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/*
Copyright (C) 2012 Sebastian Pancratz
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "ulong_extras.h"
#include "fmpz_vec.h"
#include "padic.h"
/*
Carries out the finite series evaluation for the logarithm
\begin{equation*}
\sum_{i=1}^{n} a_i x^i
= \sum_{j=0}^{\ceil{n/b} - 1} \Bigl( \sum_{i=1}^b a_{i+jb} x^i \Bigr) x^{jb}
\end{equation*}
where $a_i = 1/i$ with the choice $b = \floor{\sqrt{n}}$,
all modulo $p^N$, where also $P = p^N$.
Does not support aliasing.
*/
static void
_padic_log_rectangular_series(fmpz_t z, const fmpz_t y, slong n,
const fmpz_t p, slong N, const fmpz_t P0)
{
if (n <= 2)
{
if (n == 1)
{
fmpz_mod(z, y, P0);
}
else /* n == 2; z = y(1 + y/2) */
{
if (fmpz_is_even(y))
{
fmpz_fdiv_q_2exp(z, y, 1);
}
else /* => p and y are odd */
{
fmpz_add(z, y, P0);
fmpz_fdiv_q_2exp(z, z, 1);
}
fmpz_add_ui(z, z, 1);
fmpz_mul(z, z, y);
fmpz_mod(z, z, P0);
}
}
else
{
const slong b = n_sqrt(n);
const slong k = fmpz_fits_si(p) ? n_flog(n, fmpz_get_si(p)) : 0;
slong i, j;
fmpz_t c, f, t, P1;
fmpz *ypow;
ypow = _fmpz_vec_init(b + 1);
fmpz_init(c);
fmpz_init(f);
fmpz_init(t);
fmpz_init(P1);
fmpz_pow_ui(P1, p, N + k);
fmpz_one(ypow + 0);
for (i = 1; i <= b; i++)
{
fmpz_mul(ypow + i, ypow + (i - 1), y);
fmpz_mod(ypow + i, ypow + i, P1);
}
fmpz_zero(z);
for (j = (n + (b - 1)) / b - 1; j >= 0; j--)
{
const slong hi = FLINT_MIN(b, n - j*b);
slong w;
/* Compute inner sum in c */
fmpz_rfac_uiui(f, 1 + j*b, hi);
fmpz_zero(c);
for (i = 1; i <= hi; i++)
{
fmpz_divexact_ui(t, f, i + j*b);
fmpz_addmul(c, t, ypow + i);
}
/* Multiply c by p^k f^{-1} */
w = fmpz_remove(f, f, p);
_padic_inv(f, f, p, N);
if (w > k)
{
fmpz_pow_ui(t, p, w - k);
fmpz_divexact(c, c, t);
}
else
{
fmpz_pow_ui(t, p, k - w);
fmpz_mul(c, c, t);
}
fmpz_mul(c, c, f);
/* Set z = z y^b + c */
fmpz_mul(t, z, ypow + b);
fmpz_add(z, c, t);
fmpz_mod(z, z, P1);
}
fmpz_pow_ui(f, p, k);
fmpz_divexact(z, z, f);
fmpz_clear(c);
fmpz_clear(f);
fmpz_clear(t);
fmpz_clear(P1);
_fmpz_vec_clear(ypow, b + 1);
}
}
void _padic_log_rectangular(fmpz_t z, const fmpz_t y, slong v, const fmpz_t p, slong N)
{
const slong n = _padic_log_bound(v, N, p) - 1;
fmpz_t pN;
fmpz_init(pN);
fmpz_pow_ui(pN, p, N);
_padic_log_rectangular_series(z, y, n, p, N, pN);
fmpz_sub(z, pN, z);
fmpz_clear(pN);
}
int padic_log_rectangular(padic_t rop, const padic_t op, const padic_ctx_t ctx)
{
const fmpz *p = ctx->p;
const slong N = padic_prec(rop);
if (padic_val(op) < 0)
{
return 0;
}
else
{
fmpz_t x;
int ans;
fmpz_init(x);
padic_get_fmpz(x, op, ctx);
fmpz_sub_ui(x, x, 1);
fmpz_neg(x, x);
if (fmpz_is_zero(x))
{
padic_zero(rop);
ans = 1;
}
else
{
fmpz_t t;
slong v;
fmpz_init(t);
v = fmpz_remove(t, x, p);
fmpz_clear(t);
if (v >= 2 || (!fmpz_equal_ui(p, 2) && v >= 1))
{
if (v >= N)
{
padic_zero(rop);
}
else
{
_padic_log_rectangular(padic_unit(rop), x, v, p, N);
padic_val(rop) = 0;
_padic_canonicalise(rop, ctx);
}
ans = 1;
}
else
{
ans = 0;
}
}
fmpz_clear(x);
return ans;
}
}