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gen3.c

/* $Id: gen3.c,v 1.22.2.4 2002/04/06 12:39:10 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. */

/********************************************************************/
/**                                                                **/
/**                      GENERIC OPERATIONS                        **/
/**                         (third part)                           **/
/**                                                                **/
/********************************************************************/
#include "pari.h"

/********************************************************************/
/**                                                                **/
/**                 PRINCIPAL VARIABLE NUMBER                      **/
/**                                                                **/
/********************************************************************/

int
gvar(GEN x)
{
  long tx=typ(x),i,v,w;

  if (is_polser_t(tx)) return varn(x);
  if (tx==t_POLMOD) return varn(x[1]);
  if (is_const_t(tx) || is_qf_t(tx) || is_noncalc_t(tx)) return BIGINT;
  v=BIGINT;
  for (i=1; i < lg(x); i++) { w=gvar((GEN)x[i]); if (w<v) v=w; }
  return v;
}

int
gvar2(GEN x)
{
  long tx=typ(x),i,v,w;

  if (is_const_t(tx) || is_qf_t(tx) || is_noncalc_t(tx)) return BIGINT;
  v = BIGINT;
  switch(tx)
  {
    case t_POLMOD:
      v=gvar2((GEN)x[1]);
      w=gvar2((GEN)x[2]); if (w<v) v=w;
      return v;

    case t_SER:
      for (i=2; i < lg(x); i++) { w=gvar((GEN)x[i]); if (w<v) v=w; }
      return v;

    case t_POL:
      for (i=2; i<lgef(x); i++) { w=gvar((GEN)x[i]); if (w<v) v=w; }
      return v;
  }
  for (i=1; i<lg(x); i++) { w=gvar2((GEN)x[i]); if (w<v) v=w; }
  return v;
}

GEN
gpolvar(GEN x)
{
  if (typ(x)==t_PADIC) x = (GEN)x[2];
  else
  {
    long v=gvar(x);
    if (v==BIGINT) err(typeer,"polvar");
    x = polx[v];
  }
  return gcopy(x);
}

/*******************************************************************/
/*                                                                 */
/*                    PRECISION OF SCALAR OBJECTS                  */
/*                                                                 */
/*******************************************************************/

long
precision(GEN x)
{
  long tx=typ(x),k,l;

  if (tx==t_REAL)
  {
    k=2-(expo(x)>>TWOPOTBITS_IN_LONG);
    l=lg(x); if (l>k) k=l;
    return k;
  }
  if (tx==t_COMPLEX)
  {
    k=precision((GEN)x[1]);
    l=precision((GEN)x[2]); if (l && (!k || l<k)) k=l;
    return k;
  }
  return 0;
}

long
gprecision(GEN x)
{
  long tx=typ(x),lx=lg(x),i,k,l;

  if (is_scalar_t(tx)) return precision(x);
  switch(tx)
  {
    case t_POL: lx=lgef(x);
    case t_VEC: case t_COL: case t_MAT:
      k=VERYBIGINT;
      for (i=lontyp[tx]; i<lx; i++)
      {
        l = gprecision((GEN)x[i]);
      if (l && l<k) k = l;
      }
      return (k==VERYBIGINT)? 0: k;

    case t_RFRAC: case t_RFRACN:
    {
      k=gprecision((GEN)x[1]);
      l=gprecision((GEN)x[2]); if (l && (!k || l<k)) k=l;
      return k;
    }
    case t_QFR:
      return gprecision((GEN)x[4]);
  }
  return 0;
}

GEN
ggprecision(GEN x)
{
  long a=gprecision(x);
  return stoi(a ? (long) ((a-2)*pariK): VERYBIGINT);
}

GEN
precision0(GEN x, long n)
{
  if (n) return gprec(x,n);
  return ggprecision(x);
}

/* attention: precision p-adique absolue */
long
padicprec(GEN x, GEN p)
{
  long i,s,t,lx=lg(x),tx=typ(x);

  switch(tx)
  {
    case t_INT: case t_FRAC: case t_FRACN:
      return VERYBIGINT;

    case t_INTMOD:
      return ggval((GEN)x[1],p);

    case t_PADIC:
      if (!gegal((GEN)x[2],p))
      err(talker,"not the same prime in padicprec");
      return precp(x)+valp(x);

    case t_POL:
      lx=lgef(x);

    case t_COMPLEX: case t_QUAD: case t_POLMOD: case t_SER: case t_RFRAC:
    case t_RFRACN: case t_VEC: case t_COL: case t_MAT:
      for (s=VERYBIGINT, i=lontyp[tx]; i<lx; i++)
      {
        t = padicprec((GEN)x[i],p); if (t<s) s = t;
      }
      return s;
  }
  err(typeer,"padicprec");
  return 0; /* not reached */
}

/* Degre de x par rapport a la variable v si v>=0, par rapport a la variable
 * principale si v<0. On suppose x fraction rationnelle ou polynome.
 * Convention deg(0)=-1.
 */
long
poldegree(GEN x, long v)
{
  long tx=typ(x), av, w, d;

  if (is_scalar_t(tx)) return gcmp0(x)? -1: 0;
  switch(tx)
  {
    case t_POL:
      w = varn(x);
      if (v < 0 || v == w) return lgef(x)-3;
      if (v < w) return signe(x)? 0: -1;
      av = avma; x = gsubst(gsubst(x,w,polx[MAXVARN]),v,polx[0]);
      if (gvar(x)) { d = gcmp0(x)? -1: 0; } else d = lgef(x)-3;
      avma = av; return d;

    case t_RFRAC: case t_RFRACN:
      if (gcmp0((GEN)x[1])) return -1;
      return poldegree((GEN)x[1],v) - poldegree((GEN)x[2],v);
  }
  err(typeer,"degree");
  return 0; /* not reached  */
}

long
degree(GEN x)
{
  return poldegree(x,-1);
}

/* si v<0, par rapport a la variable principale, sinon par rapport a v.
 * On suppose que x est un polynome ou une serie.
 */
GEN
pollead(GEN x, long v)
{
  long l,tx = typ(x),av,tetpil,w;
  GEN xinit;

  if (is_scalar_t(tx)) return gcopy(x);
  w = varn(x);
  switch(tx)
  {
    case t_POL:
      if (v < 0 || v == w)
      {
      l=lgef(x);
      return (l==2)? gzero: gcopy((GEN)x[l-1]);
      }
      if (v < w) return gcopy(x);
      av = avma; xinit = x;
      x = gsubst(gsubst(x,w,polx[MAXVARN]),v,polx[0]);
      if (gvar(x)) { avma = av; return gcopy(xinit); }
      l = lgef(x); if (l == 2) { avma = av; return gzero; }
      tetpil = avma; x = gsubst((GEN)x[l-1],MAXVARN,polx[w]);
      return gerepile(av,tetpil,x);

    case t_SER:
      if (v < 0 || v == w) return (!signe(x))? gzero: gcopy((GEN)x[2]);
      if (v < w) return gcopy(x);
      av = avma; xinit = x;
      x = gsubst(gsubst(x,w,polx[MAXVARN]),v,polx[0]);
      if (gvar(x)) { avma = av; return gcopy(xinit);}
      if (!signe(x)) { avma = av; return gzero;}
      tetpil = avma; x = gsubst((GEN)x[2],MAXVARN,polx[w]);
      return gerepile(av,tetpil,x);
  }
  err(typeer,"pollead");
  return NULL; /* not reached */
}

/* returns 1 if there's a real component in the structure, 0 otherwise */
int
isinexactreal(GEN x)
{
  long tx=typ(x),lx,i;

  if (is_scalar_t(tx))
  {
    switch(tx)
    {
      case t_REAL:
        return 1;

      case t_COMPLEX: case t_QUAD:
        return (typ(x[1])==t_REAL || typ(x[2])==t_REAL);
    }
    return 0;
  }
  switch(tx)
  {
    case t_QFR: case t_QFI:
      return 0;

    case t_RFRAC: case t_RFRACN:
      return isinexactreal((GEN)x[1]) || isinexactreal((GEN)x[2]);
  }
  if (is_noncalc_t(tx)) return 0;
  lx = (tx==t_POL)? lgef(x): lg(x);
  for (i=lontyp[tx]; i<lx; i++)
    if (isinexactreal((GEN)x[i])) return 1;
  return 0;
}

int
isexactzero(GEN g)
{
  long i;
  switch (typ(g))
  {
    case t_INT:
      return !signe(g);
    case t_REAL: case t_PADIC: case t_SER:
      return 0;
    case t_INTMOD: case t_POLMOD:
      return isexactzero((GEN)g[2]);
    case t_FRAC: case t_FRACN: case t_RFRAC: case t_RFRACN:
      return isexactzero((GEN)g[1]);
    case t_COMPLEX:
      return isexactzero((GEN)g[1]) && isexactzero((GEN)g[2]);
    case t_QUAD:
      return isexactzero((GEN)g[2]) && isexactzero((GEN)g[3]);

    case t_POL:
      for (i=lgef(g)-1; i>1; i--)
        if (!isexactzero((GEN)g[i])) return 0;
      return 1;
    case t_VEC: case t_COL: case t_MAT:
      for (i=lg(g)-1; i; i--)
      if (!isexactzero((GEN)g[i])) return 0;
      return 1;
  }
  return 0;
}

int
iscomplex(GEN x)
{
  switch(typ(x))
  {
    case t_INT: case t_REAL: case t_FRAC: case t_FRACN:
      return 0;
    case t_COMPLEX:
      return !gcmp0((GEN)x[2]);
    case t_QUAD:
      return signe(mael(x,1,2)) > 0;
  }
  err(typeer,"iscomplex");
  return 0; /* not reached */
}

int
ismonome(GEN x)
{
  long i;
  if (typ(x)!=t_POL || !signe(x)) return 0;
  for (i=lgef(x)-2; i>1; i--)
    if (!isexactzero((GEN)x[i])) return 0;
  return 1;
}

/********************************************************************/
/**                                                                **/
/**                     MULTIPLICATION SIMPLE                      **/
/**                                                                **/
/********************************************************************/
#define fix_frac(z) if (signe(z[2])<0)\
{\
  setsigne(z[1],-signe(z[1]));\
  setsigne(z[2],1);\
}

/* assume z[1] was created last */
#define fix_frac_if_int(z) if (is_pm1(z[2]))\
  z = gerepileupto((long)(z+3), (GEN)z[1]);

GEN
gmulsg(long s, GEN y)
{
  long ty=typ(y),ly=lg(y),i,av,tetpil;
  GEN z,p1,p2;

  switch(ty)
  {
    case t_INT:
      return mulsi(s,y);

    case t_REAL:
      return mulsr(s,y);

    case t_INTMOD: z=cgetg(3,t_INTMOD); p2=(GEN)y[1];
      (void)new_chunk(lgefint(p2)<<2);
      p1=mulsi(s,(GEN)y[2]); avma=(long)z;
      z[2]=lmodii(p1,p2); icopyifstack(p2,z[1]); return z;

    case t_FRAC:
      if (!s) return gzero;
      z = cgetg(3,t_FRAC);
      i = cgcd(s, smodis((GEN)y[2], s));
      if (i == 1)
      {
        z[2] = licopy((GEN)y[2]);
        z[1] = lmulis((GEN)y[1], s);
      }
      else
      {
        z[2] = ldivis((GEN)y[2], i);
        z[1] = lmulis((GEN)y[1], s/i);
        fix_frac_if_int(z);
      }
      return z;

    case t_FRACN: z=cgetg(3,ty);
      z[1]=lmulsi(s,(GEN)y[1]);
      z[2]=licopy((GEN)y[2]);
      return z;

    case t_COMPLEX: z=cgetg(ly,ty);
      z[1]=lmulsg(s,(GEN)y[1]);
      z[2]=lmulsg(s,(GEN)y[2]); return z;

    case t_PADIC:
      if (!s) return gzero;
      av=avma; p1=cgetp(y); gaffsg(s,p1); tetpil=avma;
      return gerepile(av,tetpil,gmul(p1,y));

    case t_QUAD: z=cgetg(ly,ty);
      copyifstack(y[1],z[1]);
      z[2]=lmulsg(s,(GEN)y[2]);
      z[3]=lmulsg(s,(GEN)y[3]); return z;

    case t_POLMOD: z=cgetg(ly,ty);
      z[2]=lmulsg(s,(GEN)y[2]);
      copyifstack(y[1],z[1]); return z;

    case t_POL:
      if (!s || !signe(y)) return zeropol(varn(y));
      ly=lgef(y); z=cgetg(ly,t_POL); z[1]=y[1];
      for (i=2; i<ly; i++) z[i]=lmulsg(s,(GEN)y[i]);
      return normalizepol_i(z, ly);

    case t_SER:
      if (!s) return zeropol(varn(y));
      if (gcmp0(y)) return gcopy(y);
      z=cgetg(ly,ty);
      for (i=2; i<ly; i++) z[i]=lmulsg(s,(GEN)y[i]);
      z[1]=y[1]; return normalize(z);

    case t_RFRAC:
      if (!s) return zeropol(gvar(y));
      z = cgetg(3, t_RFRAC);
      i = ggcd(stoi(s),(GEN)y[2])[2];
      avma = (long)z;
      if (i == 1)
      {
        z[1]=lmulgs((GEN)y[1], s);
        z[2]= lcopy((GEN)y[2]);
      }
      else
      {
        z[1] = lmulgs((GEN)y[1], s/i);
        z[2] = ldivgs((GEN)y[2], i);
      }
      return z;

    case t_RFRACN:
      if (!s) return zeropol(gvar(y));
      z=cgetg(ly,t_RFRACN);
      z[1]=lmulsg(s,(GEN)y[1]);
      z[2]=lcopy((GEN)y[2]); return z;

    case t_VEC: case t_COL: case t_MAT:
      z=cgetg(ly,ty);
      for (i=1; i<ly; i++) z[i]=lmulsg(s,(GEN)y[i]);
      return z;
  }
  err(typeer,"gmulsg");
  return NULL; /* not reached */
}

