void _fmpq_poly_div(fmpz * Q, fmpz_t q,
const fmpz * A, const fmpz_t a, long lenA,
const fmpz * B, const fmpz_t b, long lenB)
{
long lenQ = lenA - lenB + 1;
ulong d;
const fmpz * lead = B + (lenB - 1);
if (lenB == 1)
{
_fmpq_poly_scalar_div_fmpq(Q, q, A, a, lenA, B, b);
return;
}
/*
From pseudo division over Z we have
lead^d * A = Q * B + R
and thus
{A, a} = {b * Q, a * lead^d} * {B, b} + {R, a * lead^d}.
*/
_fmpz_poly_pseudo_div(Q, &d, A, lenA, B, lenB);
/* 1. lead^d == +-1. {Q, q} = {b Q, a} up to sign */
if (d == 0UL || *lead == 1L || *lead == -1L)
{
fmpz_one(q);
_fmpq_poly_scalar_mul_fmpz(Q, q, Q, q, lenQ, b);
_fmpq_poly_scalar_div_fmpz(Q, q, Q, q, lenQ, a);
if (*lead == -1L && d % 2UL)
_fmpz_vec_neg(Q, Q, lenQ);
}
/* 2. lead^d != +-1. {Q, q} = {b Q, a lead^d} */
else
{
/*
TODO: Improve this. Clearly we do not need to compute
den = a lead^d in many cases, but can determine the GCD from
lead alone already.
*/
fmpz_t den;
fmpz_init(den);
fmpz_pow_ui(den, lead, d);
fmpz_mul(den, a, den);
fmpz_one(q);
_fmpq_poly_scalar_mul_fmpz(Q, q, Q, q, lenQ, b);
_fmpq_poly_scalar_div_fmpz(Q, q, Q, q, lenQ, den);
fmpz_clear(den);
}
}
void
_fmpq_poly_compose_series_horner(fmpz * res, fmpz_t den, const fmpz * poly1,
const fmpz_t den1, long len1, const fmpz * poly2,
const fmpz_t den2, long len2, long n)
{
if (fmpz_is_one(den2))
{
_fmpz_poly_compose_series(res, poly1, len1, poly2, len2, n);
fmpz_set(den, den1);
_fmpq_poly_canonicalise(res, den, n);
}
else if (n == 1)
{
fmpz_set(res, poly1);
fmpz_set(den, den1);
_fmpq_poly_canonicalise(res, den, 1);
}
else
{
long i = len1 - 1;
long lenr;
fmpz_t tden;
fmpz * t = _fmpz_vec_init(n);
fmpz_init(tden);
_fmpz_vec_zero(res, n);
lenr = len2;
_fmpq_poly_scalar_mul_fmpz(res, den, poly2, den2, len2, poly1 + i);
_fmpq_poly_scalar_div_fmpz(res, den, res, den, len2, den1);
i--;
_fmpq_poly_add(res, den, res, den, len2, poly1 + i, den1, 1);
_fmpq_poly_canonicalise(res, den, lenr);
while (i > 0)
{
i--;
if (lenr + len2 - 1 < n)
{
_fmpq_poly_mul(t, tden, res, den, lenr, poly2, den2, len2);
lenr = lenr + len2 - 1;
}
else
{
_fmpq_poly_mullow(t, tden, res, den, lenr,
poly2, den2, len2, n);
lenr = n;
}
_fmpq_poly_canonicalise(t, tden, lenr);
_fmpq_poly_add(res, den, t, tden, lenr, poly1 + i, den1, 1);
}
_fmpq_poly_canonicalise(res, den, n);
_fmpz_vec_clear(t, n);
fmpz_clear(tden);
}
}
void _fmpq_poly_divrem(fmpz * Q, fmpz_t q, fmpz * R, fmpz_t r,
const fmpz * A, const fmpz_t a, slong lenA,
const fmpz * B, const fmpz_t b, slong lenB, const fmpz_preinvn_t inv)
{
slong lenQ = lenA - lenB + 1;
slong lenR = lenB - 1;
ulong d;
const fmpz * lead = B + (lenB - 1);
if (lenB == 1)
{
_fmpq_poly_scalar_div_fmpq(Q, q, A, a, lenA, B, b);
fmpz_one(r);
return;
}
/*
From pseudo division over Z we have
lead^d * A = Q * B + R
and thus
{A, a} = {b * Q, a * lead^d} * {B, b} + {R, a * lead^d}.
*/
_fmpz_poly_pseudo_divrem(Q, R, &d, A, lenA, B, lenB, inv);
/* Determine the actual length of R */
for ( ; lenR != 0 && fmpz_is_zero(R + (lenR - 1)); lenR--) ;
/* 1. lead^d == +-1. {Q, q} = {b Q, a}, {R, r} = {R, a} up to sign */
if (d == UWORD(0) || *lead == WORD(1) || *lead == WORD(-1))
{
fmpz_one(q);
_fmpq_poly_scalar_mul_fmpz(Q, q, Q, q, lenQ, b);
_fmpq_poly_scalar_div_fmpz(Q, q, Q, q, lenQ, a);
fmpz_one(r);
if (lenR > 0)
_fmpq_poly_scalar_div_fmpz(R, r, R, r, lenR, a);
if (*lead == WORD(-1) && d % UWORD(2))
{
_fmpz_vec_neg(Q, Q, lenQ);
_fmpz_vec_neg(R, R, lenR);
}
}
/* 2. lead^d != +-1. {Q, q} = {b Q, a lead^d}, {R, r} = {R, a lead^d} */
else
{
/*
TODO: Improve this. Clearly we do not need to compute
den = a lead^d in many cases, but can determine the GCD from
lead alone already.
*/
fmpz_t den;
fmpz_init(den);
fmpz_pow_ui(den, lead, d);
fmpz_mul(den, a, den);
fmpz_one(q);
_fmpq_poly_scalar_mul_fmpz(Q, q, Q, q, lenQ, b);
_fmpq_poly_scalar_div_fmpz(Q, q, Q, q, lenQ, den);
fmpz_one(r);
if (lenR > 0)
_fmpq_poly_scalar_div_fmpz(R, r, R, r, lenR, den);
fmpz_clear(den);
}
}
