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_tanh_series(mp_ptr f, mp_srcptr h, long n, nmod_t mod)
{
mp_ptr t, u;
t = _nmod_vec_init(n);
u = _nmod_vec_init(n);
_nmod_vec_add(t, h, h, n, mod);
_nmod_poly_exp_series(u, t, n, mod);
_nmod_vec_set(t, u, n);
t[0] = 0UL;
u[0] = 2UL;
_nmod_poly_div_series(f, t, u, n, mod);
_nmod_vec_free(t);
_nmod_vec_free(u);
}
void
_nmod_poly_cosh_series(mp_ptr f, mp_srcptr h, long n, nmod_t mod)
{
mp_ptr g, T, U, hprime;
g = _nmod_vec_init(n);
T = _nmod_vec_init(n);
U = _nmod_vec_init(n);
hprime = _nmod_vec_init(n);
_nmod_poly_derivative(hprime, h, n, mod); hprime[n-1] = 0UL;
__nmod_poly_exp_series_prealloc(f, g, h, hprime, T, U, n, mod, 1);
_nmod_vec_add(f, f, g, n, mod);
_nmod_vec_scalar_mul_nmod(f, f, n, n_invmod(2UL, mod.n), mod);
_nmod_vec_free(hprime);
_nmod_vec_free(g);
_nmod_vec_free(T);
_nmod_vec_free(U);
}
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
*/
}
}
void
_nmod_poly_div_divconquer_recursive(mp_ptr Q, mp_ptr W, mp_ptr V,
mp_srcptr A, mp_srcptr B, long lenB, nmod_t mod)
{
if (lenB <= NMOD_DIV_DIVCONQUER_CUTOFF)
{
_nmod_poly_div_basecase(Q, V, A, 2 * lenB - 1, B, lenB, mod);
}
else
{
const long n2 = lenB / 2;
const long 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_ptr q1 = Q + n2;
mp_ptr q2 = Q;
mp_ptr d1q1 = q2 + n2 - (n1 - 1);
mp_ptr d2q1, 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 which is ignored
and a piece of length n2 at BQ.
*/
if (n2 > n1 - 1)
W1[0] = d2q1[n1 - 1];
_nmod_vec_add(W1 + n2 - (n1 - 1), 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), t, n2, mod);
p2 = t - (n2 - 1);
/*
Compute q2 = t div d3, a 2 n2 - 1 by n2 division, so q2 will have
length n2;
*/
_nmod_poly_div_divconquer_recursive(q2, W2, V, p2, d3, n2, mod);
/*
Note Q = q1 x^n2 + q2
*/
}
}
void
_nmod_poly_compose_divconquer(mp_ptr res, mp_srcptr poly1, long len1,
mp_srcptr poly2, long len2, nmod_t mod)
{
long i, j, k, n;
long * hlen, alloc, powlen;
mp_ptr v, * h, pow, temp;
if (len1 == 1)
{
res[0] = poly1[0];
return;
}
if (len2 == 1)
{
res[0] = _nmod_poly_evaluate_nmod(poly1, len1, poly2[0], mod);
return;
}
if (len1 == 2)
{
_nmod_poly_compose_horner(res, poly1, len1, poly2, len2, mod);
return;
}
/* Initialisation */
hlen = (long *) flint_malloc(((len1 + 1) / 2) * sizeof(long));
for (k = 1; (2 << k) < len1; k++) ;
hlen[0] = hlen[1] = ((1 << k) - 1) * (len2 - 1) + 1;
for (i = k - 1; i > 0; i--)
{
long 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 = _nmod_vec_init(alloc + 2 * powlen);
h = (mp_ptr *) flint_malloc(((len1 + 1) / 2) * sizeof(mp_ptr));
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] != 0L)
{
_nmod_vec_scalar_mul_nmod(h[i], poly2, len2, poly1[j + 1], mod);
h[i][0] = n_addmod(h[i][0], poly1[j], mod.n);
hlen[i] = len2;
}
else if (poly1[j] != 0L)
{
h[i][0] = poly1[j];
hlen[i] = 1;
}
}
if ((len1 & 1L))
{
if (poly1[j] != 0L)
{
h[i][0] = poly1[j];
hlen[i] = 1;
}
}
_nmod_poly_mul(pow, poly2, len2, poly2, len2, mod);
powlen = 2 * len2 - 1;
for (n = (len1 + 1) / 2; n > 2; n = (n + 1) / 2)
{
if (hlen[1] > 0)
{
long templen = powlen + hlen[1] - 1;
_nmod_poly_mul(temp, pow, powlen, h[1], hlen[1], mod);
_nmod_poly_add(h[0], temp, templen, h[0], hlen[0], mod);
hlen[0] = FLINT_MAX(hlen[0], templen);
}
for (i = 1; i < n / 2; i++)
{
if (hlen[2*i + 1] > 0)
{
_nmod_poly_mul(h[i], pow, powlen, h[2*i + 1], hlen[2*i + 1], mod);
hlen[i] = hlen[2*i + 1] + powlen - 1;
} else
hlen[i] = 0;
_nmod_poly_add(h[i], h[i], hlen[i], h[2*i], hlen[2*i], mod);
hlen[i] = FLINT_MAX(hlen[i], hlen[2*i]);
}
if ((n & 1L))
{
mpn_copyi(h[i], h[2*i], hlen[2*i]);
hlen[i] = hlen[2*i];
}
_nmod_poly_mul(temp, pow, powlen, pow, powlen, mod);
powlen += powlen - 1;
{
mp_ptr t = pow;
pow = temp;
temp = t;
}
}
_nmod_poly_mul(res, pow, powlen, h[1], hlen[1], mod);
_nmod_vec_add(res, res, h[0], hlen[0], mod);
_nmod_vec_clear(v);
flint_free(h);
flint_free(hlen);
}
