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/* mpc_rootofunity -- primitive root of unity.
Copyright (C) 2012, 2016 INRIA
This file is part of GNU MPC.
GNU MPC is free software; you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License as published by the
Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
more details.
You should have received a copy of the GNU Lesser General Public License
along with this program. If not, see http://www.gnu.org/licenses/ .
*/
#include "mpc-impl.h"
#include <assert.h>
static unsigned long int
gcd (unsigned long int a, unsigned long int b)
{
if (b == 0)
return a;
else return gcd (b, a % b);
}
/* put in rop the value of exp(2*i*pi*k/n) rounded according to rnd */
int
mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_t rnd)
{
unsigned long int g;
mpq_t kn;
mpfr_t t, s, c;
mpfr_prec_t prec;
int inex_re, inex_im;
mpfr_rnd_t rnd_re, rnd_im;
if (n == 0) {
/* Compute exp (0 + i*inf). */
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
return MPC_INEX (0, 0);
}
/* Eliminate common denominator. */
k %= n;
g = gcd (k, n);
k /= g;
n /= g;
/* We assume that only n=1, 2, 3, 4, 6 and 12 may yield exact results
and treat them separately; n=8 is also treated here for efficiency
reasons. */
if (n == 1)
return mpc_set_ui_ui (rop, 1, 0, rnd);
else if (n == 2)
return mpc_set_si_si (rop, -1, 0, rnd);
else if (n == 4)
if (k == 1)
return mpc_set_ui_ui (rop, 0, 1, rnd);
else
return mpc_set_si_si (rop, 0, -1, rnd);
else if (n == 3 || n == 6) {
inex_re = mpfr_set_si (mpc_realref (rop), (n == 3 ? -1 : 1),
MPC_RND_RE (rnd));
/* inverse the rounding mode for -sqrt(3)/2 for zeta_3^2 and zeta_6^5 */
rnd_im = MPC_RND_IM (rnd);
if (k != 1)
rnd_im = INV_RND (rnd_im);
inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 3, rnd_im);
mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
if (k != 1) {
mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
inex_im = -inex_im;
}
return MPC_INEX (inex_re, inex_im);
}
else if (n == 12) {
/* inverse the rounding mode for -sqrt(3)/2 for zeta_12^5 and zeta_12^7 */
rnd_re = MPC_RND_RE (rnd);
if (k == 5 || k == 7)
rnd_re = INV_RND (rnd_re);
inex_re = mpfr_sqrt_ui (mpc_realref (rop), 3, rnd_re);
inex_im = mpfr_set_si (mpc_imagref (rop), (k == 1 || k == 5 ? 1 : -1),
MPC_RND_IM (rnd));
mpc_div_2ui (rop, rop, 1u, MPC_RNDNN);
if (k == 5 || k == 7) {
mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
inex_re = -inex_re;
}
return MPC_INEX (inex_re, inex_im);
}
else if (n == 8) {
rnd_re = MPC_RND_RE (rnd);
if (k == 3 || k == 5)
rnd_re = INV_RND (rnd_re);
rnd_im = MPC_RND_IM (rnd);
if (k == 5 || k == 7)
rnd_im = INV_RND (rnd_im);
inex_re = mpfr_sqrt_ui (mpc_realref (rop), 2, rnd_re);
inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 2, rnd_im);
mpc_div_2ui (rop, rop, 1u, MPC_RNDNN);
if (k == 3 || k == 5) {
mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
inex_re = -inex_re;
}
if (k == 5 || k == 7) {
mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
inex_im = -inex_im;
}
return MPC_INEX (inex_re, inex_im);
}
prec = MPC_MAX_PREC(rop);
mpfr_init2 (t, 67); /* see the argument at the end of the following loop */
mpfr_init2 (s, 67);
mpfr_init2 (c, 67);
mpq_init (kn);
mpq_set_ui (kn, k, n);
mpq_mul_2exp (kn, kn, 1);
do {
prec += mpc_ceil_log2 (prec) + 5;
mpfr_set_prec (t, prec);
mpfr_set_prec (s, prec);
mpfr_set_prec (c, prec);
mpfr_const_pi (t, MPFR_RNDN); /* error 0.5 ulp */
mpfr_mul_q (t, t, kn, MPFR_RNDN); /* error 2*1.5+0.5=3.5 ulp */
mpfr_sin_cos (s, c, t, MPFR_RNDN);
/* error (1.5*2^{Exp (t) - Exp (s resp. c)} + 0.5) ulp
We have 0<t<2*pi, so Exp (t) <= 3.
Unfortunately s or c can be close to 0, but with n<2^64,
we lose at most about 64 bits:
Where the minimum of s and c over all primitive n-th roots of
unity is reached depends on n mod 4.
To simplify the argument, we will consider the 4*n-th roots of
unity, which contain the n-th roots of unity and which are
symmmetrically distributed with respect to the real and imaginary
axes, so that it becomes enough to consider only s for k=1.
With n<2^64, the absolute values of all s or c are at least
sin (2 * pi * 2^(-64) / 4) > 2^(-64) of exponent at least -63.
So the error is bounded above by
(1.5*2^66+0.5) ulp < 2^67 ulp.
To obtain a more precise bound for smaller n, which is useful
especially at small precision, we may use the error bound of
(1.5*2^(3 - Exp (s or c)) + 0.5) ulp
< 2^(4 - Exp (s or c)) ulp, since Exp (s or c) is at most 0. */
}
while ( !mpfr_can_round (c, prec - (4 - mpfr_get_exp (c)),
MPFR_RNDN, MPFR_RNDZ,
MPC_PREC_RE(rop) + (MPC_RND_RE(rnd) == MPFR_RNDN))
|| !mpfr_can_round (s, prec - (4 - mpfr_get_exp (s)),
MPFR_RNDN, MPFR_RNDZ,
MPC_PREC_IM(rop) + (MPC_RND_IM(rnd) == MPFR_RNDN)));
inex_re = mpfr_set (mpc_realref(rop), c, MPC_RND_RE(rnd));
inex_im = mpfr_set (mpc_imagref(rop), s, MPC_RND_IM(rnd));
mpfr_clear (t);
mpfr_clear (s);
mpfr_clear (c);
mpq_clear (kn);
return MPC_INEX(inex_re, inex_im);
}
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