1/* mpc_rootofunity -- primitive root of unity. 2 3Copyright (C) 2012, 2016 INRIA 4 5This file is part of GNU MPC. 6 7GNU MPC is free software; you can redistribute it and/or modify it under 8the terms of the GNU Lesser General Public License as published by the 9Free Software Foundation; either version 3 of the License, or (at your 10option) any later version. 11 12GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY 13WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS 14FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for 15more details. 16 17You should have received a copy of the GNU Lesser General Public License 18along 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 24static unsigned long 25gcd (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 */ 33int 34mpc_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