自学教程：C++ typ函数代码示例

GENnffactormod(GEN nf, GEN x, GEN pr){  long j, l, vx = varn(x), vn;  pari_sp av = avma;  GEN F, E, rep, xrd, modpr, T, p;  nf = checknf(nf);  vn = varn(nf[1]);  if (typ(x)!=t_POL) pari_err(typeer,"nffactormod");  if (varncmp(vx,vn) >= 0)    pari_err(talker,"polynomial variable must have highest priority in nffactormod");  modpr = nf_to_ff_init(nf, &pr, &T, &p);  xrd = modprX(x, nf, modpr);  rep = FqX_factor(xrd,T,p);  settyp(rep, t_MAT);  F = gel(rep,1); l = lg(F);  E = gel(rep,2); settyp(E, t_COL);  for (j = 1; j < l; j++) {    gel(F,j) = modprX_lift(gel(F,j), modpr);    gel(E,j) = stoi(E[j]);  }  return gerepilecopy(av, rep);}

voidarginit(void){	int i;	lastadj = Fadj;	lastverb = Fverb;	indchar = typ(TIND, types[TCHAR]);	memset(flagbits, Fnone, sizeof(flagbits));	for(i='0'; i<='9'; i++)		argflag(i, Fignor);	argflag('.', Fignor);	argflag('#', Fignor);	argflag('u', Fignor);	argflag('+', Fignor);	argflag('-', Fignor);	argflag('*', Fstar);	argflag('l', Fl);	argflag('o', Fverb);	flagbits['x'] = flagbits['o'];	flagbits['X'] = flagbits['o'];}

static GENbound_for_coeff(long m, GEN rr, GEN *maxroot){  long i,r1, lrr=lg(rr);  GEN p1,b1,b2,B,M, C = matpascal(m-1);  for (r1=1; r1 < lrr; r1++)    if (typ(rr[r1]) != t_REAL) break;  r1--;  rr = gabs(rr,0); *maxroot = vecmax(rr);  for (i=1; i<lrr; i++)    if (gcmp(gel(rr,i), gen_1) < 0) gel(rr,i) = gen_1;  for (b1=gen_1,i=1; i<=r1; i++) b1 = gmul(b1, gel(rr,i));  for (b2=gen_1    ; i<lrr; i++) b2 = gmul(b2, gel(rr,i));  B = gmul(b1, gsqr(b2)); /* Mahler measure */  M = cgetg(m+2, t_VEC); gel(M,1) = gel(M,2) = gen_0; /* unused */  for (i=1; i<m; i++)  {    p1 = gadd(gmul(gcoeff(C, m, i+1), B),/* binom(m-1, i)   */              gcoeff(C, m, i));          /* binom(m-1, i-1) */    gel(M,i+2) = ceil_safe(p1);  }  return M;}

static GENF2xqE_dbl_slope(GEN P, GEN a, GEN T, GEN *slope){  GEN x, y, Q;  if (ell_is_inf(P)) return ellinf();  x = gel(P,1); y = gel(P,2);  if (typ(a)==t_VECSMALL)  {    GEN a2 = a;    if (!lgpol(gel(P,1))) return ellinf();    *slope = F2x_add(x, F2xq_div(y, x, T));    Q = cgetg(3,t_VEC);    gel(Q, 1) = F2x_add(F2xq_sqr(*slope, T), F2x_add(*slope, a2));    gel(Q, 2) = F2x_add(F2xq_mul(*slope, F2x_add(x, gel(Q, 1)), T), F2x_add(y, gel(Q, 1)));  }  else  {    GEN a3 = gel(a,1), a4 = gel(a,2), a3i = gel(a,3);    *slope = F2xq_mul(F2x_add(a4, F2xq_sqr(x, T)), a3i, T);    Q = cgetg(3,t_VEC);    gel(Q, 1) = F2xq_sqr(*slope, T);    gel(Q, 2) = F2x_add(F2xq_mul(*slope, F2x_add(x, gel(Q, 1)), T), F2x_add(y, a3));  }  return Q;}

static voidcommands(long n){  long i;  entree *ep;  char **t_L;  pari_stack s_L;  pari_stack_init(&s_L, sizeof(*t_L), (void**)&t_L);  for (i = 0; i < functions_tblsz; i++)    for (ep = functions_hash[i]; ep; ep = ep->next)    {      long m;      switch (EpVALENCE(ep))      {        case EpVAR:          if (typ((GEN)ep->value) == t_CLOSURE) break;          /* fall through */        case EpNEW: continue;      }      m = ep->menu;      if (m == n || (n < 0 && m && m <= MAX_SECTION))        pari_stack_pushp(&s_L, (void*)ep->name);    }  pari_stack_pushp(&s_L, NULL);  print_fun_list(t_L, term_height()-4);  pari_stack_delete(&s_L);}

