static trace_data *
init_trace(trace_data *T, GEN S, nflift_t *L, GEN q)
{
long e = gexpo(S), i,j, l,h;
GEN qgood, S1, invd;
if (e < 0) return NULL; /* S = 0 */
qgood = int2n(e - 32); /* single precision check */
if (cmpii(qgood, q) > 0) q = qgood;
S1 = gdivround(S, q);
if (gcmp0(S1)) return NULL;
invd = ginv(itor(L->den, DEFAULTPREC));
T->dPinvS = gmul(L->iprk, S);
l = lg(S);
h = lg(T->dPinvS[1]);
T->PinvSdbl = (double**)cgetg(l, t_MAT);
init_dalloc();
for (j = 1; j < l; j++)
{
double *t = dalloc(h * sizeof(double));
GEN c = gel(T->dPinvS,j);
pari_sp av = avma;
T->PinvSdbl[j] = t;
for (i=1; i < h; i++) t[i] = rtodbl(mpmul(invd, gel(c,i)));
avma = av;
}
T->d = L->den;
T->P1 = gdivround(L->prk, q);
T->S1 = S1; return T;
}
/* We want to be able to reconstruct x, |x|^2 < C, from x mod pr^k */
static double
bestlift_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)));
}
/* Naive recombination of modular factors: combine up to maxK modular
* factors, degree <= klim and divisible by hint
*
* target = polynomial we want to factor
* famod = array of modular factors. Product should be congruent to
* target/lc(target) modulo p^a
* For true factors: S1,S2 <= p^b, with b <= a and p^(b-a) < 2^31 */
static GEN
nfcmbf(nfcmbf_t *T, GEN p, long a, long maxK, long klim)
{
GEN pol = T->pol, nf = T->nf, famod = T->fact, dn = T->dn;
GEN bound = T->bound;
GEN nfpol = gel(nf,1);
long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1, dnf = degpol(nfpol);
GEN res = cgetg(3, t_VEC);
pari_sp av0 = avma;
GEN pk = gpowgs(p,a), pks2 = shifti(pk,-1);
GEN ind = cgetg(lfamod+1, t_VECSMALL);
GEN degpol = cgetg(lfamod+1, t_VECSMALL);
GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
GEN listmod = cgetg(lfamod+1, t_COL);
GEN fa = cgetg(lfamod+1, t_COL);
GEN lc = absi(leading_term(pol)), lt = is_pm1(lc)? NULL: lc;
GEN C2ltpol, C = T->L->topowden, Tpk = T->L->Tpk;
GEN Clt = mul_content(C, lt);
GEN C2lt = mul_content(C,Clt);
const double Bhigh = get_Bhigh(lfamod, dnf);
trace_data _T1, _T2, *T1, *T2;
pari_timer ti;
TIMERstart(&ti);
if (maxK < 0) maxK = lfamod-1;
C2ltpol = C2lt? gmul(C2lt,pol): pol;
{
GEN q = ceil_safe(sqrtr(T->BS_2));
GEN t1,t2, ltdn, lt2dn;
GEN trace1 = cgetg(lfamod+1, t_MAT);
GEN trace2 = cgetg(lfamod+1, t_MAT);
ltdn = mul_content(lt, dn);
lt2dn= mul_content(ltdn, lt);
for (i=1; i <= lfamod; i++)
{
pari_sp av = avma;
GEN P = gel(famod,i);
long d = degpol(P);
degpol[i] = d; P += 2;
t1 = gel(P,d-1);/* = - S_1 */
t2 = gsqr(t1);
if (d > 1) t2 = gsub(t2, gmul2n(gel(P,d-2), 1));
/* t2 = S_2 Newton sum */
t2 = typ(t2)!=t_INT? FpX_rem(t2, Tpk, pk): modii(t2, pk);
if (lt)
{
if (typ(t2)!=t_INT) {
t1 = FpX_red(gmul(ltdn, t1), pk);
t2 = FpX_red(gmul(lt2dn,t2), pk);
} else {
t1 = remii(mulii(ltdn, t1), pk);
t2 = remii(mulii(lt2dn,t2), pk);
}
}
gel(trace1,i) = gclone( nf_bestlift(t1, NULL, T->L) );
gel(trace2,i) = gclone( nf_bestlift(t2, NULL, T->L) ); avma = av;
}
T1 = init_trace(&_T1, trace1, T->L, q);
T2 = init_trace(&_T2, trace2, T->L, q);
for (i=1; i <= lfamod; i++) {
gunclone(gel(trace1,i));
gunclone(gel(trace2,i));
}
}
degsofar[0] = 0; /* sentinel */
/* ind runs through strictly increasing sequences of length K,
* 1 <= ind[i] <= lfamod */
nextK:
if (K > maxK || 2*K > lfamod) goto END;
if (DEBUGLEVEL > 3)
fprintferr("\n### K = %d, %Z combinations\n", K,binomial(utoipos(lfamod), K));
setlg(ind, K+1); ind[1] = 1;
i = 1; curdeg = degpol[ind[1]];
for(;;)
{ /* try all combinations of K factors */
