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/*
Copyright (C) 2026 Fredrik Johansson
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 "fmpz.h"
#include "fmpz_vec.h"
#include "fmpz_poly.h"
#include "fmpq_poly.h"
/* Adapted from _gr_poly_compose_series_kinoshita_li (see comments in that
function for more details about the algorithm) with the only difference that
we track a common denominator for each polynomial.
This file was developed with the assistance of Claude Sonnet 4.6.*/
void
_fmpq_poly_compose_series_kinoshita_li(fmpz *res, fmpz_t rden,
const fmpz *poly1, const fmpz_t den1, slong len1,
const fmpz *poly2, const fmpz_t den2, slong len2,
slong n)
{
slong k, L;
FLINT_ASSERT(len1 >= 1);
FLINT_ASSERT(len2 >= 1);
FLINT_ASSERT(n >= 1);
/* ---- Trivial cases ---- */
if (len1 == 1 || n == 1)
{
_fmpz_vec_set(res, poly1, 1);
_fmpz_vec_zero(res + 1, n - 1);
fmpz_set(rden, den1);
_fmpq_poly_canonicalise(res, rden, 1);
return;
}
/* ---- Level count L = ceil(log2(n)) ---- */
L = 0;
{ slong tmp = n; while (tmp > 1) { tmp = (tmp + 1) / 2; L++; } }
/* ---- Level parameters ---- */
slong *xn_arr = flint_malloc((L + 1) * sizeof(slong));
slong *ylo_arr = flint_malloc((L + 1) * sizeof(slong));
slong *ydeg_arr = flint_malloc((L + 1) * sizeof(slong));
slong *qoff_arr = flint_malloc((L + 1) * sizeof(slong));
xn_arr[0] = n;
ylo_arr[0] = n - 1;
ydeg_arr[0] = 2;
slong qchain_total = 0;
for (k = 0; k < L; k++)
{
qoff_arr[k] = qchain_total;
qchain_total += xn_arr[k] * ydeg_arr[k];
xn_arr[k + 1] = (xn_arr[k] + 1) / 2;
ylo_arr[k + 1] = (ylo_arr[k] >= ydeg_arr[k] - 1)
? (ylo_arr[k] - (ydeg_arr[k] - 1)) : 0;
ydeg_arr[k + 1] = FLINT_MIN(2 * ydeg_arr[k] - 1, n);
}
qoff_arr[L] = qchain_total;
qchain_total += xn_arr[L] * ydeg_arr[L];
/* ---- Qchain numerators and per-level denominators ---- */
fmpz *Qchain_num = _fmpz_vec_init(qchain_total);
fmpz *Qchain_den = _fmpz_vec_init(L + 1); /* one denominator per level */
/* ---- Initialise Q_0 = 1 - y*g(x) ---- */
{
fmpz *Q0_num = Qchain_num + qoff_arr[0];
fmpz_set(Q0_num + 0, den2);
slong glen = FLINT_MIN(len2, n);
for (slong i = 0; i < glen; i++)
fmpz_neg(Q0_num + n + i, poly2 + i);
fmpz_set(Qchain_den + 0, den2);
_fmpq_poly_canonicalise(Q0_num, Qchain_den + 0, n * 2);
}
/* ================================================================
DOWNWARD PASS: Graeffe steps k = 0, ..., L-1.
y-major KS with stride s = 2*ydeg - 1.
x^i y^j maps to flat index i*s + j.
================================================================ */
for (k = 0; k < L; k++)
{
slong xn = xn_arr[k];
slong ydeg = ydeg_arr[k];
slong xnh = xn_arr[k + 1];
slong ydA = ydeg_arr[k + 1];
slong s = 2 * ydeg - 1;
slong qks_len = (xn - 1) * s + ydeg;
slong rks_len = s * (2 * xnh - 1);
fmpz *Qk_num = Qchain_num + qoff_arr[k];
fmpz *Qk1_num = Qchain_num + qoff_arr[k + 1];
fmpz *Qks_num = _fmpz_vec_init(qks_len);
fmpz *Qneg_ks_num = _fmpz_vec_init(qks_len);
fmpz *Rks_num = _fmpz_vec_init(rks_len);
fmpz_t Qks_den, Rks_den;
fmpz_init_set(Qks_den, Qchain_den + k);
fmpz_init(Rks_den);
/* Pack Q and Q(-x,y): x^i y^j -> index i*s + j.