GENbnflogdegree(GEN nf, GEN A, GEN ell){  pari_sp av = avma;  GEN AZ, A0Z, NA0;  long vAZ;  if (typ(ell) != t_INT) pari_err_TYPE("bnflogdegree", ell);  nf = checknf(nf);  A = idealhnf(nf, A);  AZ = gcoeff(A,1,1);  vAZ = Z_pvalrem(AZ, ell, &A0Z);  if (is_pm1(A0Z))    NA0 = gen_1;  else    (void)Z_pvalrem(idealnorm(nf,A), ell, &NA0);  if (vAZ)  {    GEN Aell = ZM_hnfmodid(A, powiu(ell,vAZ));    GEN S = idealprimedec(nf, ell), T;    long l, i, s = 0;    T = padicfact(nf, S, 100);    l = lg(S);    for (i = 1; i < l; i++)    {      GEN P = gel(S,i);      long v = idealval(nf, Aell, P);      if (v) s += v * ftilde(nf, P, gel(T,i));    }    if (s) NA0 = gmul(NA0, gpowgs(ell1(ell), s));  }  return gerepileupto(av, NA0);}

static GENpolsimplify(GEN x){  long i,lx = lg(x);  for (i=2; i<lx; i++)    if (typ(x[i]) == t_POL) gel(x,i) = constant_term(gel(x,i));  return x;}

static GENF2xqE_neg_i(GEN P, GEN a){  GEN LHS;  if (ell_is_inf(P)) return P;  LHS = typ(a)==t_VECSMALL ? gel(P,1): gel(a,1);  return mkvec2(gel(P,1), F2x_add(LHS, gel(P,2)));}

/* min(v, v(Log_p Norm_{F_/p/Q_p}(x))) */static longvlognorm(GEN nf, GEN T, GEN x, GEN p, long v){  GEN a = nf_to_scalar_or_alg(nf, x);  GEN N = RgXQ_norm(a, T);  if (typ(N) != t_PADIC) N = cvtop(N, p, v);  return minss(v, valp( Qp_log(N) ));}

static longvnorm_x(GEN nf, GEN x, GEN ell){  x = nf_to_scalar_or_alg(nf,x);  if (typ(x) != t_POL) return 0;  x = Q_primpart(x);  return Q_pval(nfnorm(nf,x), ell);}

GENF2xqE_neg(GEN P, GEN a, GEN T){  GEN LHS;  (void) T;  if (ell_is_inf(P)) return ellinf();  LHS = typ(a)==t_VECSMALL ? gel(P,1): gel(a,1);  return mkvec2(gcopy(gel(P,1)), F2x_add(LHS, gel(P,2)));}

longnfislocalpower(GEN nf, GEN pr, GEN a, GEN n){  pari_sp av = avma;  long r;  if (typ(n) != t_INT) pari_err_TYPE("nfislocalpower",n);  r = nfislocalpower_i(nf, pr, a, n);  avma = av; return r;}

/* assume x is squarefree */intnfissplit(GEN nf, GEN x){  pari_sp av = avma;  long l;  if (typ(x) != t_POL) pari_err(typeer, "nfissplit");  l = lg(nfsqff(checknf(nf), x, 2));  avma = av; return l != 1;}

GENstrtor(const char *s, long prec){  pari_sp av = avma;  GEN y = int_read_dec(&s);  y = real_read(av, &s, y, prec);  if (typ(y) == t_REAL) return y;  return gerepileuptoleaf(av, itor(y, prec));}

static longvtilde_prec_x(GEN nf, GEN x, GEN ell){  long i, l, v;  GEN G;  if (typ(x) != t_MAT) return vnorm_x(nf,x,ell);  G = gel(x,1); l = lg(G); v = 0;  for (i = 1; i < l; i++) v = maxss(v, vnorm_x(nf,gel(G,i),ell));  return v;}

static GENnf_to_Zq(GEN x, GEN T, GEN pk, GEN pks2, GEN proj){  GEN y;  if (typ(x) != t_COL) return centermodii(x, pk, pks2);  y = gmul(proj, x);  if (!T) return centermodii(y, pk, pks2);  y = RgV_to_RgX(y, varn(T));  return centermod_i(FpX_rem(y, T, pk), pk, pks2);}