/********************************************************************/
/**                                                                **/
/**                       DIVISION SIMPLE                          **/
/**                                                                **/
/********************************************************************/

GEN
gdivgs(GEN x, long s)
{
  static long court[] = { evaltyp(t_INT) | m_evallg(3),0,0 };
  long tx=typ(x),lx=lg(x),i,av;
  GEN z,p1;

  if (!s) err(gdiver2);
  switch(tx)
  {
    case t_INT:
      av=avma; z=dvmdis(x,s,&p1);
      if (p1==gzero) return z;

      i = cgcd(s, smodis(x,s));
      avma=av; z=cgetg(3,t_FRAC);
      if (i == 1)
        x = icopy(x);
      else
      {
        s /= i;
        x = divis(x, i);
      }
      z[1]=(long)x; z[2]=lstoi(s);
      fix_frac(z); return z;

    case t_REAL:
      return divrs(x,s);

    case t_FRAC: z=cgetg(3,tx);
      i = cgcd(s, smodis((GEN)x[1], s));
      if (i == 1)
      {
        z[2] = lmulsi(s, (GEN)x[2]);
        z[1] = licopy((GEN)x[1]);
      }
      else
      {
        z[2] = lmulsi(s/i, (GEN)x[2]);
        z[1] = ldivis((GEN)x[1], i);
      }
      fix_frac(z);
      fix_frac_if_int(z); return z;

    case t_FRACN: z = cgetg(3,tx);
      z[1]=licopy((GEN)x[1]);
      z[2]=lmulsi(s,(GEN)x[2]);
      fix_frac(z); return z;

    case t_COMPLEX: z=cgetg(lx,tx);
      z[1]=ldivgs((GEN)x[1],s);
      z[2]=ldivgs((GEN)x[2],s);
      return z;

    case t_QUAD: z=cgetg(lx,tx);
      copyifstack(x[1],z[1]);
      for (i=2; i<4; i++) z[i]=ldivgs((GEN)x[i],s);
      return z;

    case t_POLMOD: z=cgetg(lx,tx);
      copyifstack(x[1],z[1]);
      z[2]=ldivgs((GEN)x[2],s);
      return z;

    case t_POL: lx=lgef(x); z=cgetg(lx,tx);
      for (i=2; i<lx; i++) z[i]=ldivgs((GEN)x[i],s);
      z[1]=x[1]; return z;

    case t_SER: z=cgetg(lx,tx);
      for (i=2; i<lx; i++) z[i]=ldivgs((GEN)x[i],s);
      z[1]=x[1]; return z;

    case t_RFRAC:
      av = avma;
      p1 = ggcd(stoi(s),(GEN)x[1]);
      if (typ(p1) == t_INT)
      {
        avma = av;
        z = cgetg(3, t_RFRAC);
        i = p1[2];
        if (i == 1)
        {
          z[1] = lcopy((GEN)x[1]);
          z[2] = lmulsg(s,(GEN)x[2]);
        }
        else
        {
          z[1] = ldivgs((GEN)x[1], i);
          z[2] = lmulgs((GEN)x[2], s/i);
        }
      }
      else /* t_FRAC */
      {
        z = cgetg(3, t_RFRAC);
        z[1] = ldiv((GEN)x[1], p1);
        z[2] = lmul((GEN)x[2], gdivsg(s,p1));
        z = gerepileupto(av, gcopy(z));
      }
      return z;

    case t_RFRACN: z=cgetg(3,t_RFRACN);
      z[2]=lmulsg(s,(GEN)x[2]);
      z[1]=lcopy((GEN)x[1]); return z;

    case t_VEC: case t_COL: case t_MAT: z=cgetg(lx,tx);
      for (i=1; i<lx; i++) z[i]=ldivgs((GEN)x[i],s);
      return z;
  }
  affsi(s,court); return gdiv(x,court);
}

/*******************************************************************/
/*                                                                 */
/*                    MODULO GENERAL                               */
/*                                                                 */
/*******************************************************************/

GEN
gmod(GEN x, GEN y)
{
  long av,tetpil,i, tx=typ(x), ty = typ(y);
  GEN z,p1;

  if (is_matvec_t(tx))
  {
    i=lg(x); z=cgetg(i,tx);
    for (i--; i; i--) z[i]=lmod((GEN)x[i],y);
    return z;
  }
  switch(ty)
  {
    case t_INT:
      switch(tx)
      {
      case t_INT:
        return modii(x,y);

      case t_INTMOD: z=cgetg(3,tx);
          z[1]=lmppgcd((GEN)x[1],y);
        z[2]=lmodii((GEN)x[2],(GEN)z[1]); return z;

      case t_FRAC: case t_FRACN:
        av=avma; if (tx==t_FRACN) x=gred(x);
        p1=mulii((GEN)x[1],mpinvmod((GEN)x[2],y));
        tetpil=avma; return gerepile(av,tetpil,modii(p1,y));

      case t_QUAD: z=cgetg(4,tx);
          copyifstack(x[1],z[1]);
        z[2]=lmod((GEN)x[2],y);
          z[3]=lmod((GEN)x[3],y); return z;

      case t_PADIC:
        {
          long p1[] = {evaltyp(t_INTMOD)|m_evallg(3),0,0};
          p1[1] = (long)y;
          p1[2] = lgeti(lgefint(y));
          gaffect(x,p1); return (GEN)p1[2];
        }
      case t_POLMOD: case t_POL:
        return gzero;

      default: err(operf,"%",tx,ty);
      }

    case t_REAL: case t_FRAC: case t_FRACN:
      switch(tx)
      {
      case t_INT: case t_REAL: case t_FRAC: case t_FRACN:
        av=avma; p1 = gfloor(gdiv(x,y)); p1 = gneg_i(gmul(p1,y));
          tetpil=avma; return gerepile(av,tetpil,gadd(x,p1));

      case t_POLMOD: case t_POL:
        return gzero;

      default: err(operf,"%",tx,ty);
      }

    case t_POL:
      if (is_scalar_t(tx))
      {
        if (tx!=t_POLMOD || varn(x[1]) > varn(y))
          return (lgef(y) < 4)? gzero: gcopy(x);
      if (varn(x[1])!=varn(y)) return gzero;
        z=cgetg(3,t_POLMOD);
        z[1]=lgcd((GEN)x[1],y);
        z[2]=lres((GEN)x[2],(GEN)z[1]); return z;
      }
      switch(tx)
      {
      case t_POL:
        return gres(x,y);

      case t_RFRAC: case t_RFRACN:
        av=avma; if (tx==t_RFRACN) x=gred_rfrac(x);
        p1=gmul((GEN)x[1],ginvmod((GEN)x[2],y)); tetpil=avma;
          return gerepile(av,tetpil,gres(p1,y));

        case t_SER:
          if (ismonome(y) && varn(x) == varn(y))
          {
            long d = degree(y);
            if (lg(x)-2 + valp(x) < d) err(operi,"%",tx,ty);
            av = avma; 
            return gerepileupto(av, gmod(gtrunc(x), y));
          }
      default: err(operf,"%",tx,ty);
      }
  }
  err(operf,"%",tx,ty);
  return NULL; /* not reached */
}

GEN
gmodulsg(long x, GEN y)
{
  GEN z;

  switch(typ(y))
  {
    case t_INT: z=cgetg(3,t_INTMOD);
      z[1]=labsi(y); z[2]=lmodsi(x,y); return z;

    case t_POL: z=cgetg(3,t_POLMOD);
      z[1]=lcopy(y); z[2]=lstoi(x); return z;
  }
  err(operf,"%",t_INT,typ(y)); return NULL; /* not reached */
}

GEN
gmodulss(long x, long y)
{
  GEN z=cgetg(3,t_INTMOD);

  y=labs(y); z[1]=lstoi(y); z[2]=lstoi(x % y); return z;
}

static GEN 
specialmod(GEN x, GEN y)
{
  GEN z = gmod(x,y);
  if (gvar(z) < varn(y)) z = gzero;
  return z;
}

GEN
gmodulo(GEN x,GEN y)
{
  long tx=typ(x),l,i;
  GEN z;

  if (is_matvec_t(tx))
  {
    l=lg(x); z=cgetg(l,tx);
    for (i=1; i<l; i++) z[i] = lmodulo((GEN)x[i],y);
    return z;
  }
  switch(typ(y))
  {
    case t_INT:
      if (tx!=t_INT && !is_frac_t(tx) && tx!=t_PADIC) break;
      z=cgetg(3,t_INTMOD);
      if (!signe(y)) err(talker,"zero modulus in gmodulo");
      y = gclone(y); setsigne(y,1);
      z[1]=(long)y;
      z[2]=lmod(x,y); return z;

    case t_POL: z=cgetg(3,t_POLMOD);
      z[1] = lclone(y);
      if (is_scalar_t(tx)) { z[2]=lcopy(x); return z; }
      if (tx!=t_POL && !is_rfrac_t(tx) && tx!=t_SER) break;
      z[2]=(long)specialmod(x,y); return z;
  }
  err(operf,"%",tx,typ(y)); return NULL; /* not reached */
}

GEN
gmodulcp(GEN x,GEN y)
{
  long tx=typ(x),l,i;
  GEN z;

  if (is_matvec_t(tx))
  {
    l=lg(x); z=cgetg(l,tx);
    for (i=1; i<l; i++) z[i] = lmodulcp((GEN)x[i],y);
    return z;
  }
  switch(typ(y))
  {
    case t_INT:
      if (tx!=t_INT && !is_frac_t(tx) && tx!=t_PADIC) break;
      z=cgetg(3,t_INTMOD);
      z[1]=labsi(y);
      z[2]=lmod(x,y); return z;

    case t_POL: z=cgetg(3,t_POLMOD);
      z[1]=lcopy(y);
      if (is_scalar_t(tx))
      {
        z[2] = (lgef(y) > 3)? lcopy(x): lmod(x,y);
        return z;
      }
      if (tx!=t_POL && !is_rfrac_t(tx) && tx!=t_SER) break;
      z[2]=(long)specialmod(x,y); return z;
  }
  err(operf,"%",tx,typ(y)); return NULL; /* not reached */
}

GEN
Mod0(GEN x,GEN y,long flag)
{
  switch(flag)
  {
    case 0: return gmodulcp(x,y);
    case 1: return gmodulo(x,y);
    default: err(flagerr,"Mod");
  }
  return NULL; /* not reached */
}

/*******************************************************************/
/*                                                                 */
/*                 DIVISION ENTIERE GENERALE                       */
/*            DIVISION ENTIERE AVEC RESTE GENERALE                 */
/*                                                                 */
/*******************************************************************/

