1 /* mpc_rootofunity -- primitive root of unity.
2
3 Copyright (C) 2012, 2016 INRIA
4
5 This file is part of GNU MPC.
6
7 GNU MPC is free software; you can redistribute it and/or modify it under
8 the terms of the GNU Lesser General Public License as published by the
9 Free Software Foundation; either version 3 of the License, or (at your
10 option) any later version.
11
12 GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY
13 WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
14 FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
15 more details.
16
17 You should have received a copy of the GNU Lesser General Public License
18 along with this program. If not, see http://www.gnu.org/licenses/ .
19 */
20
21 #include <stdio.h> /* for MPC_ASSERT */
22 #include "mpc-impl.h"
23
24 static unsigned long
gcd(unsigned long a,unsigned long b)25 gcd (unsigned long a, unsigned long b)
26 {
27 if (b == 0)
28 return a;
29 else return gcd (b, a % b);
30 }
31
32 /* put in rop the value of exp(2*i*pi*k/n) rounded according to rnd */
33 int
mpc_rootofunity(mpc_ptr rop,unsigned long n,unsigned long k,mpc_rnd_t rnd)34 mpc_rootofunity (mpc_ptr rop, unsigned long n, unsigned long k, mpc_rnd_t rnd)
35 {
36 unsigned long g;
37 mpq_t kn;
38 mpfr_t t, s, c;
39 mpfr_prec_t prec;
40 int inex_re, inex_im;
41 mpfr_rnd_t rnd_re, rnd_im;
42
43 if (n == 0) {
44 /* Compute exp (0 + i*inf). */
45 mpfr_set_nan (mpc_realref (rop));
46 mpfr_set_nan (mpc_imagref (rop));
47 return MPC_INEX (0, 0);
48 }
49
50 /* Eliminate common denominator. */
51 k %= n;
52 g = gcd (k, n);
53 k /= g;
54 n /= g;
55
56 /* Now 0 <= k < n and gcd(k,n)=1. */
57
58 /* We assume that only n=1, 2, 3, 4, 6 and 12 may yield exact results
59 and treat them separately; n=8 is also treated here for efficiency
60 reasons. */
61 if (n == 1)
62 {
63 /* necessarily k=0 thus we want exp(0)=1 */
64 MPC_ASSERT (k == 0);
65 return mpc_set_ui_ui (rop, 1, 0, rnd);
66 }
67 else if (n == 2)
68 {
69 /* since gcd(k,n)=1, necessarily k=1, thus we want exp(i*pi)=-1 */
70 MPC_ASSERT (k == 1);
71 return mpc_set_si_si (rop, -1, 0, rnd);
72 }
73 else if (n == 4)
74 {
75 /* since gcd(k,n)=1, necessarily k=1 or k=3, thus we want
76 exp(2*i*pi/4)=i or exp(2*i*pi*3/4)=-i */
77 MPC_ASSERT (k == 1 || k == 3);
78 if (k == 1)
79 return mpc_set_ui_ui (rop, 0, 1, rnd);
80 else
81 return mpc_set_si_si (rop, 0, -1, rnd);
82 }
83 else if (n == 3 || n == 6)
84 {
85 MPC_ASSERT ((n == 3 && (k == 1 || k == 2)) ||
86 (n == 6 && (k == 1 || k == 5)));
87 /* for n=3, necessarily k=1 or k=2: -1/2+/-1/2*sqrt(3)*i;
88 for n=6, necessarily k=1 or k=5: 1/2+/-1/2*sqrt(3)*i */
89 inex_re = mpfr_set_si (mpc_realref (rop), (n == 3 ? -1 : 1),
90 MPC_RND_RE (rnd));
91 /* inverse the rounding mode for -sqrt(3)/2 for zeta_3^2 and zeta_6^5 */
92 rnd_im = MPC_RND_IM (rnd);
93 if (k != 1)
94 rnd_im = INV_RND (rnd_im);
95 inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 3, rnd_im);
