void
_acb_poly_evaluate_rectangular(acb_t y, acb_srcptr poly,
slong len, const acb_t x, slong prec)
{
slong i, j, m, r;
acb_ptr xs;
acb_t s, t, c;
if (len < 3)
{
if (len == 0)
{
acb_zero(y);
}
else if (len == 1)
{
acb_set_round(y, poly + 0, prec);
}
else if (len == 2)
{
acb_mul(y, x, poly + 1, prec);
acb_add(y, y, poly + 0, prec);
}
return;
}
m = n_sqrt(len) + 1;
r = (len + m - 1) / m;
xs = _acb_vec_init(m + 1);
acb_init(s);
acb_init(t);
acb_init(c);
_acb_vec_set_powers(xs, x, m + 1, prec);
acb_set(y, poly + (r - 1) * m);
for (j = 1; (r - 1) * m + j < len; j++)
acb_addmul(y, xs + j, poly + (r - 1) * m + j, prec);
for (i = r - 2; i >= 0; i--)
{
acb_set(s, poly + i * m);
for (j = 1; j < m; j++)
acb_addmul(s, xs + j, poly + i * m + j, prec);
acb_mul(y, y, xs + m, prec);
acb_add(y, y, s, prec);
}
_acb_vec_clear(xs, m + 1);
acb_clear(s);
acb_clear(t);
acb_clear(c);
}
void
acb_hypgeom_pfq_sum_rs(acb_t res, acb_t term, acb_srcptr a, slong p,
acb_srcptr b, slong q, const acb_t z, slong n, slong prec)
{
acb_ptr zpow;
acb_t s, t, u;
slong i, j, k, m;
mag_t B, C;
if (n == 0)
{
acb_zero(res);
acb_one(term);
return;
}
if (n < 0)
abort();
m = n_sqrt(n);
m = FLINT_MIN(m, 150);
mag_init(B);
mag_init(C);
acb_init(s);
acb_init(t);
acb_init(u);
zpow = _acb_vec_init(m + 1);
_acb_vec_set_powers(zpow, z, m + 1, prec);
mag_one(B);
for (k = n; k >= 0; k--)
{
j = k % m;
if (k < n)
acb_add(s, s, zpow + j, prec);
if (k > 0)
{
if (p > 0)
{
acb_add_ui(u, a, k - 1, prec);
for (i = 1; i < p; i++)
{
acb_add_ui(t, a + i, k - 1, prec);
acb_mul(u, u, t, prec);
}
if (k < n)
acb_mul(s, s, u, prec);
acb_get_mag(C, u);
mag_mul(B, B, C);
}
if (q > 0)
{
acb_add_ui(u, b, k - 1, prec);
for (i = 1; i < q; i++)
{
acb_add_ui(t, b + i, k - 1, prec);
acb_mul(u, u, t, prec);
}
if (k < n)
acb_div(s, s, u, prec);
acb_get_mag_lower(C, u);
mag_div(B, B, C);
}
if (j == 0 && k < n)
{
acb_mul(s, s, zpow + m, prec);
}
}
}
acb_get_mag(C, z);
mag_pow_ui(C, C, n);
mag_mul(B, B, C);
acb_zero(term);
if (_acb_vec_is_real(a, p) && _acb_vec_is_real(b, q) && acb_is_real(z))
arb_add_error_mag(acb_realref(term), B);
else
acb_add_error_mag(term, B);
acb_set(res, s);
mag_clear(B);
mag_clear(C);
acb_clear(s);
acb_clear(t);
acb_clear(u);
_acb_vec_clear(zpow, m + 1);
}
void
acb_rising2_ui_rs(acb_t u, acb_t v,
const acb_t x, ulong n, ulong m, long prec)
{
if (n == 0)
{
acb_zero(v);
acb_one(u);
}
else if (n == 1)
{
acb_set(u, x);
acb_one(v);
}
else
{
long wp;
ulong i, j, a, b;
acb_ptr xs;
acb_t S, T, U, V;
fmpz *A, *B;
wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
if (m == 0)
{
ulong m1, m2;
m1 = 0.6 * pow(wp, 0.4);
m2 = n_sqrt(n);
m = FLINT_MIN(m1, m2);
}
m = FLINT_MAX(m, 1);
xs = _acb_vec_init(m + 1);
A = _fmpz_vec_init(2 * m + 1);
B = A + (m + 1);
acb_init(S);
acb_init(T);
acb_init(U);
acb_init(V);
_acb_vec_set_powers(xs, x, m + 1, wp);
for (i = 0; i < n; i += m)
{
a = i;
b = FLINT_MIN(n, a + m);
if (a == 0 || b != a + m)
{
_gamma_rf_bsplit(A, a, b);
}
else
{
fmpz tt = m;
_fmpz_poly_taylor_shift(A, &tt, m + 1);
}
_fmpz_poly_derivative(B, A, b - a + 1);
acb_set_fmpz(S, A);
for (j = 1; j <= b - a; j++)
acb_addmul_fmpz(S, xs + j, A + j, wp);
acb_set_fmpz(T, B);
for (j = 1; j < b - a; j++)
acb_addmul_fmpz(T, xs + j, B + j, wp);
if (i == 0)
{
acb_set(U, S);
acb_set(V, T);
}
else
{
acb_mul(V, V, S, wp);
acb_addmul(V, U, T, wp);
acb_mul(U, U, S, wp);
}
}
acb_set(u, U);
acb_set(v, V);
_acb_vec_clear(xs, m + 1);
_fmpz_vec_clear(A, 2 * m + 1);
acb_clear(S);
acb_clear(T);
acb_clear(U);
acb_clear(V);
}
}
void
acb_rising_ui_rs(acb_t y, const acb_t x, ulong n, ulong m, slong prec)
{
acb_ptr xs;
acb_t t, u, v;
ulong i, k, rem;
fmpz_t c, h;
fmpz *s, *d;
slong wp;
if (n == 0)
{
acb_one(y);
return;
}
if (n == 1)
{
acb_set_round(y, x, prec);
return;
}
wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
acb_init(t);
acb_init(u);
acb_init(v);
fmpz_init(c);
fmpz_init(h);
if (m == 0)
{
ulong m1, m2;
m1 = 0.2 * pow(2.0 * wp, 0.4);
m2 = n_sqrt(n);
m = FLINT_MIN(m1, m2);
}
m = FLINT_MIN(m, n);
m = FLINT_MAX(m, 1);
xs = _acb_vec_init(m + 1);
d = _fmpz_vec_init(m * m);
s = _fmpz_vec_init(m + 1);
_acb_vec_set_powers(xs, x, m + 1, wp);
rising_difference_polynomial(s, d, m);
/* tail */
rem = m;
while (rem + m <= n)
rem += m;
acb_one(y);
for (k = rem; k < n; k++)
{
acb_add_ui(t, xs + 1, k, wp);
acb_mul(y, y, t, wp);
}
/* initial rising factorial */
acb_zero(t);
for (i = 1; i <= m; i++)
acb_addmul_fmpz(t, xs + i, s + i, wp);
acb_mul(y, y, t, wp);
/* the leading coefficient is always the same */
acb_mul_fmpz(xs + m - 1, xs + m - 1, d + m - 1 + 0, wp);
for (k = 0; k + 2 * m <= n; k += m)
{
for (i = 0; i < m - 1; i++)
{
fmpz_set_ui(h, k);
_fmpz_poly_evaluate_horner_fmpz(c, d + i * m, m - i, h);
if (i == 0)
acb_add_fmpz(t, t, c, wp);
else
acb_addmul_fmpz(t, xs + i, c, wp);
}
acb_add(t, t, xs + m - 1, wp);
acb_mul(y, y, t, wp);
}
acb_set_round(y, y, prec);
acb_clear(t);
acb_clear(u);
acb_clear(v);
_acb_vec_clear(xs, m + 1);
_fmpz_vec_clear(d, m * m);
_fmpz_vec_clear(s, m + 1);
fmpz_clear(c);
fmpz_clear(h);
}
