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/*
Copyright (C) 2020 Fredrik Johansson
This file is part of Calcium.
Calcium 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 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "flint/fmpz_poly_factor.h"
#include "arb_fmpz_poly.h"
#include "qqbar.h"
int
_qqbar_fast_detect_simple_principal_surd(const qqbar_t x)
{
slong d;
d = qqbar_degree(x);
if (d == 1)
return 0;
if (fmpz_sgn(QQBAR_COEFFS(x)) > 0)
return 0;
if (!_fmpz_vec_is_zero(QQBAR_COEFFS(x) + 1, d - 1))
return 0;
/* Slow exact version, but we only want a fast check here. */
/* return qqbar_is_real(x) && qqbar_sgn_re(x) > 0; */
if (arb_is_zero(acb_imagref(QQBAR_ENCLOSURE(x))))
{
if (arb_is_positive(acb_realref(QQBAR_ENCLOSURE(x))))
return 1;
return 0;
}
if (!arb_contains_zero(acb_imagref(QQBAR_ENCLOSURE(x))))
return 0;
/* The imaginary part enclosure may not be exactly zero; we
can still use the enclosure if it is precise enough to guarantee
that there are no collisions with the conjugate roots. */
if (acb_rel_accuracy_bits(QQBAR_ENCLOSURE(x)) > FLINT_BIT_COUNT(d) + 5)
return arb_is_positive(acb_realref(QQBAR_ENCLOSURE(x)));
return 0;
}
void
qqbar_root_ui(qqbar_t res, const qqbar_t x, ulong n)
{
if (n == 0)
{
flint_printf("qqbar_root_ui: n >= 1 is required");
return;
}
else if (n == 1 || qqbar_is_zero(x) || qqbar_is_one(x))
{
qqbar_set(res, x);
}
else
{
slong i, d, prec, found;
fmpz_poly_t H;
fmpz_poly_factor_t fac;
acb_t z, w, t;
int pure_real;
d = qqbar_degree(x);
if (FLINT_BIT_COUNT(n) + FLINT_BIT_COUNT(d) > 30)
{
flint_printf("qqbar_root_ui: ludicrously high degree %wd * %wu", d, n);
return;
}
/* handle principal roots of positive rational numbers */
/* todo: could also handle conjugates of such roots */
if ((d == 1 && (n == 2 || qqbar_sgn_re(x) > 0)) || _qqbar_fast_detect_simple_principal_surd(x))
{
fmpq_t t;
fmpq_init(t);
fmpz_neg(fmpq_numref(t), QQBAR_COEFFS(x));
fmpz_set(fmpq_denref(t), QQBAR_COEFFS(x) + d);
qqbar_fmpq_root_ui(res, t, d * n);
fmpq_clear(t);
return;
}
/* special-case roots of unity */
/* todo: also specialize rational multiples of roots of unity */
{
slong p;
ulong q;
if (qqbar_is_root_of_unity(&p, &q, x))
{
if (2 * p > q)
p -= q;
qqbar_root_of_unity(res, p, q * n);
return;
}
}
fmpz_poly_init(H);
fmpz_poly_factor_init(fac);
acb_init(z);
acb_init(w);
acb_init(t);
for (i = d; i >= 0; i--)
{
fmpz_poly_set_coeff_fmpz(H, i * n, QQBAR_COEFFS(x) + i);
}
fmpz_poly_factor(fac, H);
acb_set(z, QQBAR_ENCLOSURE(x));
pure_real = qqbar_is_real(x);
for (prec = QQBAR_DEFAULT_PREC / 2; ; prec *= 2)
{
_qqbar_enclosure_raw(z, QQBAR_POLY(x), z, prec);
if (pure_real)
arb_zero(acb_imagref(z));
acb_root_ui(w, z, n, prec);
/* Look for potential roots -- we want exactly one */
found = -1;
for (i = 0; i < fac->num && found != -2; i++)
{
arb_fmpz_poly_evaluate_acb(t, fac->p + i, w, prec);
if (acb_contains_zero(t))
{
if (found == -1)
found = i;
else
found = -2;
}
}
/* Check if the enclosure is good enough */
if (found >= 0)
{
if (_qqbar_validate_uniqueness(t, fac->p + found, w, 2 * prec))
{
fmpz_poly_set(QQBAR_POLY(res), fac->p + found);
acb_set(QQBAR_ENCLOSURE(res), t);
break;
}
}
}
fmpz_poly_clear(H);
fmpz_poly_factor_clear(fac);
acb_clear(z);
acb_clear(w);
acb_clear(t);
}
}
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