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/*
Copyright (C) 2010 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 "longlong.h"
#include "fmpz.h"
#include "fmpz_vec.h"
#include "fmpz_poly.h"
/*
Assumptions.
Suppose that $len1 \geq 3$ and $len2 \geq 2$.
Definitions.
Define a sequence $(n_i)$ by $n_1 = \ceil{len1 / 2}$,
$n_2 = \ceil{n_1 / 2}$, etc. all the way to
$n_K = \ceil{n_{K-1} / 2} = 2$. Thus, $K = \ceil{\log_2 len1} - 1$.
Note that we can write $n_i = \ceil{len1 / 2^i}$.
Rough description (of the allocation process, or the algorithm).
Step 1.
For $0 \leq i < n_1$, set h[i] to something of length at most len2.
Set pow to $poly2^2$.
Step n.
For $0 \leq i < n_n$, set h[i] to something of length at most the length
of $poly2^{2^n - 1}$.
Set pow to $poly^{2^n}$.
Step K.
For $0 \leq i < n_K$, set h[i] to something of length at most the length
of $poly2^{2^K - 1}$.
Set pow to $poly^{2^K}$.
Analysis of the space requirements.
Let $S$ be the over all space we need, measured in number of coefficients.
Then
\begin{align*}
S & = 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
+ \sum_{i=1}^{K-1} (n_i - n_{i+1}) \bigl[(2^i - 1) (len2 - 1) + 1\bigr] \\
& = 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
+ (len2 - 1) \sum_{i=1}^{K-1} (n_i - n_{i+1}) (2^i - 1) + n_1 - n_K.
\end{align*}
If $K = 1$, or equivalently $len1$ is 3 or 4, then $S = 2 \times len2$.
Otherwise, we can bound $n_i - n_{i+1}$ from above as follows. For any
non-negative integer $x$,
\begin{equation*}
\ceil{x / 2^i} - \ceil{x / 2^{i+1}} \leq x/2^i - x/2^{i+1} = x / 2^{i+1}.
\end{equation*}
Thus,
\begin{align*}
S & \leq 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
+ (len2 - 1) \times len1 \times \sum_{i=1}^{K-1} (1/2 - 1/2^{i+1}) \\
& \leq 2 \times \bigl[ (2^K - 1) (len2 - 1) + 1 \bigr]
+ (len2 - 1) \times len1 \times (K/2 + 1).
\end{align*}
*/
void
_fmpz_poly_compose_divconquer(fmpz * res, const fmpz * poly1, slong len1,
const fmpz * poly2, slong len2)
{
slong i, j, k, n;
slong *hlen, alloc, powlen;
fmpz *v, **h, *pow, *temp;
if (len1 <= 2 || len2 <= 1)
{
if (len1 == 1)
fmpz_set(res, poly1);
else if (len2 == 1)
_fmpz_poly_evaluate_fmpz(res, poly1, len1, poly2);
else /* len1 == 2 */
_fmpz_poly_compose_horner(res, poly1, len1, poly2, len2);
return;
}
/* Initialisation */
hlen = (slong *) flint_malloc(((len1 + 1) / 2) * sizeof(slong));
k = FLINT_CLOG2(len1) - 1;
hlen[0] = hlen[1] = ((1 << k) - 1) * (len2 - 1) + 1;
for (i = k - 1; i > 0; i--)
{
slong hi = (len1 + (1 << i) - 1) / (1 << i);
for (n = (hi + 1) / 2; n < hi; n++)
hlen[n] = ((1 << i) - 1) * (len2 - 1) + 1;
}
powlen = (1 << k) * (len2 - 1) + 1;
alloc = 0;
for (i = 0; i < (len1 + 1) / 2; i++)
alloc += hlen[i];
v = _fmpz_vec_init(alloc + 2 * powlen);
h = (fmpz **) flint_malloc(((len1 + 1) / 2) * sizeof(fmpz *));
h[0] = v;
for (i = 0; i < (len1 - 1) / 2; i++)
{
h[i + 1] = h[i] + hlen[i];
hlen[i] = 0;
}
hlen[(len1 - 1) / 2] = 0;
pow = v + alloc;
temp = pow + powlen;
/* Let's start the actual work */
for (i = 0, j = 0; i < len1 / 2; i++, j += 2)
{
if (poly1[j + 1] != WORD(0))
{
_fmpz_vec_scalar_mul_fmpz(h[i], poly2, len2, poly1 + j + 1);
fmpz_add(h[i], h[i], poly1 + j);
hlen[i] = len2;
}
else if (poly1[j] != WORD(0))
{
fmpz_set(h[i], poly1 + j);
hlen[i] = 1;
}
}
if ((len1 & WORD(1)))
{
if (poly1[j] != WORD(0))
{
fmpz_set(h[i], poly1 + j);
hlen[i] = 1;
}
}
_fmpz_poly_sqr(pow, poly2, len2);
powlen = 2 * len2 - 1;
for (n = (len1 + 1) / 2; n > 2; n = (n + 1) / 2)
{
if (hlen[1] > 0)
{
slong templen = powlen + hlen[1] - 1;
_fmpz_poly_mul(temp, pow, powlen, h[1], hlen[1]);
_fmpz_poly_add(h[0], temp, templen, h[0], hlen[0]);
hlen[0] = FLINT_MAX(hlen[0], templen);
}
for (i = 1; i < n / 2; i++)
{
if (hlen[2*i + 1] > 0)
{
_fmpz_poly_mul(h[i], pow, powlen, h[2*i + 1], hlen[2*i + 1]);
hlen[i] = hlen[2*i + 1] + powlen - 1;
} else
hlen[i] = 0;
_fmpz_poly_add(h[i], h[i], hlen[i], h[2*i], hlen[2*i]);
hlen[i] = FLINT_MAX(hlen[i], hlen[2*i]);
}
if ((n & WORD(1)))
{
_fmpz_vec_set(h[i], h[2*i], hlen[2*i]);
hlen[i] = hlen[2*i];
}
_fmpz_poly_sqr(temp, pow, powlen);
powlen += powlen - 1;
{
fmpz * t = pow;
pow = temp;
temp = t;
}
}
_fmpz_poly_mul(res, pow, powlen, h[1], hlen[1]);
_fmpz_vec_add(res, res, h[0], hlen[0]);
_fmpz_vec_clear(v, alloc + 2 * powlen);
flint_free(h);
flint_free(hlen);
}
void
fmpz_poly_compose_divconquer(fmpz_poly_t res,
const fmpz_poly_t poly1, const fmpz_poly_t poly2)
{
const slong len1 = poly1->length;
const slong len2 = poly2->length;
slong lenr;
if (len1 == 0)
{
fmpz_poly_zero(res);
return;
}
if (len1 == 1 || len2 == 0)
{
fmpz_poly_set_fmpz(res, poly1->coeffs);
return;
}
lenr = (len1 - 1) * (len2 - 1) + 1;
if (res != poly1 && res != poly2)
{
fmpz_poly_fit_length(res, lenr);
_fmpz_poly_compose_divconquer(res->coeffs, poly1->coeffs, len1,
poly2->coeffs, len2);
_fmpz_poly_set_length(res, lenr);
_fmpz_poly_normalise(res);
}
else
{
fmpz_poly_t t;
fmpz_poly_init2(t, lenr);
_fmpz_poly_compose_divconquer(t->coeffs, poly1->coeffs, len1,
poly2->coeffs, len2);
_fmpz_poly_set_length(t, lenr);
_fmpz_poly_normalise(t);
fmpz_poly_swap(res, t);
fmpz_poly_clear(t);
}
}