rtodbl(mplog(thickness)),&FF,&GG);
  globalu = gsub(polx[varn(p)], mygprec(globalu,bitprec3));
  FF = shiftpol(FF,globalu);
  GG = shiftpol(GG,globalu);

  gr = ginv(gr);
  bitprec2 += gexpo(FF)+gexpo(GG)-gexpo(q);
  *F = scalepol(FF,gr,bitprec2);
  *G = scalepol(GG,gr,bitprec2);
}

/* put in F and G two polynomials such that |P-FG|<2^(-bitprec)|P|,
where the maximum modulus of the roots of p is < 0.5 */
static int
split_0_2(GEN p, long bitprec, GEN *F, GEN *G)
{
  GEN q,b,FF,GG;
  long n=lgef(p)-3,k,bitprec2,i, eq;
  double aux = mylog2((GEN)p[n+1]) - mylog2((GEN)p[n+2]);

  /* beware double overflow */
  if (aux >= 0 && (aux > 1e4 || exp2(aux) > 2.5*n)) return 0;

  step4 = 1;
  
  aux = (aux < -300)? 0.: (double) n*log2(1 + exp2(aux)/(double)n);
  bitprec2=bitprec+1+(long) (log2((double)n)+aux);

  q=mygprec(p,bitprec2);
  b=gdivgs(gdiv((GEN)q[n+1],(GEN)q[n+2]),-n);
  q=shiftpol(q,gadd(polx[varn(p)],b));

  k=0; eq=gexpo(q);
  while
      (k <= n/2 && (gexpo((GEN)q[k+2]) < -(bitprec2+2*(n-k)+eq)
       || gcmp0((GEN)q[k+2]))) k++;
  if (k>0)
  {
    if (k>n/2) k=n/2;
    bitprec2+=(k<<1);
    FF=cgetg(k+3,t_POL); FF[1]=evalsigne(1)|evalvarn(varn(p))|evallgef(k+3);
    for (i=0; i<k; i++) FF[i+2]=zero;
    FF[k+2]=(long) myrealun(bitprec2);
    GG=cgetg(n-k+3,t_POL); GG[1]=evalsigne(1)|evalvarn(varn(p))|evallgef(n-k+3);
    for (i=0; i<=n-k; i++) GG[i+2]=q[i+k+2];
    b = gsub(polx[varn(p)],mygprec(b,bitprec2));
  }
  else
  {
    split_1(q,bitprec2,&FF,&GG);
    bitprec2 = bitprec+gexpo(FF)+gexpo(GG)-gexpo(p)+(long)aux+1;
    b = gsub(polx[varn(p)],mygprec(b,bitprec2));
    FF = mygprec(FF,bitprec2);
  }
  GG = mygprec(GG,bitprec2);
  *F = shiftpol(FF,b);
  *G = shiftpol(GG,b); return 1;
}

/* put in F and G two polynomials such that |P-FG|<2^(-bitprec)|P|,
where the maximum modulus of the roots of p is <2 */
static void
split_0_1(GEN p, long bitprec, GEN *F, GEN *G)
{
  GEN q,FF,GG;
  long n=lgef(p)-3,bitprec2,normp;

  if  (split_0_2(p,bitprec,F,G)) return;

  normp = gexpo(p);
  scalepol2n(p,2); /* p <- 4^(-n) p(4*x) */
  bitprec2 = bitprec+2*n+gexpo(p)-normp;
  q=mygprec(p,bitprec2);
  split_1(q,bitprec2,&FF,&GG);
  scalepol2n(FF,-2); scalepol2n(GG,-2);
  bitprec2=bitprec+gexpo(FF)+gexpo(GG)-normp;
  *F=mygprec(FF,bitprec2); *G=mygprec(GG,bitprec2);
}

