308 lines
7.6 KiB
C
308 lines
7.6 KiB
C
/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2011 Fredrik Johansson
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******************************************************************************/
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#include "arith.h"
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static const int mod4_tab[8] = { 2, 1, 3, 0, 0, 3, 1, 2 };
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static const int gcd24_tab[24] = {
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24, 1, 2, 3, 4, 1, 6, 1, 8, 3, 2, 1,
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12, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1
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};
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static mp_limb_t
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n_sqrtmod_2exp(mp_limb_t a, int k)
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{
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mp_limb_t x;
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int i;
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if (a == 0 || k == 0)
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return 0;
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if (k == 1)
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return 1;
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if (k == 2)
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{
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if (a == 1)
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return 1;
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return 0;
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}
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x = 1;
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for (i = 3; i < k; i++)
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x += (a - x * x) / 2;
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if (k < FLINT_BITS)
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x &= ((UWORD(1) << k) - 1);
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return x;
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}
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static mp_limb_t
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n_sqrtmod_ppow(mp_limb_t a, mp_limb_t p, int k, mp_limb_t pk, mp_limb_t pkinv)
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{
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mp_limb_t r, t;
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int i;
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r = n_sqrtmod(a, p);
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if (r == 0)
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return r;
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i = 1;
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while (i < k)
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{
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t = n_mulmod2_preinv(r, r, pk, pkinv);
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t = n_submod(t, a, pk);
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t = n_mulmod2_preinv(t, n_invmod(n_addmod(r, r, pk), pk), pk, pkinv);
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r = n_submod(r, t, pk);
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i *= 2;
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}
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return r;
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}
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void
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trigprod_mul_prime_power(trig_prod_t prod, mp_limb_t k, mp_limb_t n,
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mp_limb_t p, int exp)
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{
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mp_limb_t m, mod, inv;
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if (k <= 3)
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{
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if (k == 0)
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{
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prod->prefactor = 0;
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}
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else if (k == 2 && (n % 2 == 1))
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{
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prod->prefactor *= -1;
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}
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else if (k == 3)
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{
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switch (n % 3)
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{
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case 0:
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prod->prefactor *= 2;
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prod->cos_p[prod->n] = 1;
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prod->cos_q[prod->n] = 18;
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break;
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case 1:
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prod->prefactor *= -2;
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prod->cos_p[prod->n] = 7;
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prod->cos_q[prod->n] = 18;
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break;
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case 2:
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prod->prefactor *= -2;
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prod->cos_p[prod->n] = 5;
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prod->cos_q[prod->n] = 18;
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break;
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}
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prod->n++;
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}
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return;
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}
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/* Power of 2 */
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if (p == 2)
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{
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mod = 8 * k;
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inv = n_preinvert_limb(mod);
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m = n_submod(1, n_mod2_preinv(24 * n, mod, inv), mod);
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m = n_sqrtmod_2exp(m, exp + 3);
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m = n_mulmod2_preinv(m, n_invmod(3, mod), mod, inv);
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prod->prefactor *= n_jacobi(-1, m);
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if (exp % 2 == 1)
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prod->prefactor *= -1;
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prod->sqrt_p *= k;
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prod->cos_p[prod->n] = (mp_limb_signed_t)(k - m);
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prod->cos_q[prod->n] = 2 * k;
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prod->n++;
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return;
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}
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/* Power of 3 */
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if (p == 3)
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{
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mod = 3 * k;
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inv = n_preinvert_limb(mod);
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m = n_submod(1, n_mod2_preinv(24 * n, mod, inv), mod);
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m = n_sqrtmod_ppow(m, p, exp + 1, mod, inv);
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m = n_mulmod2_preinv(m, n_invmod(8, mod), mod, inv);
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prod->prefactor *= (2 * n_jacobi_unsigned(m, 3));
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if (exp % 2 == 0)
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prod->prefactor *= -1;
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prod->sqrt_p *= k;
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prod->sqrt_q *= 3;
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prod->cos_p[prod->n] = (mp_limb_signed_t)(3 * k - 8 * m);
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prod->cos_q[prod->n] = 6 * k;
