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arb/fmprb/sin_cos_pi_fmpq_algebraic.c

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/*=============================================================================
This file is part of ARB.
ARB is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
ARB is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Fredrik Johansson
******************************************************************************/
#include "fmprb.h"
#include "fmprb_poly.h"
#include "elefun.h"
/* include minpoly code here until it appears in a flint release */
#include "fmpz_poly.h"
#include "ulong_extras.h"
/* Use a lookup table for small n. We skip 53, 59 and 61, as the
coefficients do not fit in 16 bits. */
#define MINPOLY_TAB_NUM 65
#define MINPOLY_TAB_MAX_LEN 24
static const char
minpoly_len_tab[MINPOLY_TAB_NUM] = {
1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4, 10, 5, 7, 6,
12, 5, 11, 7, 10, 7, 15, 5, 16, 9, 11, 9, 13, 7, 19, 10, 13, 9, 21, 7,
22, 11, 13, 12, 24, 9, 22, 11, 17, 13, 27, 10, 21, 13, 19, 15, 30, 9,
31, 16, 19, 17
};
static const short
minpoly_tab[MINPOLY_TAB_NUM][MINPOLY_TAB_MAX_LEN] = {
{1},
{-2, 1},
{2, 1},
{1, 1},
{0, 1},
{-1, 1, 1},
{-1, 1},
{-1, -2, 1, 1},
{-2, 0, 1},
{1, -3, 0, 1},
{-1, -1, 1},
{1, 3, -3, -4, 1, 1},
{-3, 0, 1},
{-1, 3, 6, -4, -5, 1, 1},
{1, -2, -1, 1},
{1, 4, -4, -1, 1},
{2, 0, -4, 0, 1},
{1, -4, -10, 10, 15, -6, -7, 1, 1},
{-1, -3, 0, 1},
{1, 5, -10, -20, 15, 21, -7, -8, 1, 1},
{5, 0, -5, 0, 1},
{1, -8, 8, 6, -6, -1, 1},
{-1, 3, 3, -4, -1, 1},
{-1, -6, 15, 35, -35, -56, 28, 36, -9, -10, 1, 1},
{1, 0, -4, 0, 1},
{-1, 5, 25, -5, -50, 1, 35, 0, -10, 0, 1},
{-1, -3, 6, 4, -5, -1, 1},
{1, 9, 0, -30, 0, 27, 0, -9, 0, 1},
{-7, 0, 14, 0, -7, 0, 1},
{-1, 7, 28, -56, -126, 126, 210, -120, -165, 55, 66, -12, -13, 1, 1},
{1, -4, -4, 1, 1},
{-1, -8, 28, 84, -126, -252, 210, 330, -165, -220, 66, 78, -13, -14, 1, 1},
{2, 0, -16, 0, 20, 0, -8, 0, 1},
{1, -12, 12, 43, -43, -34, 34, 10, -10, -1, 1},
{1, 4, -10, -10, 15, 6, -7, -1, 1},
{1, 8, -40, -46, 110, 71, -113, -43, 54, 11, -12, -1, 1},
{-3, 0, 9, 0, -6, 0, 1},
{-1, 9, 45, -120, -330, 462, 924, -792, -1287, 715, 1001, -364, -455, 105,
120, -16, -17, 1, 1},
{-1, 5, 10, -20, -15, 21, 7, -8, -1, 1},
{1, 12, -12, -79, 79, 103, -103, -53, 53, 12, -12, -1, 1},
{1, 0, -12, 0, 19, 0, -8, 0, 1},
{1, -10, -55, 165, 495, -792, -1716, 1716, 3003, -2002, -3003, 1365, 1820,
-560, -680, 136, 153, -18, -19, 1, 1},
{1, 8, 8, -6, -6, 1, 1},
{1, 11, -55, -220, 495, 1287, -1716, -3432, 3003, 5005, -3003, -4368,