/* put in F and G two polynomials such that |P-FG|<2^(-bitprec)|P| */
static void
split_0(GEN p, long bitprec, GEN *F, GEN *G)
{
  GEN FF,GG,q,R;
  long n=lgef(p)-3,k=0,i;

  while (gexpo((GEN)p[k+2]) < -bitprec && k <= n/2) k++;
  if (k>0)
  {
    if (k>n/2) k=n/2;
    FF=cgetg(k+3,t_POL);
    FF[1]=evalsigne(1) | evalvarn(varn(p)) | evallgef(k+3);
    for (i=0; i<k; i++) FF[i+2] = zero;
    FF[k+2]=(long) myrealun(bitprec);
    GG=cgetg(n-k+3,t_POL);
    GG[1]=evalsigne(1) | evalvarn(varn(p)) | evallgef(n-k+3);
    for (i=0; i<=n-k; i++) GG[i+2]=p[i+k+2];
  }
  else
  {
    R = max_modulus(p,0.05);
    if (gexpo(R)<1 && gtodouble(R)<1.9) split_0_1(p,bitprec,&FF,&GG);
    else
    {
      q = polrecip_i(p);
      R = max_modulus(q,0.05);
      if (gexpo(R)<1 && gtodouble(R)<1.9)
      {
      split_0_1(q,bitprec,&FF,&GG);
      FF=polrecip(FF); GG=polrecip(GG);
      }
      else
      {
      step4 = 0;
      split_2(p,bitprec,1.2837,&FF,&GG);
      }
    }
  }
  *F=FF; *G=GG;
}

/********************************************************************/
/**                                                                **/
/**     CALCUL A POSTERIORI DE L'ERREUR ABSOLUE SUR LES RACINES    **/
/**                                                                **/
/********************************************************************/

static GEN
root_error(long n, long k, GEN roots_pol, GEN sigma, GEN shatzle)
{
  GEN rho,d,eps,epsbis,eps2,prod,aux,rap=NULL;
  long i,j,m;

  d=cgetg(n+1,t_VEC);
  for (i=1; i<=n; i++)
  {
    if (i!=k)
    {
      aux=gsub((GEN)roots_pol[i],(GEN)roots_pol[k]);
      d[i]=(long) gabs(mygprec(aux,31),4);
    }
  }
  rho=gabs(mygprec((GEN)roots_pol[k],31),4);
  if (gcmp(rho,dbltor(1.))==-1) rho=gun;
  eps=gmul(rho,shatzle);
  aux=gmul(gpowgs(rho,n),sigma);

  for (j=1; j<=2 || (j<=5 && gcmp(rap,dbltor(1.2))==1); j++)
  {
    m=n; prod=gun;
    epsbis=gdivgs(gmulgs(eps,5),4);
    for (i=1; i<=n; i++)
    {
      if (i!=k && gcmp((GEN)d[i],epsbis)==1)
      {
      m--;
      prod=gmul(prod,gsub((GEN)d[i],eps));
      }
    }
    eps2=gdiv(gmul2n(aux,2*m-2),prod);
    eps2=gpui(eps2,dbltor(1./m),4);
    rap=gdiv(eps,eps2); eps=eps2;
  }
  return eps;
}

/* round a complex or real number x to an absolute value of 2^(-e) */
static GEN
mygprec_absolute(GEN x, long bitprec)
{
  long tx=typ(x),e;
  GEN y;

  switch(tx)
  {
    case t_REAL:
      e=gexpo(x);
      if (e<-bitprec || !signe(x)) { y=dbltor(0.); setexpo(y,-bitprec); }
      else y=mygprec(x,bitprec+e);
      break;
    case t_COMPLEX:
      if (gexpo((GEN)x[2])<-bitprec)
      y=mygprec_absolute((GEN)x[1],bitprec);
      else
      {
      y=cgetg(3,t_COMPLEX);
      y[1]=(long) mygprec_absolute((GEN)x[1],bitprec);
      y[2]=(long) mygprec_absolute((GEN)x[2],bitprec);
      }
      break;

    default: y=mygprec(x,bitprec);
  }
  return y;
}

static long
a_posteriori_errors(GEN p, GEN roots_pol, long err)
{
  GEN sigma,overn,shatzle,x;
  long i,n=lgef(p)-3,e,e_max;

