void
__nmod_poly_exp_series_prealloc(mp_ptr f, mp_ptr g, mp_srcptr h,
mp_srcptr hprime, mp_ptr T, mp_ptr U, long n, nmod_t mod, int extend)
{
long m, m2, l;
if (n < NMOD_NEWTON_EXP_CUTOFF)
{
_nmod_poly_exp_series_basecase(f, h, n, n, mod);
_nmod_poly_inv_series_basecase(g, f, extend ? n : (n + 1) / 2, mod);
return;
}
m = (n + 1) / 2;
m2 = (m + 1) / 2;
l = m - 1; /* shifted for derivative */
/* f := exp(h) + O(x^m), g := exp(-h) + O(x^m2) */
__nmod_poly_exp_series_prealloc(f, g, h, hprime, T, U, m, mod, 0);
/* g := exp(-h) + O(x^m) */
_nmod_poly_mullow(T, f, m, g, m2, m, mod);
_nmod_poly_mullow(g + m2, g, m2, T + m2, m - m2, m - m2, mod);
_nmod_vec_neg(g + m2, g + m2, m - m2, mod);
/* U := h' + g (f' - f h') + O(x^(n-1))
Note: should replace h' by h' mod x^(m-1) */
_nmod_vec_zero(f + m, n - m);
_nmod_poly_mullow(T, f, n, hprime, n, n, mod); /* should be mulmid */
_nmod_poly_derivative(U, f, n, mod); /* should skip low terms */
_nmod_vec_sub(U + l, U + l, T + l, n - l, mod);
_nmod_poly_mullow(T + l, g, n - m, U + l, n - m, n - m, mod);
_nmod_vec_add(U + l, hprime + l, T + l, n - m, mod);
/* f := f + f * (h - int U) + O(x^n) = exp(h) + O(x^n) */
_nmod_poly_integral(U, U, n, mod); /* should skip low terms */
_nmod_vec_sub(U + m, h + m, U + m, n - m, mod);
_nmod_poly_mullow(f + m, f, n - m, U + m, n - m, n - m, mod);
/* g := exp(-h) + O(x^n) */
if (extend)
{
_nmod_poly_mullow(T, f, n, g, m, n, mod);
_nmod_poly_mullow(g + m, g, m, T + m, n - m, n - m, mod);
_nmod_vec_neg(g + m, g + m, n - m, mod);
}
}
void
_nmod_poly_atanh_series(mp_ptr g, mp_srcptr h, long n, nmod_t mod)
{
mp_ptr t, u;
t = _nmod_vec_init(n);
u = _nmod_vec_init(n);
/* atanh(h(x)) = integral(h'(x)/(1-h(x)^2)) */
_nmod_poly_mullow(u, h, n, h, n, n, mod);
_nmod_vec_neg(u, u, n, mod); u[0] = 1UL;
_nmod_poly_derivative(t, h, n, mod); t[n-1] = 0UL;
_nmod_poly_div_series(g, t, u, n, mod);
_nmod_poly_integral(g, g, n, mod);
_nmod_vec_free(t);
_nmod_vec_free(u);
}
void
_nmod_poly_asin_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)
{
mp_ptr t, u;
t = _nmod_vec_init(n);
u = _nmod_vec_init(n);
/* asin(h(x)) = integral(h'(x)/sqrt(1-h(x)^2)) */
_nmod_poly_mullow(u, h, n, h, n, n, mod);
_nmod_vec_neg(u, u, n, mod); u[0] = UWORD(1);
_nmod_poly_invsqrt_series(t, u, n, mod);
_nmod_poly_derivative(u, h, n, mod); u[n-1] = UWORD(0);
_nmod_poly_mullow(g, t, n, u, n, n, mod);
_nmod_poly_integral(g, g, n, mod);
_nmod_vec_clear(t);
_nmod_vec_clear(u);
}
slong _nmod_poly_xgcd_hgcd(mp_ptr G, mp_ptr S, mp_ptr T,
mp_srcptr A, slong lenA, mp_srcptr B, slong lenB,
nmod_t mod)
{
const slong cutoff = FLINT_BIT_COUNT(mod.n) <= 8 ?