1820, 2380, -680, -816, 153, 171, -19, -20, 1, 1},
{-11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1},
{1, -12, -36, 31, 105, -27, -112, 9, 54, -1, -12, 0, 1},
{1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1},
{-1, -12, 66, 286, -715, -2002, 3003, 6435, -6435, -11440, 8008, 12376,
-6188, -8568, 3060, 3876, -969, -1140, 190, 210, -21, -22, 1, 1},
{1, 0, -16, 0, 20, 0, -8, 0, 1},
{-1, 14, 49, -371, -196, 2072, 294, -5147, -210, 7007, 77, -5733, -14,
2940, 1, -952, 0, 189, 0, -21, 0, 1},
{-1, -5, 25, 5, -50, -1, 35, 0, -10, 0, 1},
{1, 16, -16, -188, 188, 526, -526, -596, 596, 339, -339, -103, 103, 16,
-16, -1, 1},
{13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1},
{0},
{-1, 9, 0, -30, 0, 27, 0, -9, 0, 1},
{1, 12, -108, -151, 951, 877, -2891, -2058, 4489, 2442, -4080, -1639,
2289, 650, -801, -151, 170, 19, -20, -1, 1},
{1, 0, -24, 0, 86, 0, -104, 0, 53, 0, -12, 0, 1},
{1, -20, 20, 265, -265, -989, 989, 1519, -1519, -1198, 1198, 531, -531,
-134, 134, 18, -18, -1, 1},
{-1, -7, 28, 56, -126, -126, 210, 120, -165, -55, 66, 12, -13, -1, 1},
{0},
{1, 0, -8, 0, 14, 0, -7, 0, 1},
{0},
{1, -8, -28, 84, 126, -252, -210, 330, 165, -220, -66, 78, 13, -14, -1, 1},
{1, 24, 72, -170, -534, 405, 1385, -459, -1782, 276, 1287, -90, -546, 15,
135, -1, -18, 0, 1},
{2, 0, -64, 0, 336, 0, -672, 0, 660, 0, -352, 0, 104, 0, -16, 0, 1},
};
/* Recurrence for coefficients in rescaled Chebyshev polynomials */
#define CHEB_NEXT(y, x, m, k) \
fmpz_mul2_uiui(y, x, m - 2*k + 1, m - 2*k + 2); \
fmpz_divexact2_uiui(y, y, k, m - k); \
fmpz_neg(y, y); \
/* Computes the monic integer polynomial
n odd: 2 (T(s+1,x/2) - T(s,x/2)), s = (n - 1) / 2
n even: 2 (T(s+1,x/2) - T(s-1,x/2)), s = n / 2 */
static void
chebyshev_sum(fmpz * a, ulong n)
{
ulong s, k, m;
if (n == 1)
{
fmpz_set_si(a, -2);
fmpz_one(a + 1);
return;
}
if (n == 2)
{
fmpz_set_si(a, -4);
fmpz_zero(a + 1);
fmpz_one(a + 2);
return;
}
s = n / 2;
m = s + 1;
fmpz_one(a + m);
for (k = 1; k <= m / 2; k++)
{
CHEB_NEXT(a + m - 2 * k, a + m - 2 * k + 2, m, k);
}
if (n % 2 == 1)
{
m = s;
fmpz_set_si(a + m, -1);
for (k = 1; k <= m / 2; k++)
{
CHEB_NEXT(a + m - 2 * k, a + m - 2 * k + 2, m, k);
}
}
else
{
m = s - 1;
/* Use the top coefficient as scratch space. */
for (k = 1; k <= m / 2; k++)
{
CHEB_NEXT(a + m + 2, a + m + 2, m, k);
fmpz_sub(a + m - 2*k, a + m - 2*k, a + m + 2);
}
for (k = 1 - (m % 2); k < m + 2; k += 2)
fmpz_zero(a + k);
fmpz_sub_ui(a + m, a + m, 1);
/* Set the top coefficient again. */
fmpz_one(a + m + 2);
}
}
#define MUL_TMP(P, Plen, T, Tlen) \
