void
arb_rising_fmpq_ui(arb_t y, const fmpq_t x, ulong n, long prec)
{
if (n == 0)
{
arb_one(y);
}
else if (n == 1)
{
arb_set_fmpq(y, x, prec);
}
else
{
long wp;
wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
bsplit(y, fmpq_numref(x), fmpq_denref(x), 0, n, wp);
if (fmpz_is_one(fmpq_denref(x)))
{
arb_set_round(y, y, prec);
}
else
{
arb_t t;
arb_init(t);
arb_set_fmpz(t, fmpq_denref(x));
arb_pow_ui(t, t, n, wp);
arb_div(y, y, t, prec);
arb_clear(t);
}
}
}
void
acb_rising2_ui_bs(acb_t u, acb_t v, const acb_t x, ulong n, slong prec)
{
if (n == 0)
{
acb_zero(v);
acb_one(u);
}
else if (n == 1)
{
acb_set(u, x);
acb_one(v);
}
else
{
acb_t t;
slong wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
acb_init(t); /* support aliasing */
acb_set(t, x);
bsplit(v, u, t, 0, n, wp);
acb_clear(t);
}
}
void
acb_pow_fmpz_binexp(acb_t y, const acb_t b, const fmpz_t e, long prec)
{
long i, wp, bits;
if (-2L <= *e && *e <= 4L)
{
if (*e == 0L)
{
acb_one(y);
}
else if (*e == 1L)
{
acb_set_round(y, b, prec);
}
else if (*e == -1L)
{
acb_inv(y, b, prec);
}
else if (*e == 2L)
{
acb_mul(y, b, b, prec);
}
else if (*e == 3L)
{
acb_cube(y, b, prec);
}
else if (*e == 4L)
{
acb_mul(y, b, b, prec);
acb_mul(y, y, y, prec);
}
else
{
acb_inv(y, b, prec);
acb_mul(y, y, y, prec);
}
return;
}
if (fmpz_sgn(e) < 0)
{
fmpz_t f;
fmpz_init(f);
fmpz_neg(f, e);
acb_pow_fmpz_binexp(y, b, f, prec + 2);
acb_inv(y, y, prec);
fmpz_clear(f);
return;
}
if (!COEFF_IS_MPZ(*e) && ((*e) % 3 == 0))
{
fmpz e3 = (*e) / 3;
acb_pow_fmpz_binexp(y, b, &e3, prec);
acb_cube(y, y, prec);
return;
}
if (y == b)
{
acb_t t;
acb_init(t);
acb_set(t, b);
acb_pow_fmpz_binexp(y, t, e, prec);
acb_clear(t);
return;
}
acb_set(y, b);
bits = fmpz_bits(e);
wp = ARF_PREC_ADD(prec, bits);
for (i = bits - 2; i >= 0; i--)
{
acb_mul(y, y, y, wp);
if (fmpz_tstbit(e, i))
acb_mul(y, y, b, wp);
}
}
void
arb_rising2_ui_rs(arb_t u, arb_t v,
const arb_t x, ulong n, ulong m, slong prec)
{
if (n == 0)
{
arb_zero(v);
arb_one(u);
}
else if (n == 1)
{
arb_set(u, x);
arb_one(v);
}
else
{
slong wp;
ulong i, j, a, b;
arb_ptr xs;
arb_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 = _arb_vec_init(m + 1);
A = _fmpz_vec_init(2 * m + 1);
B = A + (m + 1);
arb_init(S);
arb_init(T);
arb_init(U);
arb_init(V);
_arb_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);
arb_set_fmpz(S, A);
for (j = 1; j <= b - a; j++)
arb_addmul_fmpz(S, xs + j, A + j, wp);
arb_set_fmpz(T, B);
for (j = 1; j < b - a; j++)
arb_addmul_fmpz(T, xs + j, B + j, wp);
if (i == 0)
{
arb_set(U, S);
arb_set(V, T);
}
else
{
arb_mul(V, V, S, wp);
arb_addmul(V, U, T, wp);
arb_mul(U, U, S, wp);
}
}
arb_set(u, U);
arb_set(v, V);
_arb_vec_clear(xs, m + 1);
_fmpz_vec_clear(A, 2 * m + 1);
arb_clear(S);
arb_clear(T);
arb_clear(U);
arb_clear(V);
}
}
void arb_fib_fmpz(arb_t f, const fmpz_t n, slong prec)
{
arb_t t, u;
slong wp, sign, i;
if (fmpz_sgn(n) < 0)
{
fmpz_t m;
fmpz_init(m);
fmpz_neg(m, n);
arb_fib_fmpz(f, m, prec);
if (fmpz_is_even(m))
arb_neg(f, f);
fmpz_clear(m);
return;
}
if (fmpz_cmp_ui(n, 4) <= 0)
{
ulong x = fmpz_get_ui(n);
arb_set_ui(f, x - (x > 1));
return;
}
wp = ARF_PREC_ADD(prec, 3 * fmpz_bits(n));
arb_init(u);
arb_init(t);
arb_set_ui(f, UWORD(1));
arb_set_ui(u, UWORD(1));
sign = -1;
for (i = fmpz_flog_ui(n, UWORD(2)) - 1; i > 0; i--)
{
arb_mul(t, f, f, wp);
arb_add(f, f, u, wp);
arb_mul_2exp_si(f, f, -1);
arb_mul(f, f, f, wp);
arb_mul_2exp_si(f, f, 1);
arb_submul_ui(f, t, 3, wp);
arb_sub_si(f, f, 2 * sign, wp);
arb_mul_ui(u, t, 5, wp);
arb_add_si(u, u, 2 * sign, wp);
sign = 1;
if (fmpz_tstbit(n, i))
{
arb_set(t, f);
arb_add(f, f, u, wp);
arb_mul_2exp_si(f, f, -1);
arb_mul_2exp_si(t, t, 1);
arb_add(u, f, t, wp);
sign = -1;
}
}
if (fmpz_tstbit(n, 0))
{
arb_add(f, f, u, wp);
arb_mul_2exp_si(f, f, -1);
arb_mul(f, f, u, wp);
arb_sub_si(f, f, sign, prec);
}
else
{
arb_mul(f, f, u, prec);
}
arb_clear(u);
arb_clear(t);
}
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);
}
