125 lines
3.4 KiB
C
125 lines
3.4 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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******************************************************************************/
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#include "fmpz_mat.h"
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/*
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Assumes that \code{mat} is an $n \times n$ matrix and sets \code{(cp,n+1)}
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to its characteristic polynomial.
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Employs a division-free algorithm using $O(n^4)$ ring operations.
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*/
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void _fmpz_mat_charpoly(fmpz *cp, const fmpz_mat_t mat)
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{
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const slong n = mat->r;
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if (n == 0)
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{
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fmpz_one(cp);
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}
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else if (n == 1)
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{
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fmpz_neg(cp + 0, fmpz_mat_entry(mat, 0, 0));
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fmpz_one(cp + 1);
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}
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else
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{
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slong i, j, k, t;
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fmpz *a, *A, *s;
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a = _fmpz_vec_init(n * n);
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A = a + (n - 1) * n;
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_fmpz_vec_zero(cp, n + 1);
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fmpz_neg(cp + 0, fmpz_mat_entry(mat, 0, 0));
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for (t = 1; t < n; t++)
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{
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for (i = 0; i <= t; i++)
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{
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fmpz_set(a + 0 * n + i, fmpz_mat_entry(mat, i, t));
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}
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fmpz_set(A + 0, fmpz_mat_entry(mat, t, t));
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for (k = 1; k < t; k++)
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{
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for (i = 0; i <= t; i++)
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{
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s = a + k * n + i;
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fmpz_zero(s);
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for (j = 0; j <= t; j++)
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{
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fmpz_addmul(s, fmpz_mat_entry(mat, i, j), a + (k - 1) * n + j);
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}
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}
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fmpz_set(A + k, a + k * n + t);
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}
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fmpz_zero(A + t);
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for (j = 0; j <= t; j++)
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{
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fmpz_addmul(A + t, fmpz_mat_entry(mat, t, j), a + (t - 1) * n + j);
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}
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for (k = 0; k <= t; k++)
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{
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for (j = 0; j < k; j++)
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{
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fmpz_submul(cp + k, A + j, cp + (k - j - 1));
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}
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fmpz_sub(cp + k, cp + k, A + k);
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}
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}
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/* Shift all coefficients up by one */
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for (i = n; i > 0; i--)
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{
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fmpz_swap(cp + i, cp + (i - 1));
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}
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fmpz_one(cp + 0);
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_fmpz_poly_reverse(cp, cp, n + 1, n + 1);
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_fmpz_vec_clear(a, n * n);
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}
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}
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void fmpz_mat_charpoly(fmpz_poly_t cp, const fmpz_mat_t mat)
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{
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if (mat->r != mat->c)
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{
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flint_printf("Exception (fmpz_mat_charpoly). Non-square matrix.\n");
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abort();
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}
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fmpz_poly_fit_length(cp, mat->r + 1);
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_fmpz_poly_set_length(cp, mat->r + 1);
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_fmpz_mat_charpoly(cp->coeffs, mat);
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}
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