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/*
Copyright (C) 2021 Fredrik Johansson
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "fmpz_factor.h"
#include "qqbar.h"
/* Write |n| = A * B^2
factoring = 0 -> no factoring except for powers of two
factoring = 1 -> complete factoring
factoring = 2 -> smooth factoring
*/
static void
_fmpz_factor_square_root(fmpz_t A, fmpz_t B, const fmpz_t n, int factoring)
{
if (factoring == 0)
{
slong v;
v = fmpz_val2(n);
fmpz_abs(A, n);
fmpz_one(B);
if (v >= 2)
{
fmpz_tdiv_q_2exp(A, A, v - (v & 1));
fmpz_mul_2exp(B, B, v / 2);
}
}
else
{
fmpz_factor_t fac;
fmpz_t t;
slong i;
fmpz_factor_init(fac);
if (factoring == 1)
fmpz_factor(fac, n);
else
fmpz_factor_smooth(fac, n, 32, -1); /* 32 bit factors, -1 => no primality test */
fmpz_one(A);
fmpz_one(B);
fmpz_init(t);
for (i = 0; i < fac->num; i++)
{
if (fac->exp[i] == 1)
{
fmpz_mul(A, A, fac->p + i);
}
else if (fac->exp[i] == 2)
{
fmpz_mul(B, B, fac->p + i);
}
else
{
fmpz_pow_ui(t, fac->p + i, fac->exp[i] / 2);
fmpz_mul(B, B, t);
if (fac->exp[i] % 2)
fmpz_mul(A, A, fac->p + i);
}
}
fmpz_factor_clear(fac);
fmpz_clear(t);
}
}
void
qqbar_get_quadratic(fmpz_t res_a, fmpz_t res_b, fmpz_t res_c, fmpz_t res_q, const qqbar_t x, int factoring)
{
const fmpz *a, *b, *c;
fmpz_t D;
if (qqbar_degree(x) == 1)
{
fmpz_zero(res_b);
fmpz_zero(res_c);
_qqbar_get_fmpq(res_a, res_q, x);
return;
}
if (qqbar_degree(x) != 2)
{
flint_throw(FLINT_ERROR, "qqbar_get_quadratic: degree 1 or 2 is required\n");
}
a = QQBAR_COEFFS(x) + 2;
b = QQBAR_COEFFS(x) + 1;
c = QQBAR_COEFFS(x) + 0;
/* x = (-b +/- sqrt(b^2 - 4ac))/(2a) */
fmpz_init(D);
fmpz_mul(D, a, c);
fmpz_mul_2exp(D, D, 2);
fmpz_submul(D, b, b);
/* -D is a square <=> element of Q(i) */
if (fmpz_is_square(D))
{
fmpz_sqrt(D, D);
fmpz_set_si(res_c, -1);
if (qqbar_sgn_im(x) > 0)
fmpz_set(res_b, D);
else
fmpz_neg(res_b, D);
fmpz_neg(res_a, b);
fmpz_mul_2exp(res_q, a, 1);
fmpz_gcd(D, res_a, res_b);
fmpz_gcd(D, D, res_q);
if (!fmpz_is_one(D))
{
fmpz_divexact(res_a, res_a, D);
fmpz_divexact(res_b, res_b, D);
fmpz_divexact(res_q, res_q, D);
}
}
else
{
fmpz_t A, B;
fmpz_neg(D, D);
fmpz_init(A);
fmpz_init(B);
/*
sqrt(|D|) = A * B^2 => sqrt(D) = sqrt(A) * B (D > 0)
sqrt(-A) * B (D < 0)
*/
_fmpz_factor_square_root(A, B, D, factoring);
if (fmpz_sgn(D) < 0)
fmpz_neg(A, A);
fmpz_set(res_c, A);
/* x = (-b +/- B*sqrt(A))/(2a) */
fmpz_neg(res_a, b);
fmpz_mul_2exp(res_q, a, 1);
/* determine the correct sign */
if (fmpz_sgn(D) < 0)
{
if (qqbar_sgn_im(x) > 0)
fmpz_set(res_b, B);
else
fmpz_neg(res_b, B);
}
else if (fmpz_is_zero(b))
{
if (qqbar_sgn_re(x) > 0)
fmpz_set(res_b, B);
else
fmpz_neg(res_b, B);
}
else
{
arb_t r1, r2;
slong prec;
arb_init(r1);
arb_init(r2);
for (prec = 64; ; prec *= 2)
{
arb_sqrt_fmpz(r1, A, prec);
arb_mul_fmpz(r1, r1, B, prec);
arb_add_fmpz(r2, r1, b, prec);
arb_neg(r2, r2);
arb_sub_fmpz(r1, r1, b, prec);
arb_div_fmpz(r1, r1, a, prec);
arb_div_fmpz(r2, r2, a, prec);
arb_mul_2exp_si(r1, r1, -1);
arb_mul_2exp_si(r2, r2, -1);
if (arb_overlaps(r1, acb_realref(QQBAR_ENCLOSURE(x))) &&
!arb_overlaps(r2, acb_realref(QQBAR_ENCLOSURE(x))))
{
fmpz_set(res_b, B);
break;
}
if (arb_overlaps(r2, acb_realref(QQBAR_ENCLOSURE(x))) &&
!arb_overlaps(r1, acb_realref(QQBAR_ENCLOSURE(x))))
{
fmpz_neg(res_b, B);
break;
}
}
arb_clear(r1);
arb_clear(r2);
}
fmpz_gcd(D, res_a, res_b);
fmpz_gcd(D, D, res_q);
if (!fmpz_is_one(D))
{
fmpz_divexact(res_a, res_a, D);
fmpz_divexact(res_b, res_b, D);
fmpz_divexact(res_q, res_q, D);
}
fmpz_clear(A);
fmpz_clear(B);
}
fmpz_clear(D);
}