GEN
gdivent(GEN x, GEN y)
{
  long tx=typ(x),ty=typ(y);

  if (tx == t_INT)
  {
    if (ty == t_INT) return truedvmdii(x,y,NULL);
    if (ty!=t_POL) err(typeer,"gdivent");
    return gzero;
  }
  if (ty != t_POL) err(typeer,"gdivent");
  if (tx == t_POL) return gdeuc(x,y);
  if (! is_scalar_t(tx)) err(typeer,"gdivent");
  return gzero;
}

GEN
gdiventres(GEN x, GEN y)
{
  long tx=typ(x),ty=typ(y);
  GEN z = cgetg(3,t_COL);

  if (tx==t_INT)
  {
    if (ty==t_INT)
    {
      z[1]=(long)truedvmdii(x,y,(GEN*)(z+2));
      return z;
    }
    if (ty!=t_POL) err(typeer,"gdiventres");
    z[1]=zero; z[2]=licopy(x); return z;
  }
  if (ty != t_POL) err(typeer,"gdiventres");
  if (tx == t_POL)
  {
    z[1]=ldivres(x,y,(GEN *)(z+2));
    return z;
  }
  if (! is_scalar_t(tx)) err(typeer,"gdiventres");
  z[1]=zero; z[2]=lcopy(x); return z;
}

GEN
gdivmod(GEN x, GEN y, GEN *pr)
{
  long ty,tx=typ(x);

  if (tx==t_INT)
  {
    ty=typ(y);
    if (ty==t_INT) return dvmdii(x,y,pr);
    if (ty==t_POL) { *pr=gcopy(x); return gzero; }
    err(typeer,"gdivmod");
  }
  if (tx!=t_POL) err(typeer,"gdivmod");
  return poldivres(x,y,pr);
}

/* When x and y are integers, compute the quotient x/y, rounded to the
 * nearest integer. If there is a tie, the quotient is rounded towards zero.
 * If x and y are not both integers, same as gdivent.
 */
GEN
gdivround(GEN x, GEN y)
{
  long tx=typ(x),ty=typ(y);

  if (tx==t_INT)
  {
    if (ty==t_INT)
    {
      long av = avma, av1,fl;
      GEN r, q=dvmdii(x,y,&r); /* q = x/y rounded towards 0, sqn(r)=sgn(x) */

      if (r==gzero) return q;
      av1 = avma;
      fl = absi_cmp(shifti(r,1),y);
      avma = av1; cgiv(r);
      if (fl >= 0) /* If 2*|r| >= |y| */
      {
        long sz = signe(x)*signe(y);
      if (fl || sz > 0)
          { av1=avma; q = gerepile(av,av1,addis(q,sz)); }
      }
      return q;
    }
    if (ty!=t_POL) err(typeer,"gdivround");
    return gzero;
  }
  if (ty != t_POL) err(typeer,"gdivround");
  if (tx == t_POL) return gdeuc(x,y);
  if (! is_scalar_t(tx)) err(typeer,"gdivround");
  return gzero;
}

/*******************************************************************/
/*                                                                 */
/*                       SHIFT D'UN GEN                            */
/*                                                                 */
/*******************************************************************/

/* Shift tronque si n<0 (multiplication tronquee par 2^n)  */
GEN
gshift(GEN x, long n)
{
  long i,l,lx, tx = typ(x);
  GEN y;

  switch(tx)
  {
    case t_INT:
      return shifti(x,n);
    case t_REAL:
      return shiftr(x,n);

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); y=cgetg(lx,tx); l=lontyp[tx];
      for (i=1; i<l ; i++) y[i]=x[i];
      for (   ; i<lx; i++) y[i]=lshift((GEN)x[i],n);
      return y;
  }
  return gmul2n(x,n);
}

extern GEN mulscalrfrac(GEN x, GEN y);

/* Shift vrai (multiplication exacte par 2^n) */
GEN
gmul2n(GEN x, long n)
{
  long tx,lx,i,k,l,av,tetpil;
  GEN p2,p1,y;

  tx=typ(x);
  switch(tx)
  {
    case t_INT:
      if (n>=0) return shifti(x,n);
      if (!signe(x)) return gzero;
      l = vali(x); n = -n;
      if (n<=l) return shifti(x,-n);
      y=cgetg(3,t_FRAC);
      y[1]=lshifti(x,-l);
      y[2]=lshifti(gun,n-l); return y;
      
    case t_REAL:
      return shiftr(x,n);

    case t_INTMOD:
      if (n > 0)
      {
        y=cgetg(3,t_INTMOD); p2=(GEN)x[1];
        av=avma; new_chunk(lgefint(p2) * (3 + (n>>TWOPOTBITS_IN_LONG)));
        p1 = shifti((GEN)x[2],n); avma=av;
        y[2]=lmodii(p1,p2); icopyifstack(p2,y[1]); return y;
      }
      l=avma; y=gmul2n(gun,n); tetpil=avma;
      return gerepile(l,tetpil,gmul(y,x));

    case t_FRAC: case t_FRACN:
      l = vali((GEN)x[1]);
      k = vali((GEN)x[2]);
      if (n+l-k>=0)
      {
        if (expi((GEN)x[2]) == k) /* x[2] power of 2 */
          return shifti((GEN)x[1],n-k);
        l = n-k; k = -k;
      }
      else
      {
        k = -l-n; l = -l;
      }
      y=cgetg(3,t_FRAC);
      y[1]=lshifti((GEN)x[1],l);
      y[2]=lshifti((GEN)x[2],k); return y;

    case t_QUAD: y=cgetg(4,t_QUAD);
      copyifstack(x[1],y[1]);
      for (i=2; i<4; i++) y[i]=lmul2n((GEN)x[i],n);
      return y;

    case t_POLMOD: y=cgetg(3,t_POLMOD);
      copyifstack(x[1],y[1]);
      y[2]=lmul2n((GEN)x[2],n); return y;

    case t_POL: case t_COMPLEX: case t_SER:
    case t_VEC: case t_COL: case t_MAT:
      lx = (tx==t_POL)? lgef(x): lg(x);
      y=cgetg(lx,tx); l=lontyp[tx];
      for (i=1; i<l ; i++) y[i]=x[i];
      for (   ; i<lx; i++) y[i]=lmul2n((GEN)x[i],n);
      return y;

    case t_RFRAC: av=avma; p1 = gmul2n(gun,n); tetpil = avma;
      return gerepile(av,tetpil, mulscalrfrac(p1,x));

    case t_RFRACN: y=cgetg(3,tx);
      if (n>=0)
      {
        y[1]=lmul2n((GEN)x[1],n);  y[2]=lcopy((GEN)x[2]);
      }
      else
      {
        y[2]=lmul2n((GEN)x[2],-n); y[1]=lcopy((GEN)x[1]);
      }
      return y;

    case t_PADIC:
      l=avma; y=gmul2n(gun,n); tetpil=avma;
      return gerepile(l,tetpil,gmul(y,x));
  }
  err(typeer,"gmul2n");
  return NULL; /* not reached */
}

/*******************************************************************/
/*                                                                 */
/*                      INVERSE D'UN GEN                           */
/*                                                                 */
/*******************************************************************/
extern GEN fix_rfrac_if_pol(GEN x, GEN y);

GEN
ginv(GEN x)
{
  long tx=typ(x),av,tetpil,s;
  GEN z,y,p1,p2;

  switch(tx)
  {
    case t_INT:
      if (is_pm1(x)) return icopy(x);
      if (!signe(x)) err(gdiver2);
      z=cgetg(3,t_FRAC);
      z[1] = (signe(x)<0)? lnegi(gun): un;
      z[2]=labsi(x); return z;

    case t_REAL:
      return divsr(1,x);

    case t_INTMOD: z=cgetg(3,t_INTMOD);
      icopyifstack(x[1],z[1]);
      z[2]=lmpinvmod((GEN)x[2],(GEN)x[1]); return z;

    case t_FRAC: case t_FRACN:
      s = signe(x[1]);
      if (!s) err(gdiver2);
      if (is_pm1(x[1]))
        return s>0? icopy((GEN)x[2]): negi((GEN)x[2]);
      z = cgetg(3,tx);
      z[1] = licopy((GEN)x[2]);
      z[2] = licopy((GEN)x[1]);
      if (s < 0)
      {
      setsigne(z[1],-signe(z[1]));
      setsigne(z[2],1);
      }
      return z;

    case t_COMPLEX: case t_QUAD:
      av=avma; p1=gnorm(x); p2=gconj(x); tetpil=avma;
      return gerepile(av,tetpil,gdiv(p2,p1));

    case t_PADIC: z = cgetg(5,t_PADIC);
      if (!signe(x[4])) err(gdiver2);
      z[1] = evalprecp(precp(x)) | evalvalp(-valp(x));
      icopyifstack(x[2], z[2]);
      z[3] = licopy((GEN)x[3]);
      z[4] = lmpinvmod((GEN)x[4],(GEN)z[3]); return z;

    case t_POLMOD: z=cgetg(3,t_POLMOD);
      copyifstack(x[1],z[1]);
      z[2]=linvmod((GEN)x[2],(GEN)x[1]); return z;

    case t_POL: case t_SER:
      return gdiv(gun,x);

    case t_RFRAC: case t_RFRACN:
      if (gcmp0((GEN) x[1])) err(gdiver2);
      p1 = fix_rfrac_if_pol((GEN)x[2],(GEN)x[1]);
      if (p1) return p1;
      z=cgetg(3,tx);
      z[1]=lcopy((GEN)x[2]);
      z[2]=lcopy((GEN)x[1]); return z;

    case t_QFR:
    {
      long k,l;
      l=signe(x[2]); setsigne(x[2],-l);
      k=signe(x[4]); setsigne(x[4],-k); z=redreal(x);
      setsigne(x[2],l); setsigne(x[4],k); return z;
    }
    case t_QFI:
      y=gcopy(x);
      if (!egalii((GEN)x[1],(GEN)x[2]) && !egalii((GEN)x[1],(GEN)x[3]))
      setsigne(y[2],-signe(y[2]));
      return y;
    case t_MAT:
      return (lg(x)==1)? cgetg(1,t_MAT): invmat(x);
  }
  err(typeer,"inverse");
  return NULL; /* not reached */
}

/*******************************************************************/
/*                                                                 */
/*           SUBSTITUTION DANS UN POLYNOME OU UNE SERIE            */
/*                                                                 */
/*******************************************************************/

/* Convert t_SER --> t_POL / t_RFRAC */
static GEN
gconvsp(GEN x, int flpile)
{
  long v = varn(x), av,tetpil,i;
  GEN p1,y;

  if (gcmp0(x)) return zeropol(v);
  av=avma; y=dummycopy(x); settyp(y,t_POL);
  i=lg(x)-1; while (i>1 && gcmp0((GEN)y[i])) i--;
  setlgef(y,i+1);
  p1=gpuigs(polx[v],valp(x));
  tetpil=avma; p1=gmul(p1,y);
  return flpile? gerepile(av,tetpil,p1): p1;
}

