254 lines
7.4 KiB
C
254 lines
7.4 KiB
C
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/*=============================================================================
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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) 2010 Sebastian Pancratz
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******************************************************************************/
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#include <stdlib.h>
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#include <gmp.h>
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#include "flint.h"
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#include "fmpz.h"
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#include "fmpz_vec.h"
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#include "fmpz_poly.h"
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/*
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Assumptions.
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Suppose that $len1 \geq 3$ and $len2 \geq 2$.
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Definitions.
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Define a sequence $(n_i)$ by $n_1 = \ceil{len1 / 2}$,
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$n_2 = \ceil{n_1 / 2}$, etc. all the way to
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$n_K = \ceil{n_{K-1} / 2} = 2$. Thus, $K = \ceil{\log_2 len1} - 1$.
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Note that we can write $n_i = \ceil{len1 / 2^i}$.
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Rough description (of the allocation process, or the algorithm).
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Step 1.
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For $0 \leq i < n_1$, set h[i] to something of length at most len2.
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Set pow to $poly2^2$.
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Step n.
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For $0 \leq i < n_n$, set h[i] to something of length at most the length
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of $poly2^{2^n - 1}$.
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Set pow to $poly^{2^n}$.
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Step K.
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For $0 \leq i < n_K$, set h[i] to something of length at most the length
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of $poly2^{2^K - 1}$.
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Set pow to $poly^{2^K}$.
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Analysis of the space requirements.
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Let $S$ be the over all space we need, measured in number of coefficients.
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Then
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\begin{align*}
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S & = 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
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+ \sum_{i=1}^{K-1} (n_i - n_{i+1}) \bigl[(2^i - 1) (len2 - 1) + 1\bigr] \\
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& = 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
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+ (len2 - 1) \sum_{i=1}^{K-1} (n_i - n_{i+1}) (2^i - 1) + n_1 - n_K.
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\end{align*}
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If $K = 1$, or equivalently $len1$ is 3 or 4, then $S = 2 \times len2$.
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Otherwise, we can bound $n_i - n_{i+1}$ from above as follows. For any
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non-negative integer $x$,
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\begin{equation*}
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\ceil{x / 2^i} - \ceil{x / 2^{i+1}} \leq x/2^i - x/2^{i+1} = x / 2^{i+1}.
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\end{equation*}
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Thus,
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\begin{align*}
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S & \leq 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
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+ (len2 - 1) \times len1 \times \sum_{i=1}^{K-1} (1/2 - 1/2^{i+1}) \\
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& \leq 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
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+ (len2 - 1) \times len1 \times (K/2 + 1).
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\end{align*}
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*/
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void
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_fmpz_poly_compose_divconquer(fmpz * res, const fmpz * poly1, slong len1,
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const fmpz * poly2, slong len2)
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{
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slong i, j, k, n;
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slong *hlen, alloc, powlen;
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fmpz *v, **h, *pow, *temp;
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if (len1 <= 2 || len2 <= 1)
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{
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if (len1 == 1)
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fmpz_set(res, poly1);
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else if (len2 == 1)
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_fmpz_poly_evaluate_fmpz(res, poly1, len1, poly2);
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else /* len1 == 2 */
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_fmpz_poly_compose_horner(res, poly1, len1, poly2, len2);
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return;
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}
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/* Initialisation */
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hlen = (slong *) flint_malloc(((len1 + 1) / 2) * sizeof(slong));
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k = FLINT_CLOG2(len1) - 1;
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hlen[0] = hlen[1] = ((1 << k) - 1) * (len2 - 1) + 1;
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for (i = k - 1; i > 0; i--)
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{
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slong hi = (len1 + (1 << i) - 1) / (1 << i);
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for (n = (hi + 1) / 2; n < hi; n++)
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hlen[n] = ((1 << i) - 1) * (len2 - 1) + 1;
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}
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powlen = (1 << k) * (len2 - 1) + 1;
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alloc = 0;
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for (i = 0; i < (len1 + 1) / 2; i++)
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alloc += hlen[i];
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v = _fmpz_vec_init(alloc + 2 * powlen);
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h = (fmpz **) flint_malloc(((len1 + 1) / 2) * sizeof(fmpz *));
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h[0] = v;
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for (i = 0; i < (len1 - 1) / 2; i++)
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{
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h[i + 1] = h[i] + hlen[i];
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hlen[i] = 0;