* Qneg is the same but odd-x entries are negated. */
for (slong j = 0; j < ydeg; j++)
for (slong i = 0; i < xn; i++)
{
const fmpz *src = Qk_num + j * xn + i;
if (i & 1)
{
fmpz_set(Qks_num + i * s + j, src);
fmpz_neg(Qneg_ks_num + i * s + j, src);
}
else
{
fmpz_set(Qks_num + i * s + j, src);
fmpz_set(Qneg_ks_num + i * s + j, src);
}
}
/* Product denominator = Qks_den^2 */
fmpz_mul(Rks_den, Qks_den, Qks_den);
_fmpz_poly_mullow(Rks_num, Qks_num, qks_len, Qneg_ks_num, qks_len, rks_len);
_fmpz_vec_clear(Qks_num, qks_len);
_fmpz_vec_clear(Qneg_ks_num, qks_len);
/* Unpack even-x columns -> Q_{k+1}: x^{2i} y^j at index 2*i*s + j.
* Copy only the xnh*ydA entries that are used, then canonicalise. */
for (slong j = 0; j < ydA; j++)
for (slong i = 0; i < xnh; i++)
fmpz_set(Qk1_num + j * xnh + i, Rks_num + 2 * i * s + j);
_fmpz_vec_clear(Rks_num, rks_len);
fmpz_set(Qchain_den + (k + 1), Rks_den);
_fmpq_poly_canonicalise(Qk1_num, Qchain_den + (k + 1), xnh * ydA);
fmpz_clear(Qks_den);
fmpz_clear(Rks_den);
}
/* ================================================================
BASE CASE: W = P / Q_L(0,y) mod y^n.
Q_L has x-stride 1, so Q_L_num[j] is the j-th y-coefficient.
ylo_arr[L] = 0 always, so the full length-n result is needed.
================================================================ */
slong W_alloc = 1;
while (W_alloc < n) W_alloc *= 2;
fmpz *W_num = _fmpz_vec_init(W_alloc);
fmpz_t W_den;
fmpz_init(W_den);
slong yn_W = n;
{
/* ---- Build P_num = f.reverse(n-1), P_den = den1 ---- */
slong glen = FLINT_MIN(n, len1);
fmpz *P_num = _fmpz_vec_init(glen);
_fmpz_poly_reverse(P_num, poly1, glen, glen);
slong jmax = FLINT_MIN(ydeg_arr[L], glen);
_fmpq_poly_div_series(W_num + n - glen, W_den,
P_num, den1, glen,
Qchain_num + qoff_arr[L], Qchain_den + L, jmax,
glen);
/* Low (n - glen) coefficients of W are a priori zero. */
_fmpz_vec_clear(P_num, glen);
}
/* ================================================================
UPWARD PASS: k = L-1 down to 0.
At level k, W has x-stride xn_arr[k+1] and y-width yn_W.
We compute r = Q_k(-x,y) * W(x^2,y), then extract the y-slice
[yslice_lo, yn_r) into W in place.
================================================================ */
for (k = L - 1; k >= 0; k--)
{
slong xn = xn_arr[k];
slong xnh = xn_arr[k + 1];
slong ydeg = ydeg_arr[k];
slong ylo = ylo_arr[k];
slong ylor = ylo_arr[k + 1];
slong yn_r = n - ylor;
slong s = ydeg + yn_W - 1;
slong qneg_ks_len = (xn - 1) * s + ydeg;
slong W_ks_len = 2 * (xnh - 1) * s + yn_W;
slong yslice_lo = ylo - ylor;
/*
* We only need product coefficients [yslice_lo, yslice_lo+R_rks_len).
* _fmpz_poly_mulmid(res, f, flen, g, glen, lo, hi) computes slice [lo, hi)
* of the full product into res[0..hi-lo-1], so res[k] = product coefficient lo+k.
