216 lines
6.0 KiB
C
216 lines
6.0 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 <gmp.h>
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#include "flint.h"
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#include "ulong_extras.h"
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#include "nmod_vec.h"
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#include "nmod_poly.h"
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#define NMOD_POLY_NEWTON_EXP_CUTOFF 200
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/* with inverse=1 simultaneously computes g = exp(-x) to length n
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with inverse=0 uses g as scratch space, computing
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g = exp(-x) only to length (n+1)/2 */
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static void
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_nmod_poly_exp_series_newton(mp_ptr f, mp_ptr g,
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mp_srcptr h, slong n, nmod_t mod, int inverse)
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{
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slong a[FLINT_BITS];
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slong i, m, m2, l;
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mp_ptr T, U, hprime;
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T = _nmod_vec_init(n);
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U = _nmod_vec_init(n);
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hprime = _nmod_vec_init(n);
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_nmod_poly_derivative(hprime, h, n, mod);
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hprime[n-1] = UWORD(0);
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for (i = 1; (WORD(1) << i) < n; i++);
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a[i = 0] = n;
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while (n >= NMOD_POLY_NEWTON_EXP_CUTOFF)
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a[++i] = (n = (n + 1) / 2);
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/* f := exp(h) + O(x^m), g := exp(-h) + O(x^m2) */
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_nmod_poly_exp_series_basecase(f, h, n, n, mod);
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_nmod_poly_inv_series_basecase(g, f, (n + 1) / 2, mod);
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for (i--; i >= 0; i--)
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{
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m = n; /* previous length */
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n = a[i]; /* new length */
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m2 = (m + 1) / 2;
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l = m - 1; /* shifted for derivative */
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/* g := exp(-h) + O(x^m) */
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_nmod_poly_mullow(T, f, m, g, m2, m, mod);
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_nmod_poly_mullow(g + m2, g, m2, T + m2, m - m2, m - m2, mod);
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_nmod_vec_neg(g + m2, g + m2, m - m2, mod);
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/* U := h' + g (f' - f h') + O(x^(n-1))
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Note: should replace h' by h' mod x^(m-1) */
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_nmod_vec_zero(f + m, n - m);
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_nmod_poly_mullow(T, f, n, hprime, n, n, mod); /* should be mulmid */
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_nmod_poly_derivative(U, f, n, mod); U[n - 1] = 0; /* should skip low terms */
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_nmod_vec_sub(U + l, U + l, T + l, n - l, mod);
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_nmod_poly_mullow(T + l, g, n - m, U + l, n - m, n - m, mod);
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_nmod_vec_add(U + l, hprime + l, T + l, n - m, mod);
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/* f := f + f * (h - int U) + O(x^n) = exp(h) + O(x^n) */
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_nmod_poly_integral(U, U, n, mod); /* should skip low terms */
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_nmod_vec_sub(U + m, h + m, U + m, n - m, mod);
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_nmod_poly_mullow(f + m, f, n - m, U + m, n - m, n - m, mod);
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/* g := exp(-h) + O(x^n) */
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/* not needed if we only want exp(x) */
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if (i == 0 && inverse)
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{
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_nmod_poly_mullow(T, f, n, g, m, n, mod);
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_nmod_poly_mullow(g + m, g, m, T + m, n - m, n - m, mod);
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_nmod_vec_neg(g + m, g + m, n - m, mod);
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}
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}
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_nmod_vec_clear(hprime);
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_nmod_vec_clear(T);
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_nmod_vec_clear(U);
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}
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void
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_nmod_poly_exp_expinv_series(mp_ptr f, mp_ptr g, mp_srcptr h, slong n, nmod_t mod)
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{
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if (n < NMOD_POLY_NEWTON_EXP_CUTOFF)
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{
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_nmod_poly_exp_series_basecase(f, h, n, n, mod);
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_nmod_poly_inv_series(g, f, n, mod);
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}
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else
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{
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_nmod_poly_exp_series_newton(f, g, h, n, mod, 1);
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}
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}
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void
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_nmod_poly_exp_series(mp_ptr f, mp_srcptr h, slong n, nmod_t mod)
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{
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if (n < NMOD_POLY_NEWTON_EXP_CUTOFF)
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{
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_nmod_poly_exp_series_basecase(f, h, n, n, mod);
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}
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else
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{
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mp_ptr g = _nmod_vec_init((n + 1) / 2);
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_nmod_poly_exp_series_newton(f, g, h, n, mod, 0);
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_nmod_vec_clear(g);
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}
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}
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void
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nmod_poly_exp_series(nmod_poly_t f, const nmod_poly_t h, slong n)
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{
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mp_ptr f_coeffs, h_coeffs;
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nmod_poly_t t1;
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slong hlen, k;
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nmod_poly_fit_length(f, n);
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hlen = h->length;
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if (hlen > 0 && h->coeffs[0] != UWORD(0))
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{
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flint_printf("Exception (nmod_poly_exp_series). Constant term != 0.\n");
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abort();
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}
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if (n <= 1 || hlen == 0)
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{
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if (n == 0)
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{
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nmod_poly_zero(f);
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}
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else
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{
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f->coeffs[0] = UWORD(1);
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f->length = 1;
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}
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return;
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}
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/* Handle monomials */
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for (k = 0; h->coeffs[k] == UWORD(0) && k < n - 1; k++);
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if (k == hlen - 1 || k == n - 1)
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{
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hlen = FLINT_MIN(hlen, n);
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_nmod_poly_exp_series_monomial_ui(f->coeffs,
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h->coeffs[hlen-1], hlen - 1, n, f->mod);
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f->length = n;
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_nmod_poly_normalise(f);
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return;
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}
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if (n < NMOD_POLY_NEWTON_EXP_CUTOFF)
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{
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_nmod_poly_exp_series_basecase(f->coeffs, h->coeffs, hlen, n, f->mod);
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f->length = n;
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_nmod_poly_normalise(f);
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return;
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}
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if (hlen < n)
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{
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h_coeffs = _nmod_vec_init(n);
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flint_mpn_copyi(h_coeffs, h->coeffs, hlen);
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flint_mpn_zero(h_coeffs + hlen, n - hlen);
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}
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else
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h_coeffs = h->coeffs;
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if (h == f && hlen >= n)
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{
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nmod_poly_init2(t1, h->mod.n, n);
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f_coeffs = t1->coeffs;
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}
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else
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{
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nmod_poly_fit_length(f, n);
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f_coeffs = f->coeffs;
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}
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_nmod_poly_exp_series(f_coeffs, h_coeffs, n, f->mod);
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if (h == f && hlen >= n)
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{
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nmod_poly_swap(f, t1);
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nmod_poly_clear(t1);
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}
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f->length = n;
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if (hlen < n)
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_nmod_vec_clear(h_coeffs);
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_nmod_poly_normalise(f);
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}
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