fmpz * swap; \
if (Plen >= Tlen) \
_fmpz_poly_mul(U, P, Plen, T, Tlen); \
else \
_fmpz_poly_mul(U, T, Tlen, P, Plen); \
Plen = Plen + Tlen - 1; \
swap = P; P = U; U = swap; \
void
_arb_fmpz_poly_cos_minpoly(fmpz * f, ulong n)
{
fmpz *P, *Q, *T, *U;
int *mu;
ulong Pdeg, Qdeg;
ulong Plen, Qlen, Tlen;
ulong d;
if (n < MINPOLY_TAB_NUM && minpoly_len_tab[n] <= MINPOLY_TAB_MAX_LEN)
{
for (d = 0; d < minpoly_len_tab[n]; d++)
fmpz_set_si(f + d, minpoly_tab[n][d]);
return;
}
/* Compute values of the Moebius function. We do this as a precomputation
as it allows us to bound in advance the degrees of the numerator and
denominator. */
mu = flint_calloc(n + 1, sizeof(int));
Pdeg = Qdeg = 0;
for (d = 1; d <= n; d++)
{
if (n % d == 0)
{
mu[d] = n_moebius_mu(n / d);
if (mu[d] == 1)
Pdeg += (d / 2 + 1);
else if (mu[d] == -1)
Qdeg += (d / 2 + 1);
}
}
/* We use two extra arrays as scratch space (note that Qdeg < Pdeg). */
P = _fmpz_vec_init(Pdeg + 1);
Q = _fmpz_vec_init(Pdeg + 1);
T = _fmpz_vec_init(Pdeg + 1);
U = _fmpz_vec_init(Pdeg + 1);
Plen = Qlen = 1;
fmpz_one(P);
fmpz_one(Q);
for (d = 1; d <= n; d++)
{
if (n % d == 0 && mu[d] != 0)
{
chebyshev_sum(T, d);
Tlen = d / 2 + 2;
if (mu[d] > 0)
{
MUL_TMP(P, Plen, T, Tlen);
}
else
{
MUL_TMP(Q, Qlen, T, Tlen);
}
}
}
_fmpz_poly_div(f, P, Plen, Q, Qlen);
_fmpz_vec_clear(P, Pdeg + 1);
_fmpz_vec_clear(Q, Pdeg + 1);
_fmpz_vec_clear(T, Pdeg + 1);
_fmpz_vec_clear(U, Pdeg + 1);
flint_free(mu);
}
void
arb_fmpz_poly_cos_minpoly(fmpz_poly_t f, ulong n)
{
slong len = (n < MINPOLY_TAB_NUM) ?
minpoly_len_tab[n] : n_euler_phi(n) / 2 + 1;
fmpz_poly_fit_length(f, len);
_arb_fmpz_poly_cos_minpoly(f->coeffs, n);
_fmpz_poly_set_length(f, len);
}
void
_fmprb_cos_pi_fmpq_algebraic(fmprb_t c, ulong p, ulong q, long prec)
{
/* handle simple angles using exact formulas */
if (q <= 6)
{
if (p == 0)
{
fmprb_one(c);
}
else if (q == 2) /* p/q must be 1/2 */
{
fmprb_zero(c);
}
else if (q == 3) /* p/q must be 1/3 */
{
fmprb_set_ui(c, 1);
fmprb_mul_2exp_si(c, c, -1);
}
else if (q == 4) /* p/q must be 1/4 */
{
fmprb_sqrt_ui(c, 2, prec);
fmprb_mul_2exp_si(c, c, -1);
}
else if (q == 5) /* p/q must be 1/5 or 2/5 */
{
fmprb_sqrt_ui(c, 5, prec + 3);
fmprb_add_si(c, c, (p == 1) ? 1 : -1, prec);
fmprb_mul_2exp_si(c, c, -2);
}
else if (q == 6) /* p/q must be 1/6 */
{
fmprb_sqrt_ui(c, 3, prec);
fmprb_mul_2exp_si(c, c, -1);
}
}
/* reduce even denominator */
else if (q % 2 == 0)
{
long extra = 2 * FLINT_BIT_COUNT(q) + 2;
if (4 * p <= q)
{
_fmprb_cos_pi_fmpq_algebraic(c, p, q / 2, prec + extra);