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prod->n++;
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return;
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}
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/* Power of prime greater than 3 */
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inv = n_preinvert_limb(k);
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m = n_submod(1, n_mod2_preinv(24 * n, k, inv), k);
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if (m % p == 0)
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{
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if (exp == 1)
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{
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prod->prefactor *= n_jacobi(3, k);
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prod->sqrt_p *= k;
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}
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else
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prod->prefactor = 0;
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return;
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}
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m = n_sqrtmod_ppow(m, p, exp, k, inv);
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if (m == 0)
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{
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prod->prefactor = 0;
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return;
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}
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prod->prefactor *= 2;
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prod->prefactor *= n_jacobi(3, k);
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prod->sqrt_p *= k;
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prod->cos_p[prod->n] = 4 * n_mulmod2_preinv(m, n_invmod(24, k), k, inv);
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prod->cos_q[prod->n] = k;
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prod->n++;
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}
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/*
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Solve (k2^2 * d2 * e) * n1 = (d2 * e * n + (k2^2 - 1) / d1) mod k2
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TODO: test this on 32 bit
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*/
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static mp_limb_t
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solve_n1(mp_limb_t n, mp_limb_t k1, mp_limb_t k2,
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mp_limb_t d1, mp_limb_t d2, mp_limb_t e)
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{
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mp_limb_t inv, n1, u, t[2];
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inv = n_preinvert_limb(k1);
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umul_ppmm(t[1], t[0], k2, k2);
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sub_ddmmss(t[1], t[0], t[1], t[0], UWORD(0), UWORD(1));
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mpn_divrem_1(t, 0, t, 2, d1);
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n1 = n_ll_mod_preinv(t[1], t[0], k1, inv);
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n1 = n_mod2_preinv(n1 + d2*e*n, k1, inv);
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u = n_mulmod2_preinv(k2, k2, k1, inv);
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u = n_invmod(u * d2 * e, k1);
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n1 = n_mulmod2_preinv(n1, u, k1, inv);
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return n1;
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}
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void
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arith_hrr_expsum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t n)
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{
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n_factor_t fac;
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int i;
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if (k <= 1)
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{
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prod->prefactor = k;
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return;
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}
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n_factor_init(&fac);
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n_factor(&fac, k, 0);
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/* Repeatedly factor A_k(n) into A_k1(n1)*A_k2(n2) with k1, k2 coprime */
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for (i = 0; i + 1 < fac.num && prod->prefactor != 0; i++)
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{
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mp_limb_t p, k1, k2, inv, n1, n2;
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p = fac.p[i];
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/* k = 2 * k1 with k1 odd */
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if (p == UWORD(2) && fac.exp[i] == 1)
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{
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k2 = k / 2;
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inv = n_preinvert_limb(k2);
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n2 = n_invmod(32, k2);
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n2 = n_mulmod2_preinv(n2,
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n_mod2_preinv(8*n + 1, k2, inv), k2, inv);
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n1 = ((k2 % 8 == 3) || (k2 % 8 == 5)) ^ (n & 1);
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trigprod_mul_prime_power(prod, 2, n1, 2, 1);
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k = k2;
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n = n2;
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}
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/* k = 4 * k1 with k1 odd */
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else if (p == UWORD(2) && fac.exp[i] == 2)
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{
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k2 = k / 4;
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inv = n_preinvert_limb(k2);
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n2 = n_invmod(128, k2);
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n2 = n_mulmod2_preinv(n2,
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n_mod2_preinv(8*n + 5, k2, inv), k2, inv);
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n1 = (n + mod4_tab[(k2 / 2) % 8]) % 4;
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trigprod_mul_prime_power(prod, 4, n1, 2, 2);
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prod->prefactor *= -1;
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k = k2;
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n = n2;
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}
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/* k = k1 * k2 with k1 odd or divisible by 8 */
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else
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{
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mp_limb_t d1, d2, e;
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k1 = n_pow(fac.p[i], fac.exp[i]);
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k2 = k / k1;
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d1 = gcd24_tab[k1 % 24];
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d2 = gcd24_tab[k2 % 24];
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e = 24 / (d1 * d2);
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n1 = solve_n1(n, k1, k2, d1, d2, e);
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n2 = solve_n1(n, k2, k1, d2, d1, e);
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trigprod_mul_prime_power(prod, k1, n1, fac.p[i], fac.exp[i]);
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k = k2;
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n = n2;
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}
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}
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if (fac.num != 0 && prod->prefactor != 0)
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trigprod_mul_prime_power(prod, k, n,
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fac.p[fac.num - 1], fac.exp[fac.num - 1]);
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}
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