  sigma = realun(3);
  setexpo(sigma,err+(long) log2((double) n)+1);
  overn=dbltor(1./n);
  shatzle=gdiv(gpui(sigma,overn,0),
             gsub(gpui(gsub(gun,sigma),overn,0),
                gpui(sigma,overn,0)));
  shatzle=gmul2n(shatzle,1); e_max=-pariINFINITY;
  for (i=1; i<=n; i++)
  {
    x=root_error(n,i,roots_pol,sigma,shatzle);
    e=gexpo(x); if (e>e_max) e_max=e;
    x = (GEN)roots_pol[i];
    roots_pol[i]=(long) mygprec_absolute(x,-e);
    gunclone(x);
  }
  return e_max;
}

/********************************************************************/
/**                                                                **/
/**                           MAIN                                 **/
/**                                                                **/
/********************************************************************/
static GEN
append_root(GEN roots_pol, GEN a)
{
  long l = lg(roots_pol);
  setlg(roots_pol, l+1); return (GEN)(roots_pol[l] = lclone(a));
}

/* put roots in placeholder roots_pol so that |P-L_1...L_n|<2^(-bitprec)|P|
 *  and returns  prod (x-roots_pol[i]) for i=1..degree(p) */
static GEN
split_complete(GEN p, long bitprec, GEN roots_pol)
{
  long n=lgef(p)-3,decprec,ltop;
  GEN p1,F,G,a,b,m1,m2,m;

  if (n==1)
  {
    a=gneg_i(gdiv((GEN)p[2],(GEN)p[3]));
    append_root(roots_pol,a); return p;
  }
  ltop = avma;
  if (n==2)
  {
    F=gsub(gsqr((GEN)p[3]),gmul2n(gmul((GEN)p[2],(GEN)p[4]),2));
    decprec=(long) ((double) bitprec * L2SL10)+1;
    F=gsqrt(F,decprec);
    p1 = gmul2n((GEN)p[4],1);
    a = gneg_i(gdiv(gadd(F,(GEN)p[3]), p1));
    b =        gdiv(gsub(F,(GEN)p[3]), p1);
    a = append_root(roots_pol,a);
    b = append_root(roots_pol,b); avma = ltop;
    m=gmul(gsub(polx[varn(p)],mygprec(a,3*bitprec)),
         gsub(polx[varn(p)],mygprec(b,3*bitprec)));
    return gmul(m,(GEN)p[4]);
  }
  split_0(p,bitprec,&F,&G);
  m1 = split_complete(F,bitprec,roots_pol);
  m2 = split_complete(G,bitprec,roots_pol);
  return gerepileupto(ltop, gmul(m1,m2));
}

/* compute a bound on the maximum modulus of roots of p */
static GEN
cauchy_bound(GEN p)
{
  long i,n=lgef(p)-3;
  GEN x=gzero,y,lc;

  lc=gabs((GEN)p[n+2],DEFAULTPREC); /* leading coefficient */
  lc=gdiv(dbltor(1.),lc);
  for (i=0; i<n; i++)
  {
    y=gmul(gabs((GEN) p[i+2],DEFAULTPREC),lc);
    y=gpui(y,dbltor(1./(n-i)),DEFAULTPREC);
    if (gcmp(y,x) > 0) x=y;
  }
  return x;
}

static GEN
mygprecrc_special(GEN x, long bitprec, long e)
{
  long tx=typ(x),lx,ex;
  GEN y;

  if (bitprec<=0) bitprec=0; /* should not happen */
  switch(tx)
  {
    case t_REAL:
      lx=bitprec/BITS_IN_LONG+3;
      if (lx<lg(x)) lx=lg(x);
      y=cgetr(lx); affrr(x,y); ex=-bitprec+e;
      if (!signe(x) && expo(x)>ex) setexpo(y,ex);
      break;
    case t_COMPLEX:
      y=cgetg(3,t_COMPLEX);
      y[1]=(long) mygprecrc_special((GEN)x[1],bitprec,e);
      y[2]=(long) mygprecrc_special((GEN)x[2],bitprec,e);
      break;
    default: y=gcopy(x);
  }
  return y;
}