/* assume x in Fq[X], return Tr_{Fq[X]/Fp[X]}(x), varn(X) = 0 */static GENpoltrace(GEN x, GEN Trq, GEN p){  long i,l;  GEN y;  if (typ(x) == t_INT || varn(x) != 0) return trace(x, Trq, p);  l = lg(x); y = cgetg(l,t_POL); y[1]=x[1];  for (i=2; i<l; i++) gel(y,i) = trace(gel(x,i),Trq,p);  return y;}

/* We want to be able to reconstruct x, |x|^2 < C, from x mod pr^k */static doublebestlift_bound(GEN C, long d, double alpha, GEN Npr){  const double y = 1 / (alpha - 0.25); /* = 2 if alpha = 3/4 */  double t;  if (typ(C) != t_REAL) C = gmul(C, real_1(DEFAULTPREC));  setlg(C, DEFAULTPREC);  t = rtodbl(mplog(gmul2n(divrs(C,d), 4))) * 0.5 + (d-1) * log(1.5 * sqrt(y));  return ceil((t * d) / log(gtodouble(Npr)));}

static GENunifpol0(GEN nf,GEN x,long flag){  switch(typ(x))  {    case t_INT: case t_FRAC:      return gcopy(x);    case t_POLMOD:      x = gel(x,2); /* fall through */      if (typ(x) != t_POL) return gcopy(x); /* scalar */    case t_POL:      if (!degpol(x)) return gcopy(constant_term(x));      return (flag == t_COL)? algtobasis(nf, x): gmodulo(x, gel(nf,1));    default: /* t_COL */      return (flag == t_COL)? gcopy(x): basistoalg(nf, x);  }}

/* Return a subfield, gen_0 [ change p ] or NULL [ not a subfield ] */static GENsubfield(GEN A, blockdata *B){  long N, i, j, d, lf, m = lg(A)-1;  GEN M, pe, pol, fhk, g, e, d_1_term, delta, listdelta, whichdelta;  GEN T = B->S->T, p = B->S->p, firstroot = B->S->firstroot;  pol= (GEN)B->DATA[1]; N = degpol(pol); d = N/m; /* m | N */  pe = (GEN)B->DATA[2];  fhk= (GEN)B->DATA[3];  M  = (GEN)B->DATA[8];  delta = cgetg(m+1,t_VEC);  whichdelta = cgetg(N+1, t_VECSMALL);  d_1_term = gen_0;  for (i=1; i<=m; i++)  {    GEN Ai = gel(A,i), p1 = (GEN)fhk[Ai[1]];    for (j=2; j<=d; j++)      p1 = Fq_mul(p1, (GEN)fhk[Ai[j]], T, pe);    gel(delta,i) = p1;    if (DEBUGLEVEL>2) fprintferr("delta[%ld] = %Z/n",i,p1);    /* g = prod (X - delta[i])     * if g o h = 0 (pol), we'll have h(Ai[j]) = delta[i] for all j */    /* fk[k] belongs to block number whichdelta[k] */    for (j=1; j<=d; j++) whichdelta[Ai[j]] = i;    if (typ(p1) == t_POL) p1 = constant_term(p1);    d_1_term = addii(d_1_term, p1);  }  d_1_term = centermod(d_1_term, pe); /* Tr(g) */  if (absi_cmp(d_1_term, gel(M,3)) > 0) {    if (DEBUGLEVEL>1) fprintferr("d-1 test failed/n");    return NULL;  }  g = FqV_roots_to_pol(delta, T, pe, 0);  g = centermod(polsimplify(g), pe); /* assume g in Z[X] */  if (DEBUGLEVEL>2) fprintferr("pol. found = %Z/n",g);  if (!ok_coeffs(g,M)) {    if (DEBUGLEVEL>1) fprintferr("coeff too big for pol g(x)/n");    return NULL;  }  if (!FpX_is_squarefree(g, p)) {    if (DEBUGLEVEL>1) fprintferr("changing f(x): p divides disc(g)/n");    compute_data(B);    return subfield(A, B);  }  lf = lg(firstroot); listdelta = cgetg(lf, t_VEC);  for (i=1; i<lf; i++) listdelta[i] = delta[whichdelta[firstroot[i]]];  if (DEBUGLEVEL) fprintferr("candidate = %Z/n", g);  e = embedding(g, B->DATA, B->S, B->PD->den, listdelta);  if (!e) return NULL;  if (DEBUGLEVEL) fprintferr("embedding = %Z/n", e);  return _subfield(g, e);}