NMOD_POLY_SMALL_GCD_CUTOFF : NMOD_POLY_GCD_CUTOFF;
slong lenG, lenS, lenT;
if (lenB == 1)
{
G[0] = B[0];
T[0] = 1;
lenG = 1;
lenS = 0;
lenT = 1;
}
else
{
mp_ptr q = _nmod_vec_init(lenA + lenB);
mp_ptr r = q + lenA;
slong lenq, lenr;
__divrem(q, lenq, r, lenr, A, lenA, B, lenB);
if (lenr == 0)
{
__set(G, lenG, B, lenB);
T[0] = 1;
lenS = 0;
lenT = 1;
}
else
{
mp_ptr h, j, v, w, R[4], X;
slong lenh, lenj, lenv, lenw, lenR[4];
int sgnR;
lenh = lenj = lenB;
lenv = lenw = lenA + lenB - 2;
lenR[0] = lenR[1] = lenR[2] = lenR[3] = (lenB + 1) / 2;
X = _nmod_vec_init(2 * lenh + 2 * lenv + 4 * lenR[0]);
h = X;
j = h + lenh;
v = j + lenj;
w = v + lenv;
R[0] = w + lenw;
R[1] = R[0] + lenR[0];
R[2] = R[1] + lenR[1];
R[3] = R[2] + lenR[2];
sgnR = _nmod_poly_hgcd(R, lenR, h, &lenh, j, &lenj, B, lenB, r, lenr, mod);
if (sgnR > 0)
{
_nmod_vec_neg(S, R[1], lenR[1], mod);
_nmod_vec_set(T, R[0], lenR[0]);
}
else
{
_nmod_vec_set(S, R[1], lenR[1]);
_nmod_vec_neg(T, R[0], lenR[0], mod);
}
lenS = lenR[1];
lenT = lenR[0];
while (lenj != 0)
{
__divrem(q, lenq, r, lenr, h, lenh, j, lenj);
__mul(v, lenv, q, lenq, T, lenT);
{
slong l;
_nmod_vec_swap(S, T, FLINT_MAX(lenS, lenT));
l = lenS; lenS = lenT; lenT = l;
}
__sub(T, lenT, T, lenT, v, lenv);
if (lenr == 0)
{
__set(G, lenG, j, lenj);
goto cofactor;
}
if (lenj < cutoff)
{
mp_ptr u0 = R[0], u1 = R[1];
slong lenu0 = lenr - 1, lenu1 = lenj - 1;
lenG = _nmod_poly_xgcd_euclidean(G, u0, u1, j, lenj, r, lenr, mod);
MPN_NORM(u0, lenu0);
MPN_NORM(u1, lenu1);
__mul(v, lenv, S, lenS, u0, lenu0);
__mul(w, lenw, T, lenT, u1, lenu1);
__add(S, lenS, v, lenv, w, lenw);
goto cofactor;
}
sgnR = _nmod_poly_hgcd(R, lenR, h, &lenh, j, &lenj, j,lenj, r, lenr, mod);
__mul(v, lenv, R[1], lenR[1], T, lenT);
__mul(w, lenw, R[2], lenR[2], S, lenS);
__mul(q, lenq, S, lenS, R[3], lenR[3]);
if (sgnR > 0)
__sub(S, lenS, q, lenq, v, lenv);
else
__sub(S, lenS, v, lenv, q, lenq);
__mul(q, lenq, T, lenT, R[0], lenR[0]);
if (sgnR > WORD(0))
__sub(T, lenT, q, lenq, w, lenw);
else
__sub(T, lenT, w, lenw, q, lenq);
}
__set(G, lenG, h, lenh);
cofactor:
__mul(v, lenv, S, lenS, A, lenA);
__sub(w, lenw, G, lenG, v, lenv);
__div(T, lenT, w, lenw, B, lenB);
_nmod_vec_clear(X);
}
_nmod_vec_clear(q);
}
flint_mpn_zero(S + lenS, lenB - 1 - lenS);
flint_mpn_zero(T + lenT, lenA - 1 - lenT);
return lenG;
}
void