GEN
gsubst(GEN x, long v, GEN y)
{
  long tx = typ(x), ty = typ(y), lx = lg(x), ly = lg(y);
  long l,vx,vy,e,ex,ey,tetpil,av,i,j,k,jb,limite;
  GEN t,p1,p2,z;

  if (ty==t_MAT)
  {
    if (ly==1) return cgetg(1,t_MAT);
    if (ly != lg(y[1]))
      err(talker,"forbidden substitution by a non square matrix");
  } else if (is_graphicvec_t(ty))
    err(talker,"forbidden substitution by a vector");

  if (is_scalar_t(tx))
  {
    if (tx!=t_POLMOD || v <= varn(x[1]))
    {
      if (ty==t_MAT) return gscalmat(x,ly-1);
      return gcopy(x);
    }
    av=avma;
    p1=gsubst((GEN)x[1],v,y); vx=varn(p1);
    p2=gsubst((GEN)x[2],v,y); vy=gvar(p2);
    if (typ(p1)!=t_POL)
      err(talker,"forbidden substitution in a scalar type");
    if (vy>=vx)
    {
      tetpil=avma;
      return gerepile(av,tetpil,gmodulcp(p2,p1));
    }
    lx=lgef(p2); tetpil=avma; z=cgetg(lx,t_POL); z[1]=p2[1];
    for (i=2; i<lx; i++) z[i]=lmodulcp((GEN)p2[i],p1);
    return gerepile(av,tetpil, normalizepol_i(z,lx));
  }

  switch(tx)
  {
    case t_POL:
      l=lgef(x);
      if (l==2)
        return (ty==t_MAT)? gscalmat(gzero,ly-1): gzero;

      vx=varn(x);
      if (vx<v)
      {
      av=avma; p1=polx[vx]; z= gsubst((GEN)x[l-1],v,y);
      for (i=l-1; i>2; i--) z=gadd(gmul(z,p1),gsubst((GEN)x[i-1],v,y));
      return gerepileupto(av,z);
      }
      if (ty!=t_MAT)
        return (vx>v)? gcopy(x): poleval(x,y);

      if (vx>v) return gscalmat(x,ly-1);
      if (l==3) return gscalmat((GEN)x[2],ly-1);
      av=avma; z=(GEN)x[l-1];
      for (i=l-1; i>2; i--) z=gaddmat((GEN)x[i-1],gmul(z,y));
      return gerepileupto(av,z);

    case t_SER:
      vx=varn(x);
      if (vx > v)
        return (ty==t_MAT)? gscalmat(x,ly-1): gcopy(x);
      ex = valp(x);
      if (vx < v)
      {
        if (!signe(x)) return gcopy(x);
        /* a ameliorer */
        av=avma; setvalp(x,0); p1=gconvsp(x,0); setvalp(x,ex);
        p2=gsubst(p1,v,y); tetpil=avma; z=tayl(p2,vx,lx-2);
        if (ex)
        {
          p1=gpuigs(polx[vx],ex); tetpil=avma; z=gmul(z,p1);
        }
        return gerepile(av,tetpil,z);
      }
      switch(ty) /* here vx == v */
      {
        case t_SER:
        ey=valp(y); vy=varn(y);
        if (ey<1) return zeroser(vy,ey*(ex+lx-2));
        l=(lx-2)*ey+2;
        if (ex) { if (l>ly) l=ly; }
        else
        {
          if (gcmp0(y)) l=ey+2;
          else { if (l>ly) l=ey+ly; }
        }
        if (vy!=vx)
        {
          av=avma; z = zeroser(vy,0);
          for (i=lx-1; i>=2; i--)
              z = gadd((GEN)x[i], gmul(y,z));
          if (ex) z = gmul(z, gpuigs(y,ex));
          return gerepileupto(av,z);
        }

        av=avma; limite=stack_lim(av,1);
          t=cgetg(ly,t_SER);
          z = scalarser((GEN)x[2],varn(y),l-2);
        for (i=2; i<ly; i++) t[i]=y[i];
      
        for (i=3,jb=ey; jb<=l-2; i++,jb+=ey)
        {
          for (j=jb+2; j<l; j++)
          {
            p1=gmul((GEN)x[i],(GEN)t[j-jb]);
            z[j]=ladd((GEN)z[j],p1);
          }
          for (j=l-1-jb-ey; j>1; j--)
          {
            p1=gzero;
            for (k=2; k<j; k++)
            p1=gadd(p1,gmul((GEN)t[j-k+2],(GEN)y[k]));
            p2=gmul((GEN)t[2],(GEN)y[j]);
            t[j]=ladd(p1,p2);
          }
            if (low_stack(limite, stack_lim(av,1)))
          {
            GEN *gptr[2];
            if(DEBUGMEM>1) err(warnmem,"gsubst");
            gptr[0]=&z; gptr[1]=&t; gerepilemany(av,gptr,2);
          }
        }
        if (!ex) { tetpil=avma; return gerepile(av,tetpil,gcopy(z)); }

          if (l<ly) { setlg(y,l); p1=gpuigs(y,ex); setlg(y,ly); }
          else p1=gpuigs(y,ex);
          tetpil=avma; return gerepile(av,tetpil,gmul(z,p1));

        case t_POL: case t_RFRAC: case t_RFRACN:
          if (isexactzero(y)) return scalarser((GEN)x[2],v,lx-2);
          vy=gvar(y); e=gval(y,vy);
          if (e<=0)
            err(talker,"non positive valuation in a series substitution");
        av=avma; p1=gconvsp(x,0); p2=gsubst(p1,v,y); tetpil=avma;
        return gerepile(av,tetpil,tayl(p2,vy,e*(lx-2+ex)));

        default:
          err(talker,"non polynomial or series type substituted in a series");
      }
      break;

    case t_RFRAC: case t_RFRACN: av=avma;
      p1=gsubst((GEN)x[1],v,y);
      p2=gsubst((GEN)x[2],v,y); tetpil=avma;
      return gerepile(av,tetpil,gdiv(p1,p2));

    case t_VEC: case t_COL: case t_MAT: z=cgetg(lx,tx);
      for (i=1; i<lx; i++) z[i]=lsubst((GEN)x[i],v,y);
  }
  return z;
}

/*******************************************************************/
/*                                                                 */
/*                SERIE RECIPROQUE D'UNE SERIE                     */
/*                                                                 */
/*******************************************************************/

GEN
recip(GEN x)
{
  long v=varn(x), lx = lg(x);
  long tetpil, av=avma;
  GEN p1,p2,a,y,u;

  if (typ(x)!=t_SER) err(talker,"not a series in serreverse");
  if (valp(x)!=1 || lx < 3)
    err(talker,"valuation not equal to 1 in serreverse");

  a=(GEN)x[2];
  if (gcmp1(a))
  {
    long i, j, k, mi;
    long lim=stack_lim(av, 2);

    mi = lx-1; while (mi>=3 && gcmp0((GEN)x[mi])) mi--;
    u = cgetg(lx,t_SER);
    y = cgetg(lx,t_SER);
    u[1] = y[1] = evalsigne(1) | evalvalp(1) | evalvarn(v);
    u[2] = y[2] = un;
    if (lx > 3)
    {
      u[3] = lmulsg(-2,(GEN)x[3]);
      y[3] = lneg((GEN)x[3]);
    }
    for (i=3; i<lx-1; )
    {
      for (j=3; j<i+1; j++)
      {
        p1 = (GEN)x[j];
        for (k=max(3,j+2-mi); k<j; k++)
          p1 = gadd(p1, gmul((GEN)u[k],(GEN)x[j-k+2]));
        u[j] = lsub((GEN)u[j], p1);
      }
      p1 = gmulsg(i,(GEN)x[i+1]);
      for (k=2; k<min(i,mi); k++)
      {
        p2 = gmul((GEN)x[k+1],(GEN)u[i-k+2]);
        p1 = gadd(p1, gmulsg(k,p2));
      }
      i++;
      u[i] = lneg(p1);
      y[i] = ldivgs((GEN)u[i],i-1);
      if (low_stack(lim, stack_lim(av,2)))
      {
      GEN *gptr[2];
      if(DEBUGMEM>1) err(warnmem,"recip");
      for(k=i+1; k<lx; k++) u[k]=y[k]=zero; /* dummy */
      gptr[0]=&u; gptr[1]=&y; gerepilemany(av,gptr,2);
      }
    }
    return gerepileupto(av, gcopy(y));
  }
  y = gdiv(x,a); y[2] = un; y = recip(y);
  a = gdiv(polx[v],a); tetpil = avma;
  return gerepile(av,tetpil, gsubst(y,v,a));
}

/*******************************************************************/
/*                                                                 */
/*                    DERIVATION ET INTEGRATION                    */
/*                                                                 */
/*******************************************************************/
GEN
derivpol(GEN x)
{
  long i,lx = lgef(x)-1;
  GEN y;

  if (lx<3) return gzero;
  y = cgetg(lx,t_POL);
  for (i=2; i<lx ; i++) y[i] = lmulsg(i-1,(GEN)x[i+1]);
  y[1] = x[1]; return normalizepol_i(y,i);
}

GEN
derivser(GEN x)
{
  long i,j,vx = varn(x), e = valp(x), lx = lg(x);
  GEN y;
  if (gcmp0(x)) return zeroser(vx,e-1);
  if (e)
  {
    y=cgetg(lx,t_SER); y[1] = evalsigne(1) | evalvalp(e-1) | evalvarn(vx);
    for (i=2; i<lx; i++) y[i]=lmulsg(i+e-2,(GEN)x[i]);
    return y;
  }
  i=3; while (i<lx && gcmp0((GEN)x[i])) i++;
  if (i==lx) return zeroser(vx,lx-3);
  lx--; if (lx<3) lx=3;
  lx = lx-i+3; y=cgetg(lx,t_SER);
  y[1]=evalsigne(1) | evalvalp(i-3) | evalvarn(vx);
  for (j=2; j<lx; j++) y[j]=lmulsg(j+i-4,(GEN)x[i+j-2]);
  return y;
}

GEN
deriv(GEN x, long v)
{
  long lx,vx,tx,e,i,j,l,av,tetpil;
  GEN y,p1,p2;

  tx=typ(x); if (is_const_t(tx)) return gzero;
  if (v < 0) v = gvar(x);
  switch(tx)
  {
    case t_POLMOD:
      if (v<=varn(x[1])) return gzero;
      y=cgetg(3,t_POLMOD); copyifstack(x[1],y[1]);
      y[2]=lderiv((GEN)x[2],v); return y;

    case t_POL:
      vx=varn(x); if (vx>v) return gzero;
      if (vx<v)
      {
        lx = lgef(x);
        y = cgetg(lx,t_POL);
        for (i=2; i<lx; i++) y[i] = lderiv((GEN)x[i],v);
        y[1] = evalvarn(vx);
        return normalizepol_i(y,i);
      }
      return derivpol(x);

    case t_SER:
      vx=varn(x); if (vx>v) return gzero;
      if (vx<v)
      {
        if (!signe(x)) return gcopy(x);
        lx=lg(x); e=valp(x);
      l=avma;
      for (i=2; i<lx; i++)
        {
          if (!gcmp0(deriv((GEN)x[i],v))) break;
          avma=l;
        }
      if (i==lx) return ggrando(polx[vx],e+lx-2);
      y=cgetg(lx-i+2,t_SER);
        y[1] = evalsigne(1) | evalvalp(e+i-2) | evalvarn(vx);
      for (j=2; i<lx; j++,i++) y[j]=lderiv((GEN)x[i],v);
        return y;
      }
      return derivser(x);

    case t_RFRAC: case t_RFRACN: av=avma; y=cgetg(3,tx);
      y[2]=lsqr((GEN)x[2]); l=avma;
      p1=gmul((GEN)x[2],deriv((GEN)x[1],v));
      p2=gmul(gneg_i((GEN)x[1]),deriv((GEN)x[2],v));
      tetpil=avma; p1=gadd(p1,p2);
      if (tx==t_RFRACN) { y[1]=lpile(l,tetpil,p1); return y; }
      y[1]=(long)p1; return gerepileupto(av,gred_rfrac(y));

    case t_VEC: case t_COL: case t_MAT: lx=lg(x); y=cgetg(lx,tx);
      for (i=1; i<lx; i++) y[i]=lderiv((GEN)x[i],v);
      return y;
  }
  err(typeer,"deriv");
  return NULL; /* not reached */
}