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}
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hlen[(len1 - 1) / 2] = 0;
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pow = v + alloc;
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temp = pow + powlen;
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/* Let's start the actual work */
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for (i = 0, j = 0; i < len1 / 2; i++, j += 2)
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{
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if (poly1[j + 1] != WORD(0))
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{
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_fmpz_vec_scalar_mul_fmpz(h[i], poly2, len2, poly1 + j + 1);
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fmpz_add(h[i], h[i], poly1 + j);
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hlen[i] = len2;
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}
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else if (poly1[j] != WORD(0))
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{
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fmpz_set(h[i], poly1 + j);
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hlen[i] = 1;
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}
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}
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if ((len1 & WORD(1)))
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{
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if (poly1[j] != WORD(0))
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{
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fmpz_set(h[i], poly1 + j);
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hlen[i] = 1;
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}
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}
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_fmpz_poly_sqr(pow, poly2, len2);
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powlen = 2 * len2 - 1;
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for (n = (len1 + 1) / 2; n > 2; n = (n + 1) / 2)
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{
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if (hlen[1] > 0)
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{
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slong templen = powlen + hlen[1] - 1;
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_fmpz_poly_mul(temp, pow, powlen, h[1], hlen[1]);
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_fmpz_poly_add(h[0], temp, templen, h[0], hlen[0]);
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hlen[0] = FLINT_MAX(hlen[0], templen);
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}
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for (i = 1; i < n / 2; i++)
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{
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if (hlen[2*i + 1] > 0)
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{
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_fmpz_poly_mul(h[i], pow, powlen, h[2*i + 1], hlen[2*i + 1]);
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hlen[i] = hlen[2*i + 1] + powlen - 1;
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} else
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hlen[i] = 0;
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_fmpz_poly_add(h[i], h[i], hlen[i], h[2*i], hlen[2*i]);
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hlen[i] = FLINT_MAX(hlen[i], hlen[2*i]);
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}
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if ((n & WORD(1)))
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{
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_fmpz_vec_set(h[i], h[2*i], hlen[2*i]);
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hlen[i] = hlen[2*i];
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}
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_fmpz_poly_sqr(temp, pow, powlen);
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powlen += powlen - 1;
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{
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fmpz * t = pow;
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pow = temp;
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temp = t;
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}
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}
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_fmpz_poly_mul(res, pow, powlen, h[1], hlen[1]);
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_fmpz_vec_add(res, res, h[0], hlen[0]);
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_fmpz_vec_clear(v, alloc + 2 * powlen);
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flint_free(h);
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flint_free(hlen);
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}
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void
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fmpz_poly_compose_divconquer(fmpz_poly_t res,
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const fmpz_poly_t poly1, const fmpz_poly_t poly2)
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{
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const slong len1 = poly1->length;
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const slong len2 = poly2->length;
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slong lenr;
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if (len1 == 0)
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{
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fmpz_poly_zero(res);
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return;
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}
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if (len1 == 1 || len2 == 0)
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{
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fmpz_poly_set_fmpz(res, poly1->coeffs);
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return;
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}
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lenr = (len1 - 1) * (len2 - 1) + 1;
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if (res != poly1 && res != poly2)
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{
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fmpz_poly_fit_length(res, lenr);
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_fmpz_poly_compose_divconquer(res->coeffs, poly1->coeffs, len1,
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poly2->coeffs, len2);
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_fmpz_poly_set_length(res, lenr);
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_fmpz_poly_normalise(res);
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}
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else
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{
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fmpz_poly_t t;
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fmpz_poly_init2(t, lenr);
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_fmpz_poly_compose_divconquer(t->coeffs, poly1->coeffs, len1,
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poly2->coeffs, len2);
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_fmpz_poly_set_length(t, lenr);
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_fmpz_poly_normalise(t);
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fmpz_poly_swap(res, t);
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fmpz_poly_clear(t);
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}
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}
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