for (j = i; j < K; j++)
{
degsofar[j] = curdeg;
ind[j+1] = ind[j]+1; curdeg += degpol[ind[j+1]];
}
if (curdeg <= klim && curdeg % T->hint == 0) /* trial divide */
{
GEN t, y, q, list;
pari_sp av;
av = avma;
/* d - 1 test */
if (T1)
{
t = get_trace(ind, T1);
if (rtodbl(QuickNormL2(t,DEFAULTPREC)) > Bhigh)
{
if (DEBUGLEVEL>6) fprintferr(".");
avma = av; goto NEXT;
}
}
/* d - 2 test */
if (T2)
{
t = get_trace(ind, T2);
if (rtodbl(QuickNormL2(t,DEFAULTPREC)) > Bhigh)
{
if (DEBUGLEVEL>3) fprintferr("|");
avma = av; goto NEXT;
}
}
avma = av;
y = lt; /* full computation */
for (i=1; i<=K; i++)
{
GEN q = gel(famod, ind[i]);
if (y) q = gmul(y, q);
y = FqX_centermod(q, Tpk, pk, pks2);
}
y = nf_pol_lift(y, bound, T);
if (!y)
{
if (DEBUGLEVEL>3) fprintferr("@");
avma = av; goto NEXT;
}
/* try out the new combination: y is the candidate factor */
q = RgXQX_divrem(C2ltpol, y, nfpol, ONLY_DIVIDES);
if (!q)
{
if (DEBUGLEVEL>3) fprintferr("*");
avma = av; goto NEXT;
}
/* found a factor */
list = cgetg(K+1, t_VEC);
gel(listmod,cnt) = list;
for (i=1; i<=K; i++) list[i] = famod[ind[i]];
y = Q_primpart(y);
gel(fa,cnt++) = QXQX_normalize(y, nfpol);
/* fix up pol */
pol = q;
for (i=j=k=1; i <= lfamod; i++)
{ /* remove used factors */
if (j <= K && i == ind[j]) j++;
else
{
famod[k] = famod[i];
update_trace(T1, k, i);
update_trace(T2, k, i);
degpol[k] = degpol[i]; k++;
}
}
lfamod -= K;
if (lfamod < 2*K) goto END;
i = 1; curdeg = degpol[ind[1]];
if (C2lt) pol = Q_primpart(pol);
if (lt) lt = absi(leading_term(pol));
Clt = mul_content(C, lt);
C2lt = mul_content(C,Clt);
C2ltpol = C2lt? gmul(C2lt,pol): pol;
if (DEBUGLEVEL > 2)
{
fprintferr("\n"); msgTIMER(&ti, "to find factor %Z",y);
fprintferr("remaining modular factor(s): %ld\n", lfamod);
}
continue;
}
NEXT:
for (i = K+1;;)
{
if (--i == 0) { K++; goto nextK; }
if (++ind[i] <= lfamod - K + i)
{
curdeg = degsofar[i-1] + degpol[ind[i]];
if (curdeg <= klim) break;
}
}
}
END:
if (degpol(pol) > 0)
{ /* leftover factor */
if (signe(leading_term(pol)) < 0) pol = gneg_i(pol);
if (C2lt && lfamod < 2*K) pol = QXQX_normalize(Q_primpart(pol), nfpol);
setlg(famod, lfamod+1);
gel(listmod,cnt) = shallowcopy(famod);
gel(fa,cnt++) = pol;
}
if (DEBUGLEVEL>6) fprintferr("\n");
if (cnt == 2) {
avma = av0;
gel(res,1) = mkvec(T->pol);
gel(res,2) = mkvec(T->fact);
}
else
{
setlg(listmod, cnt); setlg(fa, cnt);
gel(res,1) = fa;
gel(res,2) = listmod;
res = gerepilecopy(av0, res);
}
return res;
}
static double
todbl(GEN x) { return rtodbl(gtofp(x, LOWDEFAULTPREC)); }
engine_RawRingElementArrayOrNull rawRoots(const RingElement *p, long prec,
int unique) {
const Ring *R = p->get_ring();
const PolynomialRing *P = R->cast_to_PolynomialRing();
if (P == 0) {
ERROR("expected a polynomial ring");
return NULL;
}
const int n = P->n_vars();
if (n != 1) {
ERROR("expected a univariate polynomial ring");
return NULL;
}
const Ring *K = P->getCoefficients();
int degree = 0;
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
degree = max(degree, abs(*(t->monom)));
}
if (prec == -1) {
prec = (K->get_precision() == 0 ? 53 : K->get_precision());
}
engine_RawRingElementArrayOrNull result = nullptr;
/* Start PARI computations. */
pari_CATCH(e_STACK) {
#ifdef NDEBUG
/*
* Every time the stack is changed PARI writes a message to the file pari_errfile
* which by default is /dev/stderr. To avoid showing this message to the user we
* redirect to /dev/null before the PARI's stack is modified.
