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/*
Copyright (C) 2013 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 "acb.h"
void
acb_sqrt(acb_t y, const acb_t x, slong prec)
{
arb_t r, t, u;
slong wp;
int done;
#define a acb_realref(x)
#define b acb_imagref(x)
#define c acb_realref(y)
#define d acb_imagref(y)
if (arb_is_zero(b))
{
if (arb_is_nonnegative(a))
{
arb_sqrt(c, a, prec);
arb_zero(d);
return;
}
else if (arb_is_nonpositive(a))
{
arb_neg(d, a);
arb_sqrt(d, d, prec);
arb_zero(c);
return;
}
}
if (arb_is_zero(a))
{
if (arb_is_nonnegative(b))
{
arb_mul_2exp_si(c, b, -1);
arb_sqrt(c, c, prec);
arb_set(d, c);
return;
}
else if (arb_is_nonpositive(b))
{
arb_mul_2exp_si(c, b, -1);
arb_neg(c, c);
arb_sqrt(c, c, prec);
arb_neg(d, c);
return;
}
}
wp = prec + 4;
arb_init(r);
arb_init(t);
arb_init(u);
/* r = |a+bi| */
acb_abs(r, x, wp);
done = 0;
if (arf_sgn(arb_midref(a)) >= 0)
{
arb_add(t, r, a, wp);
if (arb_rel_accuracy_bits(t) > 8)
{
/* sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i */
arb_mul_2exp_si(u, t, 1);
arb_sqrt(u, u, wp);
arb_div(d, b, u, prec);
arb_set_round(c, u, prec);
arb_mul_2exp_si(c, c, -1);
done = 1;
}
else
{
arb_sub(u, r, a, wp);
}
}
else if (!arb_contains_zero(b))
{
arb_sub(u, r, a, wp);
if (arb_rel_accuracy_bits(u) > 8)
{
/* sqrt(a+bi) = |b|/sqrt(2*(r-a)) + sgn(b)*sqrt((r-a)/2)*i */
int sgn = arf_sgn(arb_midref(b));
arb_mul_2exp_si(t, u, 1);
arb_sqrt(t, t, wp);
arb_div(c, b, t, prec);
arb_abs(c, c);
arb_set_round(d, t, prec);
arb_mul_2exp_si(d, d, -1);
if (sgn < 0)
arb_neg(d, d);
done = 1;
}
else
{
arb_add(t, r, a, wp);
}
}
else
{
arb_add(t, r, a, wp);
arb_sub(u, r, a, wp);
}
/* t = r+a, u = r-a */
if (!done)
{
/* sqrt(a+bi) = sqrt((r+a)/2) + (b/|b|)*sqrt((r-a)/2)*i
(sign) */
arb_mul_2exp_si(t, t, -1);
arb_mul_2exp_si(u, u, -1);
arb_sqrtpos(c, t, prec);
if (arb_is_nonnegative(b))
{
arb_sqrtpos(d, u, prec);
}
else if (arb_is_nonpositive(b))
{
arb_sqrtpos(d, u, prec);
arb_neg(d, d);
}
else
{
arb_sqrtpos(t, u, wp);
arb_neg(u, t);
arb_union(d, t, u, prec);
}
}
arb_clear(r);
arb_clear(t);
arb_clear(u);
#undef a
#undef b
#undef c
#undef d
}
void
acb_sqrt_analytic(acb_t res, const acb_t z, int analytic, slong prec)
{
if (analytic && arb_contains_zero(acb_imagref(z)) &&
!arb_is_positive(acb_realref(z)))
{
acb_indeterminate(res);
}
else
{
acb_sqrt(res, z, prec);
}
}