/*******************************************************************/
/*                                                                 */
/*                    INTEGRATION FORMELLE                         */
/*                                                                 */
/*******************************************************************/

static GEN
triv_integ(GEN x, long v, long tx, long lx)
{
  GEN y=cgetg(lx,tx);
  long i;

  y[1]=x[1];
  for (i=2; i<lx; i++) y[i]=linteg((GEN)x[i],v);
  return y;
}

GEN
integ(GEN x, long v)
{
  long lx,tx,e,i,j,vx,n,av=avma,tetpil;
  GEN y,p1;

  tx = typ(x);
  if (v < 0) v = gvar(x);
  if (is_scalar_t(tx))
  {
    if (tx == t_POLMOD && v>varn(x[1]))
    {
      y=cgetg(3,t_POLMOD); copyifstack(x[1],y[1]);
      y[2]=linteg((GEN)x[2],v); return y;
    }
    if (gcmp0(x)) return gzero;

    y=cgetg(4,t_POL); y[1] = evalsigne(1) | evallgef(4) | evalvarn(v);
    y[2]=zero; y[3]=lcopy(x); return y;
  }

  switch(tx)
  {
    case t_POL:
      vx=varn(x); lx=lgef(x);
      if (lx==2) return zeropol(min(v,vx));
      if (vx>v)
      {
        y=cgetg(4,t_POL);
      y[1] = signe(x)? evallgef(4) | evalvarn(v) | evalsigne(1)
                     : evallgef(4) | evalvarn(v);
        y[2]=zero; y[3]=lcopy(x); return y;
      }
      if (vx<v) return triv_integ(x,v,tx,lx);
      y=cgetg(lx+1,tx); y[2]=zero;
      for (i=3; i<=lx; i++)
        y[i]=ldivgs((GEN)x[i-1],i-2);
      y[1] = signe(x)? evallgef(lx+1) | evalvarn(v) | evalsigne(1)
                     : evallgef(lx+1) | evalvarn(v);
      return y;

    case t_SER:
      lx=lg(x); e=valp(x); vx=varn(x);
      if (!signe(x))
      {
        if (vx == v) e++; else if (vx < v) v = vx;
        return zeroser(v,e);
      }
      if (vx>v)
      {
        y=cgetg(4,t_POL);
        y[1] = evallgef(4) | evalvarn(v) | evalsigne(1);
        y[2]=zero; y[3]=lcopy(x); return y;
      }
      if (vx<v) return triv_integ(x,v,tx,lx);
      y=cgetg(lx,tx);
      for (i=2; i<lx; i++)
      {
      j=i+e-1;
        if (!j)
      {
        if (!gcmp0((GEN)x[i])) err(inter2);
        y[i]=zero;
      }
      else y[i] = ldivgs((GEN)x[i],j);
      }
      y[1]=x[1]+1; return y;

    case t_RFRAC: case t_RFRACN:
      vx = gvar(x);
      if (vx>v)
      {
      y=cgetg(4,t_POL);
      y[1] = signe(x[1])? evallgef(4) | evalvarn(v) | evalsigne(1)
                        : evallgef(4) | evalvarn(v);
        y[2]=zero; y[3]=lcopy(x); return y;
      }
      if (vx<v)
      {
      p1=cgetg(v+2,t_VEC);
      for (i=0; i<vx; i++)   p1[i+1] = lpolx[i];
      for (i=vx+1; i<v; i++) p1[i+1] = lpolx[i];
        p1[v+1] = lpolx[vx]; p1[vx+1] = lpolx[v];
      y=integ(changevar(x,p1),vx); tetpil=avma;
      return gerepile(av,tetpil,changevar(y,p1));
      }

      tx = typ(x[1]); i = is_scalar_t(tx)? 0: lgef(x[1])-3;
      tx = typ(x[2]); j = is_scalar_t(tx)? 0: lgef(x[2])-3;
      n = i+j + 2;
      y=gdiv(gtrunc(gmul((GEN)x[2], integ(tayl(x,v,n),v))), (GEN)x[2]);
      if (!gegal(deriv(y,v),x)) err(inter2);
      if (typ(y)==t_RFRAC && lgef(y[1]) == lgef(y[2]))
      {
        GEN p2;
      tx=typ(y[1]); p1=is_scalar_t(tx)? (GEN)y[1]: leading_term(y[1]);
      tx=typ(y[2]); p2=is_scalar_t(tx)? (GEN)y[2]: leading_term(y[2]);
      y=gsub(y, gdiv(p1,p2));
      }
      return gerepileupto(av,y);

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); y=cgetg(lx,tx);
      for (i=1; i<lg(x); i++) y[i]=linteg((GEN)x[i],v);
      return y;
  }
  err(typeer,"integ");
  return NULL; /* not reached */
}

/*******************************************************************/
/*                                                                 */
/*                    PARTIES ENTIERES                             */
/*                                                                 */
/*******************************************************************/

GEN
gfloor(GEN x)
{
  GEN y;
  long i,lx, tx = typ(x);

  switch(tx)
  {
    case t_INT: case t_POL:
      return gcopy(x);

    case t_REAL:
      return mpent(x);

    case t_FRAC: case t_FRACN:
      return truedvmdii((GEN)x[1],(GEN)x[2],NULL);

    case t_RFRAC: case t_RFRACN:
      return gdeuc((GEN)x[1],(GEN)x[2]);

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); y=cgetg(lx,tx);
      for (i=1; i<lx; i++) y[i]=lfloor((GEN)x[i]);
      return y;
  }
  err(typeer,"gfloor");
  return NULL; /* not reached */
}

GEN
gfrac(GEN x)
{
  long av = avma,tetpil;
  GEN p1 = gneg_i(gfloor(x));

  tetpil=avma; return gerepile(av,tetpil,gadd(x,p1));
}

GEN
gceil(GEN x)
{
  GEN y,p1;
  long i,lx,tx=typ(x),av,tetpil;

  switch(tx)
  {
    case t_INT: case t_POL:
      return gcopy(x);

    case t_REAL:
      av=avma; y=mpent(x);
      if (!gegal(x,y))
      {
        tetpil=avma; return gerepile(av,tetpil,addsi(1,y));
      }
      return y;

    case t_FRAC: case t_FRACN:
      av=avma; y=dvmdii((GEN)x[1],(GEN)x[2],&p1);
      if (p1 != gzero && gsigne(x)>0)
      {
        cgiv(p1); tetpil=avma;
        return gerepile(av,tetpil,addsi(1,y));
      }
      return y;

    case t_RFRAC: case t_RFRACN:
      return gdeuc((GEN)x[1],(GEN)x[2]);

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); y=cgetg(lx,tx);
      for (i=1; i<lx; i++) y[i]=lceil((GEN)x[i]);
      return y;
  }
  err(typeer,"gceil");
  return NULL; /* not reached */
}

GEN
round0(GEN x, GEN *pte)
{
  if (pte) { long e; x = grndtoi(x,&e); *pte = stoi(e); }
  return ground(x);
}

GEN
ground(GEN x)
{
  GEN y,p1;
  long i,lx,tx=typ(x),av,tetpil;

  switch(tx)
  {
    case t_INT: case t_INTMOD: case t_QUAD:
      return gcopy(x);

    case t_REAL:
    {
      long ex, s = signe(x);
      if (s==0 || (ex=expo(x)) < -1) return gzero;
      if (ex < 0) return s>0? gun: negi(gun);
      av=avma; p1 = realun(3 + (ex>>TWOPOTBITS_IN_LONG));
      setexpo(p1,-1); /* p1 = 0.5 */
      p1 = addrr(x,p1); tetpil = avma;
      return gerepile(av,tetpil,mpent(p1));
    }
    case t_FRAC: case t_FRACN:
      av=avma; p1 = addii(shifti((GEN)x[2], -1), (GEN)x[1]);
      return gerepileuptoint(av, truedvmdii(p1, (GEN)x[2], NULL));

    case t_POLMOD: y=cgetg(3,t_POLMOD);
      copyifstack(x[1],y[1]);
      y[2]=lround((GEN)x[2]); return y;

    case t_COMPLEX: 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);
      y=cgetg(lx,tx);
      for (i=1; i<lontyp[tx]; i++) y[i]=x[i];
      for (   ; i<lx; i++) y[i]=lround((GEN)x[i]);
      if (tx==t_POL) return normalizepol_i(y, lx);
      if (tx==t_SER) return normalize(y);
      return y;
  }
  err(typeer,"ground");
  return NULL; /* not reached */
}

/* e = number of error bits on integral part */
GEN
grndtoi(GEN x, long *e)
{
  GEN y,p1;
  long i,tx=typ(x), lx,av,ex,e1;

  *e = -HIGHEXPOBIT;
  switch(tx)
  {
    case t_INT: case t_INTMOD: case t_QUAD:
    case t_FRAC: case t_FRACN:
      return ground(x);

    case t_REAL:
      av=avma; p1=gadd(ghalf,x); ex=expo(p1);
      if (ex<0)
      {
      if (signe(p1)>=0) { *e=expo(x); avma=av; return gzero; }
        *e=expo(addsr(1,x)); avma=av; return negi(gun);
      }
      lx=lg(x); e1 = ex - bit_accuracy(lx) + 1;
      settyp(p1,t_INT); setlgefint(p1,lx);
      y=shifti(p1,e1); if (signe(x)<0) y=addsi(-1,y);
      y = gerepileupto(av,y);

      if (e1<=0) { av=avma; e1=expo(subri(x,y)); avma=av; }
      *e=e1; return y;

    case t_POLMOD: y=cgetg(3,t_POLMOD);
      copyifstack(x[1],y[1]);
      y[2]=lrndtoi((GEN)x[2],e); return y;

    case t_COMPLEX: 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);
      y=cgetg(lx,tx);
      for (i=1; i<lontyp[tx]; i++) y[i]=x[i];
      for (   ; i<lx; i++)
      {
        y[i]=lrndtoi((GEN)x[i],&e1);
        if (e1>*e) *e=e1;
      }
      return y;
  }
  err(typeer,"grndtoi");
  return NULL; /* not reached */
}

/* e = number of error bits on integral part */
GEN
gcvtoi(GEN x, long *e)
{
  long tx=typ(x), lx,i,ex,av,e1;
  GEN y;

  *e = -HIGHEXPOBIT;
  if (tx == t_REAL)
  {
    long x0,x1;
    ex=expo(x); if (ex<0) { *e=ex; return gzero; }
    lx=lg(x); e1 = ex - bit_accuracy(lx) + 1;
    x0=x[0]; x1=x[1]; settyp(x,t_INT); setlgefint(x,lx);
    y=shifti(x,e1); x[0]=x0; x[1]=x1;
    if (e1<=0) { av=avma; e1=expo(subri(x,y)); avma=av; }
    *e=e1; return y;
  }
  if (is_matvec_t(tx))
  {
    lx=lg(x); y=cgetg(lx,tx);
    for (i=1; i<lx; i++)
    {
      y[i]=lcvtoi((GEN)x[i],&e1);
      if (e1>*e) *e=e1;
    }
    return y;
  }
  return gtrunc(x);
}

GEN
gtrunc(GEN x)
{
  long tx=typ(x),av,tetpil,i,v;
  GEN y;

  switch(tx)
  {
    case t_INT: case t_POL:
      return gcopy(x);

    case t_REAL:
      return mptrunc(x);

    case t_FRAC: case t_FRACN:
      return divii((GEN)x[1],(GEN)x[2]);

    case t_PADIC:
      if (!signe(x[4])) return gzero;
      v = valp(x);
      if (!v) return gcopy((GEN)x[4]);
      if (v>0)
      { /* here p^v is an integer */
        av=avma; y=gpuigs((GEN)x[2],v); tetpil=avma;
        return gerepile(av,tetpil, mulii(y,(GEN)x[4]));
      }
      y=cgetg(3,t_FRAC);
      y[1]=licopy((GEN)x[4]);
      y[2]=lpuigs((GEN)x[2],-v);
      return y;

    case t_RFRAC: case t_RFRACN:
      return gdeuc((GEN)x[1],(GEN)x[2]);

    case t_SER:
      return gconvsp(x,1);

    case t_VEC: case t_COL: case t_MAT:
    {
      long lx=lg(x);

      y=cgetg(lx,tx);
      for (i=1; i<lx; i++) y[i]=ltrunc((GEN)x[i]);
      return y;
    }
  }
  err(typeer,"gtrunc");
  return NULL; /* not reached */
}

GEN
trunc0(GEN x, GEN *pte)
{
  if (pte) { long e; x = gcvtoi(x,&e); *pte = stoi(e); }
  return gtrunc(x);
}

/*******************************************************************/
/*                                                                 */
/*           CONVERSIONS GEN -->  POLYNOMIALS & SERIES             */
/*                                                                 */
/*******************************************************************/

GEN
zeropol(long v)
{
  GEN x = cgetg(2,t_POL);
  x[1] = evallgef(2) | evalvarn(v); return x;
}

/* return a_(n-1) x^(n-1) + ... + a_0 */
GEN
coefs_to_pol(long n, ...)
{
  va_list ap;
  GEN x, y;
  long i;
  va_start(ap,n);
  x = cgetg(n+2, t_POL); y = x + 2;
  x[1] = evallgef(n+2) | evalvarn(0);
  for (i=n-1; i >= 0; i--) y[i] = (long) va_arg(ap, GEN);
  return normalizepol(x);
}

