/*============================================================================= This file is part of FLINT. FLINT is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. FLINT 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 General Public License for more details. You should have received a copy of the GNU General Public License along with FLINT; if not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA =============================================================================*/ /****************************************************************************** Copyright (C) 2011 Fredrik Johansson ******************************************************************************/ #include #include "arith.h" #define MAX_32BIT 58 static const int lookup_table[MAX_32BIT][28] = { {-1, 1}, {1, 1}, {1, 2}, {0, 1}, {-1, 2, 4}, {-1, 2}, {-1, -4, 4, 8}, {-1, 0, 2}, {1, -6, 0, 8}, {-1, -2, 4}, {1, 6, -12, -32, 16, 32}, {-3, 0, 4}, {-1, 6, 24, -32, -80, 32, 64}, {1, -4, -4, 8}, {1, 8, -16, -8, 16}, {1, 0, -8, 0, 8}, {1, -8, -40, 80, 240, -192, -448, 128, 256}, {-1, -6, 0, 8}, {1, 10, -40, -160, 240, 672, -448, -1024, 256, 512}, {5, 0, -20, 0, 16}, {1, -16, 32, 48, -96, -32, 64}, {-1, 6, 12, -32, -16, 32}, {-1, -12, 60, 280, -560, -1792, 1792, 4608, -2304, -5120, 1024, 2048}, {1, 0, -16, 0, 16}, {-1, 10, 100, -40, -800, 32, 2240, 0, -2560, 0, 1024}, {-1, -6, 24, 32, -80, -32, 64}, {1, 18, 0, -240, 0, 864, 0, -1152, 0, 512}, {-7, 0, 56, 0, -112, 0, 64}, {-1, 14, 112, -448, -2016, 4032, 13440, -15360, -42240, 28160, 67584, -24576, -53248, 8192, 16384}, {1, -8, -16, 8, 16}, {-1, -16, 112, 672, -2016, -8064, 13440, 42240, -42240, -112640, 67584, 159744, -53248, -114688, 16384, 32768}, {1, 0, -32, 0, 160, 0, -256, 0, 128}, {1, -24, 48, 344, -688, -1088, 2176, 1280, -2560, -512, 1024}, {1, 8, -40, -80, 240, 192, -448, -128, 256}, {1, 16, -160, -368, 1760, 2272, -7232, -5504, 13824, 5632, -12288, -2048, 4096}, {-3, 0, 36, 0, -96, 0, 64}, {-1, 18, 180, -960, -5280, 14784, 59136, -101376, -329472, 366080, 1025024, -745472, -1863680, 860160, 1966080, -524288, -1114112, 131072, 262144}, {-1, 10, 40, -160, -240, 672, 448, -1024, -256, 512}, {1, 24, -48, -632, 1264, 3296, -6592, -6784, 13568, 6144, -12288, -2048, 4096}, {1, 0, -48, 0, 304, 0, -512, 0, 256}, {1, -20, -220, 1320, 7920, -25344, -109824, 219648, 768768, -1025024, -3075072, 2795520, 7454720, -4587520, -11141120, 4456448, 10027008, -2359296, -4980736, 524288, 1048576}, {1, 16, 32, -48, -96, 32, 64}, {1, 22, -220, -1760, 7920, 41184, -109824, -439296, 768768, 2562560, -3075072, -8945664, 7454720, 19496960, -11141120, -26738688, 10027008, 22413312, -4980736, -10485760, 1048576, 2097152}, {-11, 0, 220, 0, -1232, 0, 2816, 0, -2816, 0, 1024}, {1, -24, -144, 248, 1680, -864, -7168, 1152, 13824, -512, -12288, 0, 4096}, {1, -12, -60, 280, 560, -1792, -1792, 4608, 2304, -5120, -1024, 2048}, {-1, -24, 264, 2288, -11440, -64064, 192192, 823680, -1647360, -5857280, 8200192, 25346048, -25346048, -70189056, 50135040, 127008768, -63504384, -149422080, 49807360, 110100480, -22020096, -46137344, 4194304, 8388608}, {1, 0, -64, 0, 320, 0, -512, 0, 256}, {-1, 28, 196, -2968, -3136, 66304, 18816, -658816, -53760, 3587584, 78848, -11741184, -57344, 24084480, 16384, -31195136, 0, 24772608, 0, -11010048, 0, 2097152}, {-1, -10, 100, 40, -800, -32, 2240, 0, -2560, 0, 1024}, {1, 32, -64, -1504, 3008, 16832, -33664, -76288, 152576, 173568, -347136, -210944, 421888, 131072, -262144, -32768, 65536}, {13, 0, -364, 0, 2912, 0, -9984, 0, 16640, 0, -13312, 0, 4096}, {-1, 26, 364, -2912, -21840, 96096, 512512, -1464320, -6223360, 12446720, 44808192, -65175552, -206389248, 222265344, 635043840, -508035072, -1333592064, 784465920, 1917583360, -807403520, -1857028096, 530579456, 