*/
slong R_rks_len = (xn - 1) * s + yn_r - yslice_lo;
fmpz *Qk_num = Qchain_num + qoff_arr[k];
fmpz *Qneg_ks_num = _fmpz_vec_init(qneg_ks_len);
fmpz *W_ks_num = _fmpz_vec_init(W_ks_len);
fmpz *R_rks_num = _fmpz_vec_init(R_rks_len);
fmpz_t Qneg_den, W_ks_den, R_rks_den;
fmpz_init_set(Qneg_den, Qchain_den + k);
fmpz_init_set(W_ks_den, W_den);
fmpz_init(R_rks_den);
/* Pack Qneg_k: x^i y^j -> i*s + j, odd-x entries negated. */
for (slong j = 0; j < ydeg; j++)
for (slong i = 0; i < xn; i++)
{
const fmpz *src = Qk_num + j * xn + i;
if (i & 1)
fmpz_neg(Qneg_ks_num + i * s + j, src);
else
fmpz_set(Qneg_ks_num + i * s + j, src);
}
/* Pack W(x^2): x^{2i} y^j -> 2*i*s + j. */
for (slong j = 0; j < yn_W; j++)
for (slong i = 0; i < xnh; i++)
fmpz_set(W_ks_num + 2 * i * s + j, W_num + j * xnh + i);
fmpz_mul(R_rks_den, Qneg_den, W_ks_den);
_fmpz_poly_mulmid(R_rks_num, Qneg_ks_num, qneg_ks_len,
W_ks_num, W_ks_len,
yslice_lo, yslice_lo + R_rks_len);
_fmpz_vec_clear(Qneg_ks_num, qneg_ks_len);
_fmpz_vec_clear(W_ks_num, W_ks_len);
/*
* Unpack y-slice [yslice_lo, yn_r) into W in place.
* R_rks_num[k] holds product coefficient yslice_lo + k.
* The entry R[jsrc][i] is product coefficient i*s + jsrc,
* so it sits at R_rks_num[i*s + jsrc - yslice_lo].
*/
slong new_yn_W = n - ylo;
{
slong jout = 0;
for (slong jsrc = yslice_lo; jsrc < yn_r; jsrc++, jout++)
for (slong i = 0; i < xn; i++)
fmpz_set(W_num + jout * xn + i,
R_rks_num + i * s + jsrc - yslice_lo);
}
_fmpz_vec_clear(R_rks_num, R_rks_len);
fmpz_set(W_den, R_rks_den);
_fmpq_poly_canonicalise(W_num, W_den, new_yn_W * xn);
fmpz_clear(Qneg_den);
fmpz_clear(W_ks_den);
fmpz_clear(R_rks_den);
yn_W = new_yn_W;
}
/* W_num[0..n) / W_den is the result f(g(x)) mod x^n. */
_fmpz_vec_set(res, W_num, n);
fmpz_set(rden, W_den);
/* Already canonical from the last upward step. */
_fmpz_vec_clear(W_num, W_alloc);
fmpz_clear(W_den);
_fmpz_vec_clear(Qchain_num, qchain_total);
_fmpz_vec_clear(Qchain_den, L + 1);
flint_free(xn_arr);
flint_free(ylo_arr);
flint_free(ydeg_arr);
flint_free(qoff_arr);
}
void
fmpq_poly_compose_series_kinoshita_li(fmpq_poly_t res,
const fmpq_poly_t poly1,
const fmpq_poly_t poly2,
slong n)
{
slong len1 = poly1->length;
slong len2 = poly2->length;
slong lenr;
if (len1 == 0 || n == 0)
{
fmpq_poly_zero(res);
return;
}
if (len2 != 0 && !fmpz_is_zero(poly2->coeffs))
{
flint_throw(FLINT_ERROR, "(fmpq_poly_compose_series_kinoshita_li): "
"Inner polynomial must have zero constant term.\n");
}
if (len2 == 0 || len1 == 1)
{
fmpq_poly_set(res, poly1);
fmpq_poly_truncate(res, 1);
return;
}
lenr = FLINT_MIN((len1 - 1) * (len2 - 1) + 1, n);
len1 = FLINT_MIN(len1, lenr);
len2 = FLINT_MIN(len2, lenr);
fmpq_poly_fit_length(res, lenr);
_fmpq_poly_compose_series_kinoshita_li(res->coeffs, res->den,
poly1->coeffs, poly1->den, len1,
poly2->coeffs, poly2->den, len2, lenr);
_fmpq_poly_set_length(res, lenr);
_fmpq_poly_normalise(res);
}