GEN
scalarpol(GEN x, long v)
{
  GEN y=cgetg(3,t_POL);
  y[1] = gcmp0(x)? evallgef(3) | evalvarn(v)
                 : evallgef(3) | evalvarn(v) | evalsigne(1);
  y[2]=lcopy(x); return y;
}
/*FIXME:suppress poldeg1 in polarit2.c*/
/* deg1pol(a,b,x)=a*x+b*/
GEN
deg1pol(GEN x1, GEN x0,long v)
{
  GEN x = cgetg(4,t_POL);
  x[1] = evalsigne(1) | evalvarn(v) | evallgef(4);
  x[2] = lcopy(x0); x[3] = lcopy(x1); return normalizepol_i(x,4);
}

static GEN
gtopoly0(GEN x, long v, int reverse)
{
  long tx=typ(x),lx,i,j;
  GEN y;

  if (v<0) v = 0;
  if (isexactzero(x)) return zeropol(v);
  if (is_scalar_t(tx)) return scalarpol(x,v);

  switch(tx)
  {
    case t_POL:
      y=gcopy(x); break;
    case t_SER:
      y=gconvsp(x,1);
      if (typ(y) != t_POL)
        err(talker,"t_SER with negative valuation in gtopoly");
      break;
    case t_RFRAC: case t_RFRACN:
      y=gdeuc((GEN)x[1],(GEN)x[2]); break;
    case t_QFR: case t_QFI: case t_VEC: case t_COL: case t_MAT:
      lx=lg(x);
      if (reverse)
      {
      while (lx-- && isexactzero((GEN)x[lx]));
      i=lx+2; y=cgetg(i,t_POL);
      y[1]=evallgef(i); if (!gcmp0(x)) y[1] |= evalsigne(1);
      for (j=2; j<i; j++) y[j]=lcopy((GEN)x[j-1]);
      }
      else
      {
      i=1; j=lx; while (lx-- && isexactzero((GEN)x[i++]));
      i=lx+2; y=cgetg(i,t_POL);
      y[1]=evallgef(i); if (!gcmp0(x)) y[1] |= evalsigne(1);
      lx = j-1;
      for (j=2; j<i; j++) y[j]=lcopy((GEN)x[lx--]);
      }
      break;
    default: err(typeer,"gtopoly");
      return NULL; /* not reached */
  }
  setvarn(y,v); return y;
}

GEN
gtopolyrev(GEN x, long v) { return gtopoly0(x,v,1); }

GEN
gtopoly(GEN x, long v) { return gtopoly0(x,v,0); }

GEN
zeroser(long v, long val)
{
  GEN x = cgetg(2,t_SER);
  x[1]=evalvalp(val) | evalvarn(v); return x;
}

GEN
scalarser(GEN x, long v, long prec)
{
  GEN y=cgetg(prec+2,t_SER);
  long i;

  y[1]=evalsigne(1) | evalvalp(0) | evalvarn(v);
  y[2]=lcopy(x); for (i=3; i<=prec+1; i++) y[i]=zero;
  return y;
}

GEN
gtoser(GEN x, long v)
{
  long tx=typ(x),lx,i,j,l,av,tetpil;
  GEN y,p1,p2;

  if (v<0) v = 0;
  if (tx==t_SER) { y=gcopy(x); setvarn(y,v); return y; }
  if (isexactzero(x)) return zeroser(v,precdl);
  if (is_scalar_t(tx)) return scalarser(x,v,precdl);

  switch(tx)
  {
    case t_POL:
      lx=lgef(x); i=2; while (i<lx && gcmp0((GEN)x[i])) i++;
      l=lx-i; if (precdl>l) l=precdl;
      y=cgetg(l+2,t_SER);
      y[1] = evalsigne(1) | evalvalp(i-2) | evalvarn(v);
      for (j=2; j<=lx-i+1; j++) y[j]=lcopy((GEN)x[j+i-2]);
      for (   ; j<=l+1;    j++) y[j]=zero;
      break;

    case t_RFRAC: case t_RFRACN:
      av=avma; p1=gtoser((GEN)x[1],v); p2=gtoser((GEN)x[2],v);
      tetpil=avma; return gerepile(av,tetpil,gdiv(p1,p2));

    case t_QFR: case t_QFI: case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); i=1; while (i<lx && isexactzero((GEN)x[i])) i++;
      y = cgetg(lx-i+2,t_SER);
      y[1] = evalsigne(1) | evalvalp(i-1) | evalvarn(v);
      for (j=2; j<=lx-i+1; j++) y[j]=lcopy((GEN)x[j+i-2]);
      break;

    default: err(typeer,"gtoser");
      return NULL; /* not reached */
  }
  return y;
}

GEN
gtovec(GEN x)
{
  long tx,lx,i;
  GEN y;

  if (!x) return cgetg(1,t_VEC);
  tx = typ(x);
  if (is_scalar_t(tx) || is_rfrac_t(tx) || tx==t_STR)
  {
    y=cgetg(2,t_VEC); y[1]=lcopy(x);
    return y;
  }
  if (is_graphicvec_t(tx))
  {
    lx=lg(x); y=cgetg(lx,t_VEC);
    for (i=1; i<lx; i++) y[i]=lcopy((GEN)x[i]);
    return y;
  }
  if (tx==t_POL)
  {
    lx=lgef(x); y=cgetg(lx-1,t_VEC);
    for (i=1; i<=lx-2; i++) y[i]=lcopy((GEN)x[lx-i]);
    return y;
  }
  if (tx==t_LIST)
  {
    lx=lgef(x); y=cgetg(lx-1,t_VEC); x++;
    for (i=1; i<=lx-2; i++) y[i]=lcopy((GEN)x[i]);
    return y;
  }
  if (tx == t_VECSMALL)
  {
    lx=lg(x); y=cgetg(lx,t_VEC);
    for (i=1; i<lx; i++) y[i]=lstoi(x[i]);
    return y;
  }
  if (!signe(x)) { y=cgetg(2,t_VEC); y[1]=zero; return y; }
  lx=lg(x); y=cgetg(lx-1,t_VEC); x++;
  for (i=1; i<=lx-2; i++) y[i]=lcopy((GEN)x[i]);
  return y;
}

GEN
compo(GEN x, long n)
{
  long l,tx=typ(x);

  if (tx==t_POL && n+1 >= lgef(x)) return gzero;
  if (tx==t_SER && !signe(x)) return gzero;
  if (!is_recursive_t(tx))
    err(talker, "this object doesn't have components (is a leaf)");
  l=lontyp[tx]+n-1;
  if (n<1 || l>=lg(x))
    err(talker,"nonexistent component");
  return gcopy((GEN)x[l]);
}

/* with respect to the main variable if v<0, with respect to the variable v
   otherwise. v ignored if x is not a polynomial/series. */

GEN
polcoeff0(GEN x, long n, long v)
{
  long tx=typ(x),lx,ex,w,av,tetpil;
  GEN xinit;

  if (is_scalar_t(tx)) return n? gzero: gcopy(x);

  switch(tx)
  {
    case t_QFR: case t_QFI: case t_VEC: case t_COL: case t_MAT:
      if (n<1 || n>=lg(x)) break;
      return gcopy((GEN)x[n]);

    case t_POL:
      if (n<0) return gzero;
      w=varn(x);
      if (v < 0 || v == w)
      return (n>=lgef(x)-2)? gzero: gcopy((GEN)x[n+2]);
      if (v < w) return n? gzero: gcopy(x);
      av=avma; xinit=x;
      x=gsubst(gsubst(x,w,polx[MAXVARN]),v,polx[0]);
      if (gvar(x)) { avma=av; return n? gzero: gcopy(xinit); }
      if (typ(x) == t_POL)
      {
        if (n>=lgef(x)-2) { avma=av; return gzero; }
        x = (GEN)x[n+2];
      }
      else
        x = polcoeff0(x, n, 0);
      tetpil=avma; return gerepile(av,tetpil,gsubst(x,MAXVARN,polx[w]));

    case t_SER:
      w=varn(x);
      if (v < 0 || v == w)
      {
      if (!signe(x)) return gzero;
      lx=lg(x); ex=valp(x); if (n<ex) return gzero;
      if (n>=ex+lx-2) break;
      return gcopy((GEN)x[n-ex+2]);
      }
      if (v < w) return n?  gzero: gcopy(x);
      av=avma; xinit=x;
      x=gsubst(gsubst(x,w,polx[MAXVARN]),v,polx[0]);
      if (gvar(x)) { avma=av; return n? gzero: gcopy(xinit); }
      if (gcmp0(x)) { avma=av; return gzero; }
      if (typ(x) == t_SER)
      {
        lx=lg(x); ex=valp(x); if (n<ex) return gzero;
        if (n>=ex+lx-2) break;
        x = (GEN)x[n-ex+2];
      }
      else
        x = polcoeff0(x, n, 0);
      tetpil=avma; return gerepile(av,tetpil,gsubst(x,MAXVARN,polx[w]));

    case t_RFRAC: case t_RFRACN:
      w = precdl; av = avma;
      if (v<0) v = gvar(x);
      ex = ggval((GEN)x[2], polx[v]);
      precdl = n + ex + 1; x = gtoser(x, v); precdl = w;
      return gerepileupto(av, polcoeff0(x, n, v));
  }
  err(talker,"nonexistent component in truecoeff");
  return NULL; /* not reached */
}

GEN
truecoeff(GEN x, long n)
{
  return polcoeff0(x,n,-1);
}

GEN
denom(GEN x)
{
  long lx,i,av,tetpil;
  GEN s,t;

  switch(typ(x))
  {
    case t_INT: case t_REAL: case t_INTMOD: case t_PADIC: case t_SER:
      return gun;

    case t_FRAC: case t_FRACN:
      return absi((GEN)x[2]);

    case t_COMPLEX:
      av=avma; t=denom((GEN)x[1]); s=denom((GEN)x[2]); tetpil=avma;
      return gerepile(av,tetpil,glcm(s,t));

    case t_QUAD:
      av=avma; t=denom((GEN)x[2]); s=denom((GEN)x[3]); tetpil=avma;
      return gerepile(av,tetpil,glcm(s,t));

    case t_POLMOD:
      return denom((GEN)x[2]);

    case t_RFRAC: case t_RFRACN:
      return gcopy((GEN)x[2]);

    case t_POL:
      return polun[varn(x)];

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); if (lx==1) return gun;
      av = tetpil = avma; s = denom((GEN)x[1]);
      for (i=2; i<lx; i++)
      {
        t = denom((GEN)x[i]);
        /* t != gun est volontaire */
        if (t != gun) { tetpil=avma; s=glcm(s,t); }
      }
      return gerepile(av,tetpil,s);
  }
  err(typeer,"denom");
  return NULL; /* not reached */
}

GEN
numer(GEN x)
{
  long av,tetpil;
  GEN s;

  switch(typ(x))
  {
    case t_INT: case t_REAL: case t_INTMOD:
    case t_PADIC: case t_POL: case t_SER:
      return gcopy(x);

    case t_FRAC: case t_FRACN:
      return (signe(x[2])>0)? gcopy((GEN)x[1]): gneg((GEN)x[1]);

    case t_POLMOD:
      av=avma; s=numer((GEN)x[2]); tetpil=avma;
      return gerepile(av,tetpil,gmodulcp(s,(GEN)x[1]));

    case t_RFRAC: case t_RFRACN:
      return gcopy((GEN)x[1]);

    case t_COMPLEX: case t_QUAD: case t_VEC: case t_COL: case t_MAT:
      av=avma; s=denom(x); tetpil=avma;
      return gerepile(av,tetpil,gmul(s,x));
  }
  err(typeer,"numer");
  return NULL; /* not reached */
}