96 mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
97 if (k != 1)
98 {
99 mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
100 inex_im = -inex_im;
101 }
102 return MPC_INEX (inex_re, inex_im);
103 }
104 else if (n == 12)
105 {
106 /* necessarily k=1, 5, 7, 11:
107 k=1: 1/2*sqrt(3) + 1/2*I
108 k=5: -1/2*sqrt(3) + 1/2*I
109 k=7: -1/2*sqrt(3) - 1/2*I
110 k=11: 1/2*sqrt(3) - 1/2*I */
111 MPC_ASSERT (k == 1 || k == 5 || k == 7 || k == 11);
112 /* inverse the rounding mode for -sqrt(3)/2 for zeta_12^5 and zeta_12^7 */
113 rnd_re = MPC_RND_RE (rnd);
114 if (k == 5 || k == 7)
115 rnd_re = INV_RND (rnd_re);
116 inex_re = mpfr_sqrt_ui (mpc_realref (rop), 3, rnd_re);
117 inex_im = mpfr_set_si (mpc_imagref (rop), k < 6 ? 1 : -1,
118 MPC_RND_IM (rnd));
119 mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
120 if (k == 5 || k == 7)
121 {
122 mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
123 inex_re = -inex_re;
124 }
125 return MPC_INEX (inex_re, inex_im);
126 }
127 else if (n == 8)
128 {
129 /* k=1, 3, 5 or 7:
130 k=1: (1/2*I + 1/2)*sqrt(2)
131 k=3: (1/2*I - 1/2)*sqrt(2)
132 k=5: -(1/2*I + 1/2)*sqrt(2)
133 k=7: -(1/2*I - 1/2)*sqrt(2) */
134 MPC_ASSERT (k == 1 || k == 3 || k == 5 || k == 7);
135 rnd_re = MPC_RND_RE (rnd);
136 if (k == 3 || k == 5)
137 rnd_re = INV_RND (rnd_re);
138 rnd_im = MPC_RND_IM (rnd);
139 if (k > 4)
140 rnd_im = INV_RND (rnd_im);
141 inex_re = mpfr_sqrt_ui (mpc_realref (rop), 2, rnd_re);
142 inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 2, rnd_im);
143 mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
144 if (k == 3 || k == 5)
145 {
146 mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
147 inex_re = -inex_re;
148 }
149 if (k > 4)
150 {
151 mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
152 inex_im = -inex_im;
153 }
154 return MPC_INEX (inex_re, inex_im);
155 }
156
157 prec = MPC_MAX_PREC(rop);
158
159 /* For the error analysis justifying the following algorithm,
160 see algorithms.tex. */
161 mpfr_init2 (t, 67);
162 mpfr_init2 (s, 67);
163 mpfr_init2 (c, 67);
164 mpq_init (kn);
165 mpq_set_ui (kn, k, n);
166 mpq_mul_2exp (kn, kn, 1); /* kn=2*k/n < 2 */
167
168 do {
169 prec += mpc_ceil_log2 (prec) + 5; /* prec >= 6 */
170
171 mpfr_set_prec (t, prec);
172 mpfr_set_prec (s, prec);
173 mpfr_set_prec (c, prec);
174
175 mpfr_const_pi (t, MPFR_RNDN);
176 mpfr_mul_q (t, t, kn, MPFR_RNDN);
177 mpfr_sin_cos (s, c, t, MPFR_RNDN);
178 }
179 while ( !mpfr_can_round (c, prec - (4 - mpfr_get_exp (c)),
180 MPFR_RNDN, MPFR_RNDZ,
181 MPC_PREC_RE(rop) + (MPC_RND_RE(rnd) == MPFR_RNDN))
182 || !mpfr_can_round (s, prec - (4 - mpfr_get_exp (s)),
183 MPFR_RNDN, MPFR_RNDZ,
184 MPC_PREC_IM(rop) + (MPC_RND_IM(rnd) == MPFR_RNDN)));
185
186 inex_re = mpfr_set (mpc_realref(rop), c, MPC_RND_RE(rnd));
187 inex_im = mpfr_set (mpc_imagref(rop), s, MPC_RND_IM(rnd));
188
189 mpfr_clear (t);
190 mpfr_clear (s);
191 mpfr_clear (c);
192 mpq_clear (kn);
193
194 return MPC_INEX(inex_re, inex_im);
195 }
196