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/*=============================================================================
This file is part of FLINT.
FLINT 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.
FLINT 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 FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2012 Sebastian Pancratz
******************************************************************************/
#include "fmpz_mod_poly.h"
#include "qadic.h"
extern slong _padic_log_bound(slong v, slong N, const fmpz_t p);
/*
Carries out the finite series evaluation for the logarithm
\begin{equation*}
\sum_{i=1}^{n} a_i y^i
= \sum_{j=0}^{\ceil{n/b}-1} \Bigl(\sum_{i=1}^b a_{i+jb} y^i\Bigr) y^{jb}
\end{equation*}
where $a_i = 1/i$ with the choice $b = \floor{\sqrt{n}}$,
all modulo $p^N$, where also $P = p^N$.
Assumes that $y$ is reduced modulo $p^N$.
Assumes that $z$ has space for $2d - 1$ coefficients, but
sets only the first $d$ to meaningful values on exit.
Supports aliasing between $y$ and $z$.
*/
static void
_qadic_log_rectangular_series(fmpz *z, const fmpz *y, slong len, slong n,
const fmpz *a, const slong *j, slong lena,
const fmpz_t p, slong N, const fmpz_t pN)
{
const slong d = j[lena - 1];
if (n <= 2)
{
if (n == 1) /* n == 1; z = y */
{
_fmpz_vec_set(z, y, len);
_fmpz_vec_zero(z + len, d - len);
}
else /* n == 2; z = y + y^2/2 */
{
slong i;
fmpz *t;
t = _fmpz_vec_init(2 * len - 1);
_fmpz_poly_sqr(t, y, len);
for (i = 0; i < 2 * len - 1; i++)
if (fmpz_is_even(t + i))
{
fmpz_fdiv_q_2exp(t + i, t + i, 1);
}
else /* => p and t(i) are odd */
{
fmpz_add(t + i, t + i, pN);
fmpz_fdiv_q_2exp(t + i, t + i, 1);
}
_fmpz_mod_poly_reduce(t, 2 * len - 1, a, j, lena, pN);
_fmpz_mod_poly_add(z, y, len, t, FLINT_MIN(d, 2 * len - 1), pN);
_fmpz_vec_clear(t, 2 * len - 1);
}
}
else /* n >= 3 */
{
const slong b = n_sqrt(n);
const slong k = fmpz_fits_si(p) ? n_flog(n, fmpz_get_si(p)) : 0;
slong i, h;
fmpz_t f, pNk;
fmpz *c, *t, *ypow;
c = _fmpz_vec_init(d);
t = _fmpz_vec_init(2 * d - 1);
ypow = _fmpz_vec_init((b + 1) * d + d - 1);
fmpz_init(f);
fmpz_init(pNk);
fmpz_pow_ui(pNk, p, N + k);
fmpz_one(ypow);
_fmpz_vec_set(ypow + d, y, len);
for (i = 2; i <= b; i++)
{
_fmpz_mod_poly_mul(ypow + i * d, ypow + (i - 1) * d, d, y, len, pNk);
_fmpz_mod_poly_reduce(ypow + i * d, d + len - 1, a, j, lena, pNk);
}
_fmpz_vec_zero(z, d);
for (h = (n + (b - 1)) / b - 1; h >= 0; h--)
{
const slong hi = FLINT_MIN(b, n - h*b);
slong w;
/* Compute inner sum in c */
fmpz_rfac_uiui(f, 1 + h*b, hi);
_fmpz_vec_zero(c, d);
for (i = 1; i <= hi; i++)
{
fmpz_divexact_ui(t, f, i + h*b);
_fmpz_vec_scalar_addmul_fmpz(c, ypow + i * d, d, t);
}
/* Multiply c by p^k f */
w = fmpz_remove(f, f, p);
_padic_inv(f, f, p, N + k);
if (w > k)
{
fmpz_pow_ui(t, p, w - k);
_fmpz_vec_scalar_divexact_fmpz(c, c, d, t);
}
else if (w < k)
{
fmpz_pow_ui(t, p, k - w);
_fmpz_vec_scalar_mul_fmpz(c, c, d, t);
}
_fmpz_vec_scalar_mul_fmpz(c, c, d, f);
/* Set z = z y^b + c */
_fmpz_mod_poly_mul(t, z, d, ypow + b * d, d, pNk);
_fmpz_mod_poly_reduce(t, 2 * d - 1, a, j, lena, pNk);
_fmpz_vec_add(z, c, t, d);
_fmpz_vec_scalar_mod_fmpz(z, z, d, pNk);
}
fmpz_pow_ui(f, p, k);
_fmpz_vec_scalar_divexact_fmpz(z, z, d, f);
fmpz_clear(f);
fmpz_clear(pNk);
_fmpz_vec_clear(c, d);
_fmpz_vec_clear(t, 2 * d - 1);
_fmpz_vec_clear(ypow, (b + 1) * d + d - 1);
}
}
void _qadic_log_rectangular(fmpz *z, const fmpz *y, slong v, slong len,
const fmpz *a, const slong *j, slong lena,
const fmpz_t p, slong N, const fmpz_t pN)
{
const slong d = j[lena - 1];
const slong n = _padic_log_bound(v, N, p) - 1;
_qadic_log_rectangular_series(z, y, len, n, a, j, lena, p, N, pN);
_fmpz_mod_poly_neg(z, z, d, pN);
}
int qadic_log_rectangular(qadic_t rop, const qadic_t op, const qadic_ctx_t ctx)
{
const fmpz *p = (&ctx->pctx)->p;
const slong d = qadic_ctx_degree(ctx);
const slong N = qadic_prec(rop);
const slong len = op->length;
if (op->val < 0)
{
return 0;
}
else
{
fmpz *x;
fmpz_t pN;
int alloc, ans;
x = _fmpz_vec_init(len + 1);
alloc = _padic_ctx_pow_ui(pN, N, &ctx->pctx);
/* Set x := (1 - op) mod p^N */
fmpz_pow_ui(x + len, p, op->val);
_fmpz_vec_scalar_mul_fmpz(x, op->coeffs, len, x + len);
fmpz_sub_ui(x, x, 1);
_fmpz_vec_neg(x, x, len);
_fmpz_vec_scalar_mod_fmpz(x, x, len, pN);
if (_fmpz_vec_is_zero(x, len))
{
padic_poly_zero(rop);
ans = 1;
}
else
{
const slong v = _fmpz_vec_ord_p(x, len, p);
if (v >= 2 || (*p != WORD(2) && v >= 1))
{
if (v >= N)
{
padic_poly_zero(rop);
}
else
{
padic_poly_fit_length(rop, d);
_qadic_log_rectangular(rop->coeffs, x, v, len,
ctx->a, ctx->j, ctx->len, p, N, pN);
rop->val = 0;
_padic_poly_set_length(rop, d);
_padic_poly_normalise(rop);
padic_poly_canonicalise(rop, p);
}
ans = 1;
}
else
{
ans = 0;
}
}
_fmpz_vec_clear(x, len + 1);
if (alloc)
fmpz_clear(pN);
return ans;
}
}
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