/* Lift only intmods if v does not occur in x, lift with respect to main
 * variable of x if v < 0, with respect to variable v otherwise.
 */
GEN
lift0(GEN x, long v)
{
  long lx,tx=typ(x),i;
  GEN y;

  switch(tx)
  {
    case t_INT: case t_REAL:
      return gcopy(x);

    case t_INTMOD:
      return gcopy((GEN)x[2]);

    case t_POLMOD:
      if (v < 0 || v == varn((GEN)x[1])) return gcopy((GEN)x[2]);
      y = cgetg(3,tx);
      y[1] = (long)lift0((GEN)x[1],v);
      y[2] = (long)lift0((GEN)x[2],v); return y;

    case t_SER:
      if (!signe(x)) return gcopy(x);
      lx=lg(x); y=cgetg(lx,tx); y[1]=x[1];
      for (i=2; i<lx; i++) y[i]=(long)lift0((GEN)x[i],v);
      return y;

    case t_FRAC: case t_FRACN: case t_COMPLEX: case t_RFRAC:
    case t_RFRACN: case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); y=cgetg(lx,tx);
      for (i=1; i<lx; i++) y[i]=(long)lift0((GEN)x[i],v);
      return y;

    case t_POL:
      y=cgetg(lx=lgef(x),tx); y[1]=x[1];
      for (i=2; i<lx; i++) y[i]=(long)lift0((GEN)x[i],v);
      return y;

    case t_QUAD:
      y=cgetg(4,tx); copyifstack(x[1],y[1]);
      for (i=2; i<4; i++) y[i]=(long)lift0((GEN)x[i],v);
      return y;
  }
  err(typeer,"lift");
  return NULL; /* not reached */
}

GEN
lift(GEN x)
{
  return lift0(x,-1);
}

/* same as lift, without copy. May DESTROY x. For internal use only.
   Conventions on v as for lift. */
GEN
lift_intern0(GEN x, long v)
{
  long i,lx,tx=typ(x);

  switch(tx)
  {
    case t_INT: case t_REAL:
      return x;

    case t_INTMOD:
      return (GEN)x[2];

    case t_POLMOD:
      if (v < 0 || v == varn((GEN)x[1])) return (GEN)x[2];
      x[1]=(long)lift_intern0((GEN)x[1],v);
      x[2]=(long)lift_intern0((GEN)x[2],v);
      return x;

    case t_SER: if (!signe(x)) return x; /* fall through */
    case t_FRAC: case t_FRACN: case t_COMPLEX: case t_QUAD: case t_POL:
    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 = lx-1; i>=lontyp[tx]; i--)
        x[i] = (long) lift_intern0((GEN)x[i],v);
      return x;
  }
  err(typeer,"lift_intern");
  return NULL; /* not reached */
}

/* memes conventions pour v que lift */
GEN
centerlift0(GEN x, long v)
{
  long lx,tx=typ(x),i,av;
  GEN y;

  switch(tx)
  {
    case t_INT:
      return icopy(x);

    case t_INTMOD:
      av=avma; i=cmpii(shifti((GEN)x[2],1),(GEN)x[1]); avma=av;
      return (i>0)? subii((GEN)x[2],(GEN)x[1]): icopy((GEN)x[2]);

    case t_POLMOD:
      if (v < 0 || v == varn((GEN)x[1])) return gcopy((GEN)x[2]);
      y=cgetg(3,tx);
      y[1]=(long)centerlift0((GEN)x[1],v); y[2]=(long)centerlift0((GEN)x[2],v);
      return y;

    case t_SER:
      if (!signe(x)) return gcopy(x);
      lx=lg(x); y=cgetg(lx,tx); y[1]=x[1];
      for (i=2; i<lx; i++) y[i]=(long)centerlift0((GEN)x[i],v);
      return y;

    case t_POL:
    case t_FRAC: case t_FRACN: case t_COMPLEX: case t_RFRAC:
    case t_RFRACN: case t_VEC: case t_COL: case t_MAT:
      lx = (tx==t_POL)? lgef(x): lg(x);
      y=cgetg(lx,tx); y[1]=x[1];
      for (i=lontyp[tx]; i<lx; i++) y[i]=(long)centerlift0((GEN)x[i],v);
      return y;

    case t_QUAD:
      y=cgetg(4,tx); copyifstack(x[1],y[1]);
      for (i=2; i<4; i++) y[i]=(long)centerlift0((GEN)x[i],v);
      return y;
  }
  err(typeer,"centerlift");
  return NULL; /* not reached */
}

GEN
centerlift(GEN x)
{
  return centerlift0(x,-1);
}

/*******************************************************************/
/*                                                                 */
/*                  PARTIES REELLE ET IMAGINAIRES                  */
/*                                                                 */
/*******************************************************************/

static GEN
op_ReIm(GEN f(GEN), GEN x)
{
  long lx,i,j,av, tx = typ(x);
  GEN z;

  switch(tx)
  {
    case t_POL:
      lx=lgef(x); av=avma;
      for (i=lx-1; i>=2; i--)
        if (!gcmp0(f((GEN)x[i]))) break;
      avma=av; if (i==1) return zeropol(varn(x));

      z=cgetg(i+1,tx); z[1]=evalsigne(1)|evallgef(1+i)|evalvarn(varn(x));
      for (j=2; j<=i; j++) z[j] = (long)f((GEN)x[j]);
      return z;

    case t_SER:
      if (gcmp0(x)) { z=cgetg(2,tx); z[1]=x[1]; return z; }
      lx=lg(x); av=avma;
      for (i=2; i<lx; i++)
        if (!gcmp0(f((GEN)x[i]))) break;
      avma=av; if (i==lx) return zeroser(varn(x),lx-2+valp(x));

      z=cgetg(lx-i+2,tx); z[1]=x[1]; setvalp(z, valp(x)+i-2);
      for (j=2; i<lx; j++,i++) z[j] = (long) f((GEN)x[i]);
      return z;

    case t_RFRAC: case t_RFRACN:
    {
      GEN dxb, n, d;
      av = avma; dxb = gconj((GEN)x[2]);
      n = gmul((GEN)x[1], dxb);
      d = gmul((GEN)x[2], dxb);
      return gerepileupto(av, gdiv(f(n), d));
    }

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); z=cgetg(lx,tx);
      for (i=1; i<lx; i++) z[i] = (long) f((GEN)x[i]);
      return z;
  }
  err(typeer,"greal/gimag");
  return NULL; /* not reached */
}

GEN
greal(GEN x)
{
  switch(typ(x))
  {
    case t_INT: case t_REAL: case t_FRAC: case t_FRACN:
      return gcopy(x);

    case t_COMPLEX:
      return gcopy((GEN)x[1]);

    case t_QUAD:
      return gcopy((GEN)x[2]);
  }
  return op_ReIm(greal,x);
}

GEN
gimag(GEN x)
{
  switch(typ(x))
  {
    case t_INT: case t_REAL: case t_FRAC: case t_FRACN:
      return gzero;

    case t_COMPLEX:
      return gcopy((GEN)x[2]);

    case t_QUAD:
      return gcopy((GEN)x[3]);
  }
  return op_ReIm(gimag,x);
}

/*******************************************************************/
/*                                                                 */
/*                     LOGICAL OPERATIONS                          */
/*                                                                 */
/*******************************************************************/
static long
_egal(GEN x, GEN y)
{
  long av = avma;
  long r = gegal(simplify_i(x), simplify_i(y));
  avma = av; return r;
}

GEN
glt(GEN x, GEN y) { return gcmp(x,y)<0? gun: gzero; }

GEN
gle(GEN x, GEN y) { return gcmp(x,y)<=0? gun: gzero; }

GEN
gge(GEN x, GEN y) { return gcmp(x,y)>=0? gun: gzero; }

GEN
ggt(GEN x, GEN y) { return gcmp(x,y)>0? gun: gzero; }

GEN
geq(GEN x, GEN y) { return _egal(x,y)? gun: gzero; }

GEN
gne(GEN x, GEN y) { return _egal(x,y)? gzero: gun; }

GEN
gand(GEN x, GEN y) { return gcmp0(x)? gzero: (gcmp0(y)? gzero: gun); }

GEN
gor(GEN x, GEN y) { return gcmp0(x)? (gcmp0(y)? gzero: gun): gun; }

GEN
gnot(GEN x) { return gcmp0(x)? gun: gzero; }

/*******************************************************************/
/*                                                                 */
/*                      FORMAL SIMPLIFICATIONS                     */
/*                                                                 */
/*******************************************************************/

GEN
geval(GEN x)
{
  long av,tetpil,lx,i, tx = typ(x);
  GEN y,z;

  if (is_const_t(tx)) return gcopy(x);
  if (is_graphicvec_t(tx) || tx == t_RFRACN)
  {
    lx=lg(x); y=cgetg(lx, tx);
    for (i=1; i<lx; i++) y[i] = (long)geval((GEN)x[i]);
    return y;
  }

  switch(tx)
  {
    case t_STR:
      return flisexpr(GSTR(x));

    case t_POLMOD: y=cgetg(3,tx);
      y[1]=(long)geval((GEN)x[1]);
      av=avma; z=geval((GEN)x[2]); tetpil=avma;
      y[2]=lpile(av,tetpil,gmod(z,(GEN)y[1]));
      return y;

    case t_POL:
      lx=lgef(x); if (lx==2) return gzero;
      {
#define initial_value(ep) (GEN)((ep)+1) /* cf anal.h */
        entree *ep = varentries[varn(x)];
        if (!ep) return gcopy(x);
        z = (GEN)ep->value;
        if (gegal(x, initial_value(ep))) return gcopy(z);
#undef initial_value
      }
      y=gzero; av=avma;
      for (i=lx-1; i>1; i--)
        y = gadd(geval((GEN)x[i]), gmul(z,y));
      return gerepileupto(av, y);

    case t_SER:
      err(impl, "evaluation of a power series");

    case t_RFRAC:
      return gdiv(geval((GEN)x[1]),geval((GEN)x[2]));
  }
  err(typeer,"geval");
  return NULL; /* not reached */
}

GEN
simplify_i(GEN x)
{
  long tx=typ(x),i,lx;
  GEN y;

  switch(tx)
  {
    case t_INT: case t_REAL: case t_FRAC:
    case t_INTMOD: case t_PADIC: case t_QFR: case t_QFI:
    case t_LIST: case t_STR: case t_VECSMALL:
      return x;

    case t_FRACN:
      return gred(x);

    case t_COMPLEX:
      if (isexactzero((GEN)x[2])) return simplify_i((GEN)x[1]);
      y=cgetg(3,t_COMPLEX);
      y[1]=(long)simplify_i((GEN)x[1]);
      y[2]=(long)simplify_i((GEN)x[2]); return y;

    case t_QUAD:
      if (isexactzero((GEN)x[3])) return simplify_i((GEN)x[2]);
      y=cgetg(4,t_QUAD);
      y[1]=x[1];
      y[2]=(long)simplify_i((GEN)x[2]);
      y[3]=(long)simplify_i((GEN)x[3]); return y;

    case t_POLMOD: y=cgetg(3,t_POLMOD);
      y[1]=(long)simplify_i((GEN)x[1]);
      i = typ(y[1]);
      if (i != t_POL)
      {
        if (i == t_INT) settyp(y, t_INTMOD);
        else y[1] = x[1]; /* invalid object otherwise */
      }
      y[2]=lmod(simplify_i((GEN)x[2]),(GEN)y[1]); return y;

    case t_POL:
      lx=lgef(x); if (lx==2) return gzero;
      if (lx==3) return simplify_i((GEN)x[2]);
      y=cgetg(lx,t_POL); y[1]=x[1];
      for (i=2; i<lx; i++) y[i]=(long)simplify_i((GEN)x[i]);
      return y;

    case t_SER:
      if (!signe(x)) return gcopy(x);
      lx=lg(x);
      y=cgetg(lx,t_SER); y[1]=x[1];
      for (i=2; i<lx; i++) y[i]=(long)simplify_i((GEN)x[i]);
      return y;

    case t_RFRAC: y=cgetg(3,t_RFRAC);
      y[1]=(long)simplify_i((GEN)x[1]);
      y[2]=(long)simplify_i((GEN)x[2]); return y;

    case t_RFRACN: y=cgetg(3,t_RFRAC);
      y[1]=(long)simplify_i((GEN)x[1]);
      y[2]=(long)simplify_i((GEN)x[2]); return gred_rfrac(y);

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); y=cgetg(lx,tx);
      for (i=1; i<lx; i++) y[i]=(long)simplify_i((GEN)x[i]);
      return y;
  }
  err(typeer,"simplify_i");
  return NULL; /* not reached */
}

GEN
simplify(GEN x)
{
  long av = avma;
  return gerepileupto(av, gcopy(simplify_i(x)));
}