/* return d = gcd(a,b), sets u, v such that au + bv = gcd(a,b) */GENextgcd(GEN A, GEN B, GEN *U, GEN *V){  pari_sp av = avma;  GEN ux = gen_1, vx = gen_0, a = A, b = B;  if (typ(a) != t_INT) pari_err_TYPE("extgcd",a);  if (typ(b) != t_INT) pari_err_TYPE("extgcd",b);  if (signe(a) < 0) { a = negi(a); ux = negi(ux); }  while (!gequal0(b))  {    GEN r, q = dvmdii(a, b, &r), v = vx;    vx = subii(ux, mulii(q, vx));    ux = v; a = b; b = r;  }  *U = ux;  *V = diviiexact( subii(a, mulii(A,ux)), B );  gerepileall(av, 3, &a, U, V); return a;}

static GENvtilde(GEN K, GEN x, GEN T, GEN deg, GEN ell, long prec){  GEN G, E, vG;  long i, l;  if (typ(x) != t_MAT) return vtilde_i(K,x,T,deg,ell,prec);  G = gel(x,1); vG = cgetg_copy(G, &l);  E = gel(x,2);  for (i = 1; i < l; i++) gel(vG, i) = vtilde_i(K, gel(G,i),T,deg,ell,prec);  return RgV_dotproduct(E, vG);}

/* assume A or B is a t_LIST */static GENlistconcat(GEN A, GEN B){  long i, l1, lx;  GEN L, z, L1, L2;  if (typ(A) != t_LIST) {    if (list_typ(B)!=t_LIST_RAW) pari_err_TYPE("listconcat",B);    L2 = list_data(B);    if (!L2) return mklistcopy(A);    lx = lg(L2) + 1;    z = listcreate();    list_data(z) = L = cgetg(lx, t_VEC);    for (i = 2; i < lx; i++) gel(L,i) = gcopy(gel(L2,i-1));    gel(L,1) = gcopy(A); return z;  } else if (typ(B) != t_LIST) {    if (list_typ(A)!=t_LIST_RAW) pari_err_TYPE("listconcat",A);    L1 = list_data(A);    if (!L1) return mklistcopy(B);    lx = lg(L1) + 1;    z = listcreate();    list_data(z) = L = cgetg(lx, t_VEC);    for (i = 1; i < lx-1; i++) gel(L,i) = gcopy(gel(L1,i));    gel(L,i) = gcopy(B); return z;  }  /* A, B both t_LISTs */  if (list_typ(A)!=t_LIST_RAW) pari_err_TYPE("listconcat",A);  if (list_typ(B)!=t_LIST_RAW) pari_err_TYPE("listconcat",B);  L1 = list_data(A); if (!L1) return listcopy(B);  L2 = list_data(B); if (!L2) return listcopy(A);  l1 = lg(L1);  lx = l1-1 + lg(L2);  z = cgetg(3, t_LIST);  z[1] = 0UL;  list_data(z) = L = cgetg(lx, t_VEC);  L2 -= l1-1;  for (i=1; i<l1; i++) gel(L,i) = gclone(gel(L1,i));  for (   ; i<lx; i++) gel(L,i) = gclone(gel(L2,i));  return z;}

static GENsubfieldsall(GEN nf){  pari_sp av = avma;  long N, ld, i, v0;  GEN G, pol, dg, LSB, NLSB;  poldata PD;  primedata S;  blockdata B;  /* much easier if nf is Galois (WSS) */  G = galoisconj4(nf, NULL, 1);  if (typ(G) != t_INT)  {    GEN L, S, p;    long l;    pol = get_nfpol(nf, &nf);    L = lift_intern( galoissubfields(G, 0, varn(pol)) );    l = lg(L);    S = cgetg(l, t_VECSMALL);    for (i=1; i<l; i++) S[i] = lg(gmael(L,i,1));    p = vecsmall_indexsort(S);    return gerepilecopy(av,  vecpermute(L, p));  }  subfields_poldata(nf, &PD);  pol = PD.pol;  v0 = varn(pol); N = degpol(pol);  dg = divisors(utoipos(N)); ld = lg(dg)-1;  if (DEBUGLEVEL) fprintferr("/n***** Entering subfields/n/npol = %Z/n",pol);  LSB = _subfield(pol, pol_x[0]);  if (ld > 2)  {    B.PD = &PD;    B.S  = &S;    B.N  = N;    choose_prime(&S, PD.pol, PD.dis);    for (i=2; i<ld; i++)    {      B.size  = itos(gel(dg,i));      B.d = N / B.size;      NLSB = subfields_of_given_degree(&B);      if (NLSB) { LSB = concat(LSB, NLSB); gunclone(NLSB); }    }    (void)delete_var(); /* from choose_prime */  }  LSB = shallowconcat(LSB, _subfield(pol_x[0], pol));  if (DEBUGLEVEL) fprintferr("/n***** Leaving subfields/n/n");  return fix_var(gerepilecopy(av, LSB), v0);}