/* like mygprec but keep at least the same precision as before */
static GEN
mygprec_special(GEN x, long bitprec)
{
  long tx=typ(x),lx,i,e;
  GEN y;

  switch(tx)
  {
    case t_POL:
      lx=lgef(x); y=cgetg(lx,tx); y[1]=x[1]; e=gexpo(x);
      for (i=2; i<lx; i++) y[i]=(long) mygprecrc_special((GEN)x[i],bitprec,e);
      break;

    default: y=mygprecrc_special(x,bitprec,0);
  }
  return y;
}

static GEN
all_roots(GEN p, long bitprec)
{
  GEN q,roots_pol,m;
  long bitprec0, bitprec2,n=lgef(p)-3,i,e,av;

  roots_pol=cgetg(n+1,t_VEC); av=avma;
  e = 2*gexpo(cauchy_bound(p)); if (e<0) e=0;
  bitprec0=bitprec + gexpo(p) - gexpo((GEN)p[2+n]) + (long)log2(n) + 1 + e;
  for (i=1;; i++)
  {
    setlg(roots_pol,1); 
    bitprec2 = bitprec0 + (1<<i)*n;
    q = gmul(myrealun(bitprec2), mygprec(p,bitprec2));
    m = split_complete(q,bitprec2,roots_pol);
    e = gexpo(gsub(mygprec_special(p,bitprec2),m))
      - gexpo((GEN)q[2+n])+(long) log2((double)n)+1;
    if (e<-2*bitprec2) e=-2*bitprec2; /* to avoid e=-pariINFINITY */
    if (e < 0)
    {
      e = a_posteriori_errors(q,roots_pol,e);
      if (e < -bitprec) return roots_pol;
    }
    if (DEBUGLEVEL > 7)
      fprintferr("all_roots: restarting, i = %ld, e = %ld\n", i,e);
    avma = av;
  }
}

/* true if x is an exact scalar, that is integer or rational */
static int
isexactscalar(GEN x)
{
  long tx=typ(x);
  return (tx==t_INT || is_frac_t(tx));
}

static int
isexactpol(GEN p)
{
  long i,n=lgef(p)-3;

  for (i=0; i<=n; i++)
    if (isexactscalar((GEN)p[i+2])==0) return 0;
  return 1;
}

static long
isvalidcoeff(GEN x)
{
  long tx=typ(x);

  switch(tx)
  {
    case t_INT: case t_REAL: case t_FRAC: case t_FRACN: return 1;
    case t_COMPLEX:
      if (isvalidcoeff((GEN)x[1]) && isvalidcoeff((GEN)x[2])) return 1;
  }
  return 0;
}

static long
isvalidpol(GEN p)
{
  long i,n = lgef(p);
  for (i=2; i<n; i++)
    if (!isvalidcoeff((GEN)p[i])) return 0;
  return 1;
}

static GEN
solve_exact_pol(GEN p, long bitprec)
{
  GEN S,ex,factors,roots_pol,roots_fact;
  long i,j,k,m,n,iroots;

  n=lgef(p)-3;

  iroots=0;
  roots_pol=cgetg(n+1,t_VEC); for (i=1; i<=n; i++) roots_pol[i]=zero;

  S=square_free_factorization(p);
  ex=(GEN) S[1]; factors=(GEN) S[2];
  for (i=1; i<lg(factors); i++)
  {
    roots_fact=all_roots((GEN)factors[i],bitprec);
    n=lgef(factors[i])-3; m=itos((GEN)ex[i]);
    for (j=1; j<=n; j++)
      for (k=1; k<=m; k++) roots_pol[++iroots] = roots_fact[j];
  }
  return roots_pol;
}