fmprb_add_ui(c, c, 1, prec + extra);
}
else
{
_fmprb_cos_pi_fmpq_algebraic(c, q / 2 - p, q / 2, prec + extra);
fmprb_sub_ui(c, c, 1, prec + extra);
fmprb_neg(c, c);
}
fmprb_mul_2exp_si(c, c, -1);
fmprb_sqrt(c, c, prec);
}
else
{
/* compute root of the minimal polynomial */
long start_prec, eval_extra_prec;
fmpz_poly_t poly;
fmprb_poly_t fpoly;
fmpr_t interval_bound;
fmprb_t interval;
fmpr_init(interval_bound);
fmprb_init(interval);
fmpz_poly_init(poly);
fmprb_poly_init(fpoly);
if (p % 2 == 0)
arb_fmpz_poly_cos_minpoly(poly, q);
else
arb_fmpz_poly_cos_minpoly(poly, 2 * q);
eval_extra_prec = fmpz_poly_max_bits(poly) * 2; /* heuristic */
eval_extra_prec = FLINT_ABS(eval_extra_prec);
fmprb_poly_set_fmpz_poly(fpoly, poly, FMPR_PREC_EXACT);
/* todo: smallify for accuracy */
start_prec = 100 + eval_extra_prec;
fmprb_const_pi(c, start_prec);
fmprb_mul_ui(c, c, p, start_prec);
fmprb_div_ui(c, c, q, start_prec);
fmprb_cos(c, c, start_prec);
fmprb_mul_2exp_si(c, c, 1); /* poly is for 2*cos */
if (100 + eval_extra_prec - 10 < prec)
{
fmprb_set(interval, c);
fmpr_mul_2exp_si(fmprb_radref(interval), fmprb_radref(interval), 1);
_fmprb_poly_newton_convergence_factor(interval_bound,
fpoly->coeffs, fpoly->length, interval, start_prec);
_fmprb_poly_newton_refine_root(c, fpoly->coeffs, fpoly->length,
c, interval, interval_bound, eval_extra_prec, prec);
}
fmprb_mul_2exp_si(c, c, -1);
fmpz_poly_clear(poly);
fmprb_poly_clear(fpoly);
fmpr_clear(interval_bound);
fmprb_clear(interval);
}
}
void
_fmprb_sin_pi_fmpq_algebraic(fmprb_t s, ulong p, ulong q, long prec)
{
if (q % 2 == 0)
{
p = q / 2 - p;
while ((p % 2 == 0) && (q % 2 == 0))
{
p /= 2;
q /= 2;
}
_fmprb_cos_pi_fmpq_algebraic(s, p, q, prec);
}
else
{
_fmprb_cos_pi_fmpq_algebraic(s, q - 2 * p, 2 * q, prec);
}
}
void
_fmprb_sin_cos_pi_fmpq_algebraic(fmprb_t s, fmprb_t c, ulong p, ulong q, long prec)
{
long wp;
if (q <= 6)
{
if (p == 0)
{
fmprb_one(c);
fmprb_zero(s);
return;
}
else if (q == 2) /* p/q must be 1/2 */
{
fmprb_zero(c);
fmprb_one(s);
return;
}
else if (q == 4) /* p/q must be 1/4 */
{
fmprb_sqrt_ui(c, 2, prec);
fmprb_mul_2exp_si(c, c, -1);
fmprb_set(s, c);
return;
}
}
wp = prec + 3;
/* prefer the formula with less cancellation */
if (p <= q / 4)
{
_fmprb_sin_pi_fmpq_algebraic(s, p, q, wp);
fmprb_mul(c, s, s, wp);
fmprb_sub_ui(c, c, 1, wp);
fmprb_neg(c, c);
fmprb_sqrt(c, c, prec);
}
else
{
_fmprb_cos_pi_fmpq_algebraic(c, p, q, wp);
fmprb_mul(s, c, c, wp);
fmprb_sub_ui(s, s, 1, wp);
fmprb_neg(s, s);
fmprb_sqrt(s, s, prec);
}
}
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