*/
FILE *tmp, *dev_null = fopen("/dev/null", "w");
if (dev_null != NULL) {
tmp = pari_errfile;
pari_errfile = dev_null;
}
#endif
allocatemem(0); // passing 0 will double the current stack size.
#ifdef NDEBUG
/*
* We set pari_errfile back to the default value just in case PARI crashes.
*/
if (dev_null != NULL) {
pari_errfile = tmp;
fclose(dev_null);
}
#endif
} pari_RETRY {
const pari_sp av = avma;
GEN q = cgetg(2 + degree + 1, t_POL);
setsigne(q, 1);
setvarn(q, 0);
for (int i = 0; i < degree + 1; ++i) {
gel(q, 2 + i) = gen_0;
}
switch (K->ringID()) {
case M2::ring_ZZ:
ZZ_GMP:
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
gel(q, 2 + abs(*(t->monom))) =
mpz_get_GEN(reinterpret_cast<const mpz_ptr>(t->coeff.poly_val));
}
break;
case M2::ring_QQ:
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
gel(q, 2 + abs(*(t->monom))) =
mpq_get_GEN(reinterpret_cast<const mpq_ptr>(t->coeff.poly_val));
}
break;
case M2::ring_RR:
pari_CATCH(e_OVERFLOW) {
ERROR("coefficient is NaN or Infinity");
avma = av;
return NULL;
}
pari_TRY {
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
gel(q, 2 + abs(*(t->monom))) =
dbltor(*reinterpret_cast<double *>(t->coeff.poly_val));
}
}
pari_ENDCATCH
break;
case M2::ring_CC:
pari_CATCH(e_OVERFLOW) {
ERROR("coefficient is NaN or Infinity");
avma = av;
return NULL;
}
pari_TRY {
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
GEN z = cgetg(3, t_COMPLEX);
gel(z, 1) = dbltor(reinterpret_cast<complex *>(t->coeff.poly_val)->re);
gel(z, 2) = dbltor(reinterpret_cast<complex *>(t->coeff.poly_val)->im);
gel(q, 2 + abs(*(t->monom))) = z;
}
}
pari_ENDCATCH
break;
case M2::ring_RRR:
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
gel(q, 2 + abs(*(t->monom))) =
mpfr_get_GEN(reinterpret_cast<const mpfr_ptr>(t->coeff.poly_val));
}
break;
case M2::ring_CCC:
for (Nterm *t = p->get_value(); t != NULL; t = t->next) {
gel(q, 2 + abs(*(t->monom))) =
mpc_get_GEN(reinterpret_cast<const mpc_ptr>(t->coeff.poly_val));
}
break;
case M2::ring_old:
if (K->is_ZZ()) {
goto ZZ_GMP;
}
default:
ERROR("expected coefficient ring of the form ZZ, QQ, RR or CC");
return NULL;
}
if (unique) {
q = RgX_div(q, RgX_gcd_simple(q, RgX_deriv(q)));
}
GEN roots = cleanroots(q, nbits2prec(prec));
const size_t num_roots = lg(roots) - 1;
result = getmemarraytype(engine_RawRingElementArray, num_roots);
result->len = static_cast<int>(num_roots);
ring_elem m2_root;
if (prec <= 53) {
const RingCC *CC = dynamic_cast<const RingCC *>(IM2_Ring_CCC(prec));
for (int i = 0; i < num_roots; ++i) {
const pari_sp av2 = avma;
GEN pari_root = gel(roots, 1 + i);
const complex root = {rtodbl(greal(pari_root)),
rtodbl(gimag(pari_root))};
CC->ring().to_ring_elem(m2_root, root);
result->array[i] = RingElement::make_raw(CC, m2_root);
avma = av2;
}
} else {
const RingCCC *CCC = dynamic_cast<const RingCCC *>(IM2_Ring_CCC(prec));
for (int i = 0; i < num_roots; ++i) {
const pari_sp av2 = avma;
mpc_t root;
auto root1 = reinterpret_cast<mpfc_t*>(&root);
pari_mpc_init_set_GEN(root, gel(roots, 1 + i), GMP_RNDN);
// CCC->ring().to_ring_elem(m2_root, **reinterpret_cast<mpfc_t*>(&root));
CCC->ring().to_ring_elem(m2_root, **root1);
result->array[i] = RingElement::make_raw(CCC, m2_root);
avma = av2;
}
}
/* End PARI computations. */
avma = av;
}
pari_ENDCATCH
return result;
}