/* return the roots of p with absolute error bitprec */
static GEN
roots_com(GEN p, long l)
{
  long bitprec;

  if (typ(p)!=t_POL)
  {
    if (!isvalidcoeff(p)) err(typeer,"roots");
    return cgetg(1,t_VEC); /* constant polynomial */
  }
  if (!isvalidpol(p)) err(talker,"invalid coefficients in roots");
  if (lgef(p) == 3) return cgetg(1,t_VEC); /* constant polynomial */
  if (l<3) l=3;
  bitprec=bit_accuracy(l);
  return isexactpol(p)? solve_exact_pol(p,bitprec): all_roots(p,bitprec);
}

static GEN
tocomplex(GEN x, long l)
{
  GEN y=cgetg(3,t_COMPLEX);

  y[1]=lgetr(l);
  if (typ(x) == t_COMPLEX)
    { y[2]=lgetr(l); gaffect(x,y); }
  else
    { gaffect(x,(GEN)y[1]); y[2]=(long)realzero(l); }
 return y;
}

/* Check if x is approximately real with precision e */
int
isrealappr(GEN x, long e)
{
  long tx=typ(x),lx,i;
  switch(tx)
  {
    case t_INT: case t_REAL: case t_FRAC: case t_FRACN:
      return 1;
    case t_COMPLEX:
      return (gexpo((GEN)x[2]) < e);
    case t_QUAD:
      err(impl,"isrealappr for type t_QUAD");
    case t_POL: case t_SER: case t_RFRAC: case t_RFRACN:
    case t_VEC: case t_COL: case t_MAT:
      lx = (tx==t_POL)?lgef(x): lg(x);
      for (i=lontyp[tx]; i<lx; i++)
        if (! isrealappr((GEN)x[i],e)) return 0;
      return 1;
    default: err(typeer,"isrealappr"); return 0;
  }
}

/* x,y sont de type t_COMPLEX */
static int
isconj(GEN x, GEN y, long e)
{
  long av = avma;
  long i= (gexpo( gsub((GEN)x[1],(GEN)y[1]) ) < e
        && gexpo( gadd((GEN)x[2],(GEN)y[2]) ) < e);
  avma = av; return i;
}

/* returns the vector of roots of p, with guaranteed absolute error
 * 2 ^ (- bit_accuracy(l))
 */
GEN
roots(GEN p, long l)
{
  long av,n,i,k,s,e;
  GEN c,c2,r,p1,p2,res;

  if (gcmp0(p)) err(zeropoler,"roots");
  av=avma; r=roots_com(p,l); n=lg(r);
  if (n <= 1) return r;

  if (!isreal(p))
  {
    res = cgetg(n,t_COL);
    for (i=1; i<n; i++) res[i]=(long)tocomplex((GEN)r[i],l);
    return gerepileupto(av,res);
  }
  e = 5 - bit_accuracy(l);
  p1=cgetg(n,t_COL); s=0;
  for (i=1; i<n; i++)
  {
    p2=(GEN)r[i];
    if (typ(p2) != t_COMPLEX) { p1[++s]=(long)p2; r[i]=zero; }
    else if (isrealappr(p2,e)) { p1[++s]=p2[1]; r[i]=zero; }
  }
  setlg(p1,s+1); p2=sort(p1);
  res = cgetg(n,t_COL);
  for (i=1; i<=s; i++) res[i]=(long)tocomplex((GEN)p2[i],l);
  for (i=1; i<n; i++)
  {
    c=(GEN)r[i];
    if (typ(c) == t_COMPLEX)
    {
      res[++s]=(long)tocomplex(c,l);
      for (k=i+1; k<n; k++)
      {
        c2=(GEN)r[k];
        if (typ(c2) == t_COMPLEX && isconj(c,c2,e))
        {
          res[++s]=(long)tocomplex(c2,l);
          r[k]=zero; break;
        }
      }
      if (k==n) err(bugparier,"roots (conjugates)");
    }
  }
  return gerepileupto(av,res);
}

GEN
roots0(GEN p, long flag,long l)
{
  switch(flag)
  {
    case 0: return roots(p,l);
    case 1: return rootsold(p,l);
    default: err(flagerr,"polroots");
  }
  return NULL; /* not reached */
}

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