xref: /netbsd-src/external/lgpl3/mpc/dist/src/rootofunity.c (revision 39f28e1e142c5bfb6be935a49cb55e2287fec7ea)
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