198 lines
4.9 KiB
C
198 lines
4.9 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) 2012 Sebastian Pancratz
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Copyright (C) 2012 Fredrik Johansson
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******************************************************************************/
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#include "padic.h"
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/*
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Computes the sum $1 + x + x^2 / 2$ reduced modulo $p^N$,
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where $x = p^v u$.
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Supports aliasing between \code{rop} and $u$.
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*/
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static void _padic_exp_small(fmpz_t rop, const fmpz_t u, slong v, slong n,
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const fmpz_t p, const fmpz_t pN)
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{
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if (n == 1) /* rop = 1 */
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{
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fmpz_one(rop);
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}
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else if (n == 2) /* rop = 1 + x */
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{
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fmpz_t f;
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fmpz_init(f);
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fmpz_pow_ui(f, p, v);
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fmpz_mul(rop, f, u);
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fmpz_add_ui(rop, rop, 1);
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fmpz_mod(rop, rop, pN);
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fmpz_clear(f);
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}
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else /* n == 3, rop = 1 + x + x^2 / 2 */
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{
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fmpz_t f;
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fmpz_init(f);
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fmpz_pow_ui(f, p, v);
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fmpz_mul(rop, f, u);
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fmpz_mul(f, rop, rop);
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if (fmpz_is_odd(f))
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fmpz_add(f, f, pN);
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fmpz_fdiv_q_2exp(f, f, 1);
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fmpz_add(rop, rop, f);
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fmpz_add_ui(rop, rop, 1);
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fmpz_clear(f);
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}
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}
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void _padic_exp_rectangular(fmpz_t rop, const fmpz_t u, slong v,
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const fmpz_t p, slong N)
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{
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const slong n = _padic_exp_bound(v, N, p);
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fmpz_t pN;
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fmpz_init(pN);
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fmpz_pow_ui(pN, p, N);
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if (n <= 3)
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{
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_padic_exp_small(rop, u, v, n, p, pN);
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}
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else
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{
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const slong k = fmpz_fits_si(p) ?
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(n - 1 - 1) / (fmpz_get_si(p) - 1) : 0;
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slong i, npows, nsums;
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fmpz_t c, f, s, t, sum, pNk;
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fmpz *pows;
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fmpz_init(pNk);
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fmpz_pow_ui(pNk, p, N + k);
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npows = n_sqrt(n);
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nsums = (n + npows - 1) / npows;
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fmpz_init(c);
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fmpz_init(f);
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fmpz_init(s);
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fmpz_init(t);
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fmpz_init(sum);
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/* Compute pows; pows[i] = x^i. */
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pows = _fmpz_vec_init(npows + 1);
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fmpz_one(pows + 0);
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fmpz_pow_ui(f, p, v);
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fmpz_mul(pows + 1, f, u);
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for (i = 2; i <= npows; i++)
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{
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fmpz_mul(pows + i, pows + i - 1, pows + 1);
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fmpz_mod(pows + i, pows + i, pNk);
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}
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fmpz_zero(sum);
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fmpz_one(f);
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for (i = nsums - 1; i >= 0; i--)
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{
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slong lo = i * npows;
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slong hi = FLINT_MIN(n - 1, lo + npows - 1);
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fmpz_zero(s);
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fmpz_one(c);
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for ( ; hi >= lo; hi--)
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{
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fmpz_addmul(s, pows + hi - lo, c);
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if (hi != 0)
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fmpz_mul_ui(c, c, hi);
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}
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fmpz_mul(t, pows + npows, sum);
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fmpz_mul(sum, s, f);
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fmpz_add(sum, sum, t);
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fmpz_mod(sum, sum, pNk);
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fmpz_mul(f, f, c);
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}
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/* Divide by factorial, TODO: Improve */
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/* Note exp(x) is a unit so val(sum) == val(f) */
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if (fmpz_remove(sum, sum, p))
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fmpz_remove(f, f, p);
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_padic_inv(f, f, p, N);
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fmpz_mul(rop, sum, f);
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_fmpz_vec_clear(pows, npows + 1);
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fmpz_clear(c);
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fmpz_clear(f);
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fmpz_clear(s);
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fmpz_clear(t);
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fmpz_clear(sum);
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fmpz_clear(pNk);
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}
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fmpz_mod(rop, rop, pN);
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fmpz_clear(pN);
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}
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int padic_exp_rectangular(padic_t rop, const padic_t op, const padic_ctx_t ctx)
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{
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const slong N = padic_prec(rop);
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const slong v = padic_val(op);
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const fmpz *p = ctx->p;
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if (padic_is_zero(op))
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{
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padic_one(rop);
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return 1;
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}
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if ((fmpz_equal_ui(p, 2) && v <= 1) || (v <= 0))
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{
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return 0;
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}
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else
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{
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if (v < N)
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{
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_padic_exp_rectangular(padic_unit(rop),
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padic_unit(op), padic_val(op), p, N);
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padic_val(rop) = 0;
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}
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else
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{
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padic_one(rop);
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}
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return 1;
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}
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}
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