/*******************************************************************/
/*                                                                 */
/*                EVALUATION OF SOME SIMPLE OBJECTS                */
/*                                                                 */
/*******************************************************************/
static GEN
qfeval0_i(GEN q, GEN x, long n)
{
  long i,j,av=avma;
  GEN res=gzero;

  for (i=2;i<n;i++)
    for (j=1;j<i;j++)
      res = gadd(res, gmul(gcoeff(q,i,j), mulii((GEN)x[i],(GEN)x[j])) );
  res=gshift(res,1);
  for (i=1;i<n;i++)
    res = gadd(res, gmul(gcoeff(q,i,i), sqri((GEN)x[i])) );
  return gerepileupto(av,res);
}

#if 0
static GEN
qfeval0(GEN q, GEN x, long n)
{
  long i,j,av=avma;
  GEN res=gzero;

  for (i=2;i<n;i++)
    for (j=1;j<i;j++)
      res = gadd(res, gmul(gcoeff(q,i,j), gmul((GEN)x[i],(GEN)x[j])) );
  res=gshift(res,1);
  for (i=1;i<n;i++)
    res = gadd(res, gmul(gcoeff(q,i,i), gsqr((GEN)x[i])) );
  return gerepileupto(av,res);
}
#else
static GEN
qfeval0(GEN q, GEN x, long n)
{
  long i,j,av=avma;
  GEN res = gmul(gcoeff(q,1,1), gsqr((GEN)x[1]));

  for (i=2; i<n; i++)
  {
    GEN l = (GEN)q[i];
    GEN sx = gmul((GEN)l[1], (GEN)x[1]);
    for (j=2; j<i; j++)
      sx = gadd(sx, gmul((GEN)l[j],(GEN)x[j]));
    sx = gadd(gshift(sx,1), gmul((GEN)l[i],(GEN)x[i]));

    res = gadd(res, gmul((GEN)x[i], sx));
  }
  return gerepileupto(av,res);
}
#endif

/* We assume q is a real symetric matrix */
GEN
qfeval(GEN q, GEN x)
{
  long n=lg(q);

  if (n==1)
  {
    if (typ(q) != t_MAT || lg(x) != 1)
      err(talker,"invalid data in qfeval");
    return gzero;
  }
  if (typ(q) != t_MAT || lg(q[1]) != n)
    err(talker,"invalid quadratic form in qfeval");
  if (typ(x) != t_COL || lg(x) != n)
    err(talker,"invalid vector in qfeval");

  return qfeval0(q,x,n);
}

/* the Horner-type evaluation (mul x 2/3) would force us to use gmul and not
 * mulii (more than 4 x slower for small entries). Not worth it.
 */
static GEN
qfbeval0_i(GEN q, GEN x, GEN y, long n)
{
  long i,j,av=avma;
  GEN res = gmul(gcoeff(q,1,1), mulii((GEN)x[1],(GEN)y[1]));

  for (i=2;i<n;i++)
  {
    for (j=1;j<i;j++)
    {
      GEN p1 = addii(mulii((GEN)x[i],(GEN)y[j]), mulii((GEN)x[j],(GEN)y[i]));
      res = gadd(res, gmul(gcoeff(q,i,j),p1));
    }
    res = gadd(res, gmul(gcoeff(q,i,i), mulii((GEN)x[i],(GEN)y[i])));
  }
  return gerepileupto(av,res);
}

#if 0
static GEN
qfbeval0(GEN q, GEN x, GEN y, long n)
{
  long i,j,av=avma;
  GEN res = gmul(gcoeff(q,1,1), gmul((GEN)x[1],(GEN)y[1]));

  for (i=2;i<n;i++)
  {
    for (j=1;j<i;j++)
    {
      GEN p1 = gadd(gmul((GEN)x[i],(GEN)y[j]), gmul((GEN)x[j],(GEN)y[i]));
      res = gadd(res, gmul(gcoeff(q,i,j),p1));
    }
    res = gadd(res, gmul(gcoeff(q,i,i), gmul((GEN)x[i],(GEN)y[i])));
  }
  return gerepileupto(av,res);
}
#else
static GEN
qfbeval0(GEN q, GEN x, GEN y, long n)
{
  long i,j,av=avma;
  GEN res = gmul(gcoeff(q,1,1), gmul((GEN)x[1], (GEN)y[1]));

  for (i=2; i<n; i++)
  {
    GEN l = (GEN)q[i];
    GEN sx = gmul((GEN)l[1], (GEN)y[1]);
    GEN sy = gmul((GEN)l[1], (GEN)x[1]);
    for (j=2; j<i; j++)
    {
      sx = gadd(sx, gmul((GEN)l[j],(GEN)y[j]));
      sy = gadd(sy, gmul((GEN)l[j],(GEN)x[j]));
    }
    sx = gadd(sx, gmul((GEN)l[i],(GEN)y[i]));

    sx = gmul((GEN)x[i], sx);
    sy = gmul((GEN)y[i], sy);
    res = gadd(res, gadd(sx,sy));
  }
  return gerepileupto(av,res);
}
#endif

/* We assume q is a real symetric matrix */
GEN
qfbeval(GEN q, GEN x, GEN y)
{
  long n=lg(q);

  if (n==1)
  {
    if (typ(q) != t_MAT || lg(x) != 1 || lg(y) != 1)
      err(talker,"invalid data in qfbeval");
    return gzero;
  }
  if (typ(q) != t_MAT || lg(q[1]) != n)
    err(talker,"invalid quadratic form in qfbeval");
  if (typ(x) != t_COL || lg(x) != n || typ(y) != t_COL || lg(y) != n)
    err(talker,"invalid vector in qfbeval");

  return qfbeval0(q,x,y,n);
}

/* yield X = M'.q.M, assuming q is symetric.
 * X_ij are X_ji identical, not copies
 * if flag is set, M has integer entries
 */
GEN
qf_base_change(GEN q, GEN M, int flag)
{
  long i,j, k = lg(M), n=lg(q);
  GEN res = cgetg(k,t_MAT);
  GEN (*qf)(GEN,GEN,long)  = flag? qfeval0_i:  qfeval0;
  GEN (*qfb)(GEN,GEN,GEN,long) = flag? qfbeval0_i: qfbeval0;

  if (n==1)
  {
    if (typ(q) != t_MAT || k != 1)
      err(talker,"invalid data in qf_base_change");
    return res;
  }
  if (typ(M) != t_MAT || k == 1 || lg(M[1]) != n)
    err(talker,"invalid base change matrix in qf_base_change");

  for (i=1;i<k;i++)
  {
    res[i] = lgetg(k,t_COL);
    coeff(res,i,i) = (long) qf(q,(GEN)M[i],n);
  }
  for (i=2;i<k;i++)
    for (j=1;j<i;j++)
      coeff(res,i,j)=coeff(res,j,i) = (long) qfb(q,(GEN)M[i],(GEN)M[j],n);
  return res;
}

/* compute M'.M */
GEN
gram_matrix(GEN M)
{
  long n=lg(M),i,j,k, av;
  GEN res = cgetg(n,t_MAT),p1;

  if (n==1)
  {
    if (typ(M) != t_MAT)
      err(talker,"invalid data in gram_matrix");
    return res;
  }
  if (typ(M) != t_MAT || lg(M[1]) != n)
    err(talker,"not a square matrix in gram_matrix");

  for (i=1;i<n;i++) res[i]=lgetg(n,t_COL);
  av=avma;
  for (i=1;i<n;i++)
  {
    p1 = gzero;
    for (k=1;k<n;k++)
      p1 = gadd(p1, gsqr(gcoeff(M,k,i)));
    coeff(res,i,i) = (long) gerepileupto(av,p1);
    av=avma;
  }
  for (i=2;i<n;i++)
    for (j=1;j<i;j++)
    {
      p1=gzero;
      for (k=1;k<n;k++)
      p1 = gadd(p1, gmul(gcoeff(M,k,i),gcoeff(M,k,j)));
      coeff(res,i,j)=coeff(res,j,i) = lpileupto(av,p1);
      av=avma;
    }
  return res;
}

/* return Re(x * y), x and y scalars */
GEN
mul_real(GEN x, GEN y)
{
  if (typ(x) == t_COMPLEX)
  {
    if (typ(y) == t_COMPLEX)
    {
      long av=avma, tetpil;
      GEN p1 = gmul((GEN)x[1], (GEN) y[1]);
      GEN p2 = gneg(gmul((GEN)x[2], (GEN) y[2]));
      tetpil=avma; return gerepile(av,tetpil,gadd(p1,p2));
    }
    x = (GEN)x[1];
  }
  else if (typ(y) == t_COMPLEX) y = (GEN)y[1];
  return gmul(x,y);
}

/* Compute Re(x * y), x and y matrices of compatible dimensions
 * assume lx, ly > 1, and scalar entries */
GEN
mulmat_real(GEN x, GEN y)
{
  long av,i,j,k, lx = lg(x), ly = lg(y), l = lg(x[1]);
  GEN p1, z = cgetg(ly,t_MAT);

  for (j=1; j<ly; j++)
  {
    z[j] = lgetg(l,t_COL);
    for (i=1; i<l; i++)
    {
      p1 = gzero; av=avma;
      for (k=1; k<lx; k++)
        p1 = gadd(p1, mul_real(gcoeff(x,i,k),gcoeff(y,k,j)));
      coeff(z,i,j)=lpileupto(av, p1);
    }
  }
  return z;
}

static GEN
hqfeval0(GEN q, GEN x, long n)
{
  long i,j, av=avma;
  GEN res=gzero;

  for (i=2;i<n;i++)
    for (j=1;j<i;j++)
    {
      GEN p1 = gmul((GEN)x[i],gconj((GEN)x[j]));
      res = gadd(res, mul_real(gcoeff(q,i,j),p1));
    }
  res=gshift(res,1);
  for (i=1;i<n;i++)
    res = gadd(res, gmul(gcoeff(q,i,i), gnorm((GEN)x[i])) );
  return gerepileupto(av,res);
}

/* We assume q is a hermitian complex matrix */
GEN
hqfeval(GEN q, GEN x)
{
  long n=lg(q);

  if (n==1)
  {
    if (typ(q) != t_MAT || lg(x) != 1)
      err(talker,"invalid data in hqfeval");
    return gzero;
  }
  if (typ(q) != t_MAT || lg(q[1]) != n)
    err(talker,"invalid quadratic form in hqfeval");
  if (typ(x) != t_COL || lg(x) != n)
    err(talker,"invalid vector in hqfeval");

  return hqfeval0(q,x,n);
}

GEN
poleval(GEN x, GEN y)
{
  long av,tetpil,i,j,imin,tx=typ(x);
  GEN p1,p2,p3,r,s;

  if (is_scalar_t(tx)) return gcopy(x);
  switch(tx)
  {
    case t_POL:
      i=lgef(x)-1; imin=2; break;

    case t_RFRAC: case t_RFRACN: av=avma;
      p1=poleval((GEN)x[1],y);
      p2=poleval((GEN)x[2],y); tetpil=avma;
      return gerepile(av,tetpil,gdiv(p1,p2));

    case t_VEC: case t_COL:
      i=lg(x)-1; imin=1; break;
    default: err(typeer,"poleval");
      return NULL; /* not reached */
  }
  if (i<=imin)
    return (i==imin)? gcopy((GEN)x[imin]): gzero;

  av=avma; p1=(GEN)x[i]; i--;
  if (typ(y)!=t_COMPLEX)
  {
#if 0 /* standard Horner's rule */
    for ( ; i>=imin; i--)
      p1 = gadd(gmul(p1,y),(GEN)x[i]);
#endif
    /* specific attention to sparse polynomials */
    for ( ; i>=imin; i=j-1)
    {
      for (j=i; gcmp0((GEN)x[j]); j--)
        if (j==imin)
        {
          if (i!=j) y = gpuigs(y,i-j+1);
          tetpil=avma; return gerepile(av,tetpil,gmul(p1,y));
        }
      r = (i==j)? y: gpuigs(y,i-j+1);
      p1 = gadd(gmul(p1,r), (GEN)x[j]);
    }
    return gerepileupto(av,p1);
  }

  p2=(GEN)x[i]; i--; r=gtrace(y); s=gneg_i(gnorm(y));
  for ( ; i>=imin; i--)
  {
    p3=gadd(p2,gmul(r,p1));
    p2=gadd((GEN)x[i],gmul(s,p1)); p1=p3;
  }
  p1=gmul(y,p1); tetpil=avma;
  return gerepile(av,tetpil,gadd(p1,p2));
}

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