1157627904, -201326592, -419430400, 33554432, 67108864}, {-1, 18, 0, -240, 0, 864, 0, -1152, 0, 512}, {1, 24, -432, -1208, 15216, 28064, -185024, -263424, 1149184, 1250304, -4177920, -3356672, 9375744, 5324800, -13123584, -4947968, 11141120, 2490368, -5242880, -524288, 1048576}, {1, 0, -96, 0, 1376, 0, -6656, 0, 13568, 0, -12288, 0, 4096}, {1, -40, 80, 2120, -4240, -31648, 63296, 194432, -388864, -613376, 1226752, 1087488, -2174976, -1097728, 2195456, 589824, -1179648, -131072, 262144}, {-1, -14, 112, 448, -2016, -4032, 13440, 15360, -42240, -28160, 67584, 24576, -53248, -8192, 16384} }; /* The coefficients in 2^d * \prod_{i=1}^d (x - cos(a_i)) are easily bounded using the binomial theorem. */ static slong magnitude_bound(slong d) { slong res; fmpz_t t; fmpz_init(t); fmpz_bin_uiui(t, d, d / 2); res = fmpz_bits(t); fmpz_clear(t); return FLINT_ABS(res) + d; } static void fmpz_mul_or_div_2exp(fmpz_t x, fmpz_t y, slong s) { if (s >= 0) fmpz_mul_2exp(x, y, s); else fmpz_fdiv_q_2exp(x, y, -s); } /* Balanced product of linear factors (x+alpha_i) using fixed-point arithmetic with prec bits */ static void balanced_product(fmpz * c, fmpz * alpha, slong len, slong prec) { if (len == 1) { fmpz_one(c + 1); fmpz_mul_2exp(c + 1, c + 1, prec); fmpz_set(c, alpha); } else if (len == 2) { fmpz_mul(c, alpha, alpha + 1); fmpz_fdiv_q_2exp(c, c, prec); fmpz_add(c + 1, alpha, alpha + 1); fmpz_one(c + 2); fmpz_mul_2exp(c + 2, c + 2, prec); } else { fmpz *L, *R; slong i, m; m = len / 2; L = _fmpz_vec_init(len + 2); R = L + m + 1; balanced_product(L, alpha, m, prec); balanced_product(R, alpha + m, len - m, prec); _fmpz_poly_mul(c, R, len - m + 1, L, m + 1); for (i = 0; i < len + 1; i++) fmpz_fdiv_q_2exp(c + i, c + i, prec); _fmpz_vec_clear(L, len + 2); } } void _arith_cos_minpoly(fmpz * coeffs, slong d, ulong n) { slong i, j; fmpz * alpha; fmpz_t half; mpfr_t t, u; mp_bitcnt_t prec; slong exp; if (n <= MAX_32BIT) { for (i = 0; i <= d; i++) fmpz_set_si(coeffs + i, lookup_table[n - 1][i]); return; } /* Direct formula for odd primes > 3 */ if (n_is_prime(n)) { slong s = (n - 1) / 2; switch (s % 4) { case 0: fmpz_set_si(coeffs, WORD(1)); fmpz_set_si(coeffs + 1, -s); break; case 1: fmpz_set_si(coeffs, WORD(1)); fmpz_set_si(coeffs + 1, s + 1); break; case 2: fmpz_set_si(coeffs, WORD(-1)); fmpz_set_si(coeffs + 1, s); break; case 3: fmpz_set_si(coeffs, WORD(-1)); fmpz_set_si(coeffs + 1, -s - 1); break; } for (i = 2; i <= s; i++) { slong b = (s - i) % 2; fmpz_mul2_uiui(coeffs + i, coeffs + i - 2, s+i-b, s+2-b-i); fmpz_divexact2_uiui(coeffs + i, coeffs + i, i, i-1); fmpz_neg(coeffs + i, coeffs + i); } return; } prec = magnitude_bound(d) + 5 + FLINT_BIT_COUNT(d); alpha = _fmpz_vec_init(d); fmpz_init(half); mpfr_init2(t, prec); mpfr_init2(u, prec); fmpz_one(half); fmpz_mul_2exp(half, half, prec - 1); mpfr_const_pi(t, prec); mpfr_div_ui(t, t, n, MPFR_RNDN); for (i = j = 0; j < d; i++) { if (n_gcd(n, i) == 1) { mpfr_mul_ui(u, t, 2 * i, MPFR_RNDN); mpfr_cos(u, u, MPFR_RNDN); mpfr_neg(u, u, MPFR_RNDN); exp = mpfr_get_z_2exp(_fmpz_promote(alpha + j), u); _fmpz_demote_val(alpha + j); fmpz_mul_or_div_2exp(alpha + j, alpha + j, exp + prec); j++; } } balanced_product(coeffs, alpha, d, prec); /* Scale and round */ for (i = 0; i < d + 1; i++) { slong r = d; if ((n & (n - 1)) == 0) r--; fmpz_mul_2exp(coeffs + i, coeffs + i, r); fmpz_add(coeffs + i, coeffs + i, half); fmpz_fdiv_q_2exp(coeffs + i, coeffs + i, prec); } fmpz_clear(half); mpfr_clear(t); mpfr_clear(u); _fmpz_vec_clear(alpha, d); } void arith_cos_minpoly(fmpz_poly_t poly, ulong n) { if (n == 0) { fmpz_poly_set_ui(poly, UWORD(1)); } else { slong d = (n <= 2) ? 1 : n_euler_phi(n) / 2; fmpz_poly_fit_length(poly, d + 1); _arith_cos_minpoly(poly->coeffs, d, n); _fmpz_poly_set_length(poly, d + 1); } }