_nmod_poly_divrem_divconquer_recursive(mp_ptr Q, mp_ptr BQ, mp_ptr W, mp_ptr V,
mp_srcptr A, mp_srcptr B, slong lenB, nmod_t mod)
{
if (lenB <= NMOD_DIVREM_DIVCONQUER_CUTOFF)
{
mp_ptr t = V;
mp_ptr w = t + 2*lenB - 1;
flint_mpn_copyi(t + lenB - 1, A + lenB - 1, lenB);
flint_mpn_zero(t, lenB - 1);
_nmod_poly_divrem_basecase(Q, BQ, w, t, 2 * lenB - 1, B, lenB, mod);
/* BQ = A - R */
_nmod_vec_neg(BQ, BQ, lenB - 1, mod);
}
else
{
const slong n2 = lenB / 2;
const slong n1 = lenB - n2;
mp_ptr W1 = W;
mp_ptr W2 = W + n2;
mp_srcptr p1 = A + 2 * n2;
mp_srcptr p2;
mp_srcptr d1 = B + n2;
mp_srcptr d2 = B;
mp_srcptr d3 = B + n1;
mp_srcptr d4 = B;
mp_ptr q1 = Q + n2;
mp_ptr q2 = Q;
mp_ptr dq1 = BQ + n2;
mp_ptr d1q1 = BQ + n2 - (n1 - 1);
mp_ptr d2q1, d3q2, d4q2, t;
/*
Set q1 to p1 div d1, a 2 n1 - 1 by n1 division so q1 ends up
being of length n1; low(d1q1) = d1 q1 is of length n1 - 1
*/
_nmod_poly_divrem_divconquer_recursive(q1, d1q1, W1, V, p1, d1, n1, mod);
/*
Compute bottom n1 + n2 - 1 coeffs of d2q1 = d2 q1
*/
d2q1 = W1;
_nmod_poly_mullow(d2q1, q1, n1, d2, n2, n1 + n2 - 1, mod);
/*
Compute dq1 = d1 q1 x^n2 + d2 q1, of length n1 + n2 - 1
Split it into a segment of length n1 - 1 at dq1 and a piece
of length n2 at BQ.
*/
flint_mpn_copyi(dq1, d2q1, n1 - 1);
if (n2 > n1 - 1)
BQ[0] = d2q1[n1 - 1];
_nmod_vec_add(d1q1, d1q1, d2q1 + n2, n1 - 1, mod);
/*
Compute t = A/x^n2 - dq1, which has length 2 n1 + n2 - 1, but we
are not interested in the top n1 coeffs as they will be zero, so
this has effective length n1 + n2 - 1
For the following division, we want to set {p2, 2 n2 - 1} to the
top 2 n2 - 1 coeffs of this
Since the bottom n2 - 1 coeffs of p2 are irrelevant for the
division, we in fact set {t, n2} to the relevant coeffs
*/
t = W1;
_nmod_vec_sub(t, A + n2 + (n1 - 1), BQ, n2, mod);
p2 = t - (n2 - 1);
/*
Compute q2 = t div d3, a 2 n2 - 1 by n2 division, so q2 will have
length n2; let low(d3q2) = d3 q2, of length n2 - 1
*/
d3q2 = BQ;
_nmod_poly_divrem_divconquer_recursive(q2, d3q2, W2, V, p2, d3, n2, mod);
/*
Compute d4q2 = d4 q2, of length n1 + n2 - 1
*/
d4q2 = W1;
_nmod_poly_mullow(d4q2, d4, n1, q2, n2, n1 + n2 - 1, mod);
/*
Compute dq2 = d3q2 x^n1 + d4q2, of length n1 + n2 - 1
*/
_nmod_vec_add(BQ + n1, BQ + n1, d3q2, n2 - 1, mod);
flint_mpn_copyi(BQ, d4q2, n2);
_nmod_vec_add(BQ + n2, BQ + n2, d4q2 + n2, n1 - 1, mod);
/*
Note Q = q1 x^n2 + q2, and BQ = dq1 x^n2 + dq2
*/
}
}