/* Let x be a polynomial with coefficients in Z or nf (vectors or polymods) * return the same polynomial with coefficients expressed: *  if flag=t_COL: as vectors (on the integral basis). *  if flag=t_POLMOD: as polmods. */GENunifpol(GEN nf, GEN x, long flag){  if (typ(x)==t_POL && varncmp(varn(x), varn(nf[1])) < 0)  {    long i, d = lg(x);    GEN y = cgetg(d,t_POL); y[1] = x[1];    for (i=2; i<d; i++) gel(y,i) = unifpol0(nf, gel(x,i), flag);    return y;  }  return unifpol0(nf, x, flag);}

/* assume x in Fq, return Tr_{Fq/Fp}(x) as a t_INT */static GENtrace(GEN x, GEN Trq, GEN p){  long i, l;  GEN s;  if (typ(x) == t_INT) return modii(mulii(x, gel(Trq,1)), p);  l = lg(x)-1; if (l == 1) return gen_0;  x++; s = mulii(gel(x,1), gel(Trq,1));  for (i=2; i<l; i++)    s = addii(s, mulii(gel(x,i), gel(Trq,i)));  return modii(s, p);}

// <vardeclaration>		::=	<identifier>{ ,<identifier> } : <typ>void SyntaxAnalyzer::varDeclaration(){    accept(ident);        while (symbol == comma) {        accept(comma);        accept(ident);    }        accept(colon);    typ();} // vardeclaration( )

GENshallowextract(GEN x, GEN L){  long i,j, tl = typ(L), tx = typ(x), lx = lg(x);  GEN y;  switch(tx)  {    case t_VEC:    case t_COL:    case t_MAT:    case t_VECSMALL: break;    default: pari_err_TYPE("extract",x);  }  if (tl==t_INT)  { /* extract components of x as per the bits of mask L */    long k, l, ix, iy, maxj;    GEN Ld;    if (!signe(L)) return cgetg(1,tx);    y = new_chunk(lx);    l = lgefint(L)-1; ix = iy = 1;    maxj = BITS_IN_LONG - bfffo(*int_MSW(L));    if ((l-2) * BITS_IN_LONG + maxj >= lx)      pari_err_TYPE("vecextract [mask too large]", L);    for (k = 2, Ld = int_LSW(L); k < l; k++, Ld = int_nextW(Ld))    {      ulong B = *Ld;      for (j = 0; j < BITS_IN_LONG; j++, B >>= 1, ix++)        if (B & 1) y[iy++] = x[ix];    }    { /* k = l */      ulong B = *Ld;      for (j = 0; j < maxj; j++, B >>= 1, ix++)        if (B & 1) y[iy++] = x[ix];    }    y[0] = evaltyp(tx) | evallg(iy);    return y;  }

GENshallowconcat1(GEN x){  pari_sp av = avma;  long lx, t, i;  GEN z;  switch(typ(x))  {    case t_VEC:      lx = lg(x);      if (lx==1) pari_err_DOMAIN("concat","vector","=",x,x);      break;    case t_LIST:      if (list_typ(x)!=t_LIST_RAW) pari_err_TYPE("concat",x);      if (!list_data(x)) pari_err_DOMAIN("concat","vector","=",x,x);      x = list_data(x); lx = lg(x);      break;    default:      pari_err_TYPE("concat",x);      return NULL; /* not reached */  }  if (lx==2) return gel(x,1);  z = gel(x,1); t = typ(z); i = 2;  if (is_matvec_t(t) || t == t_VECSMALL || t == t_STR)  { /* detect a "homogeneous" object: catmany is faster */    for (; i<lx; i++)      if (typ(gel(x,i)) != t) break;    z = catmany(x + 1, x + i-1, t);  }  for (; i<lx; i++) {    z = shallowconcat(z, gel(x,i));    if (gc_needed(av,3))    {      if (DEBUGMEM>1) pari_warn(warnmem,"concat: i = %ld", i);      z = gerepilecopy(av, z);    }  }  return z;}

GENgpolvar(GEN x){  long v;  if (!x) {    GEN h = hash_values(h_polvar);    return vars_to_RgXV(vars_sort_inplace(h));  }  if (typ(x)==t_PADIC) return gcopy( gel(x,2) );  v = gvar(x);  if (v==NO_VARIABLE) return gen_0;  return pol_x(v);}