mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
121 lines
2.9 KiB
C
121 lines
2.9 KiB
C
/*
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Copyright (C) 2015 Fredrik Johansson
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This file is part of Arb.
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Arb is free software: you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "arb_poly.h"
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/* Bound based on binomial theorem */
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slong
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_arb_poly_swinnerton_dyer_ui_prec(ulong n)
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{
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slong i;
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double u, N;
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N = UWORD(1) << n;
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/* u = (sum of square roots)^(2^n) */
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u = 0;
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for (i = 0; i < n; i++)
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u += sqrt(n_nth_prime(1 + i));
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u = N * log(u) * 1.44269504088897;
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/* Central binomial coefficient C(N,N/2) < 2^N / sqrt(3*N/2) */
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u += N - 0.5*(n-1) - 0.792481250360578; /* log(sqrt(3)) */
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/* experimental heuristic: the bound is 2x too large */
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return u * 0.5 + 15;
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}
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void
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_arb_poly_swinnerton_dyer_ui(arb_ptr T, ulong n, slong trunc, slong prec)
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{
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arb_ptr square_roots, tmp1, tmp2, tmp3;
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arb_t one;
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slong i, j, k, N;
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if (n == 0)
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{
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arb_zero(T);
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arb_one(T + 1);
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return;
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}
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if (prec == 0)
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prec = _arb_poly_swinnerton_dyer_ui_prec(n);
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N = WORD(1) << n;
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trunc = FLINT_MIN(trunc, N + 1);
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arb_init(one);
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arb_one(one);
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square_roots = _arb_vec_init(n);
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tmp1 = flint_malloc((N/2 + 1) * sizeof(arb_struct));
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tmp2 = flint_malloc((N/2 + 1) * sizeof(arb_struct));
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tmp3 = _arb_vec_init(N);
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for (i = 0; i < n; i++)
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arb_sqrt_ui(square_roots + i, n_nth_prime(i + 1), prec);
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/* Build linear factors */
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for (i = 0; i < N; i++)
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{
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arb_zero(T + i);
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for (j = 0; j < n; j++)
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{
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if ((i >> j) & 1)
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arb_add(T + i, T + i, square_roots + j, prec);
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else
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arb_sub(T + i, T + i, square_roots + j, prec);
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}
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}
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/* For each level... */
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for (i = 0; i < n; i++)
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{
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slong stride = UWORD(1) << i;
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for (j = 0; j < N; j += 2*stride)
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{
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for (k = 0; k < stride; k++)
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{
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tmp1[k] = T[j + k];
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tmp2[k] = T[j + stride + k];
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}
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tmp1[stride] = *one;
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tmp2[stride] = *one;
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_arb_poly_mullow(tmp3, tmp1, stride + 1, tmp2, stride + 1,
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FLINT_MIN(2 * stride, trunc), prec);
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_arb_vec_set(T + j, tmp3, FLINT_MIN(2 * stride, trunc));
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}
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}
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arb_one(T + N);
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_arb_vec_clear(square_roots, n);
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flint_free(tmp1);
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flint_free(tmp2);
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_arb_vec_clear(tmp3, UWORD(1) << n);
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arb_clear(one);
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}
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void
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arb_poly_swinnerton_dyer_ui(arb_poly_t poly, ulong n, slong prec)
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{
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slong N = WORD(1) << n;
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arb_poly_fit_length(poly, N + 1);
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_arb_poly_swinnerton_dyer_ui(poly->coeffs, n, N + 1, prec);
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_arb_poly_set_length(poly, N + 1);
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_arb_poly_normalise(poly);
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}
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