void
arb_fac2_ui(arb_t res, ulong n, long prec)
{
if (n % 2 == 0)
{
arb_fac_ui(res, n / 2, prec);
arb_mul_2exp_si(res, res, n / 2);
}
else
{
arb_t t;
arb_init(t);
arb_fac2_ui(t, n - 1, prec + 5);
arb_fac_ui(res, n, prec + 5);
arb_div(res, res, t, prec);
arb_clear(t);
}
}
void
acb_hypgeom_m(acb_t res, const acb_t a, const acb_t b, const acb_t z, int regularized, long prec)
{
long m = LONG_MAX;
long n = LONG_MAX;
if (acb_is_int(a) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(a)), 30) < 0)
{
m = arf_get_si(arb_midref(acb_realref(a)), ARF_RND_DOWN);
}
if (acb_is_int(b) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(b)), 30) < 0)
{
n = arf_get_si(arb_midref(acb_realref(b)), ARF_RND_DOWN);
}
/* terminating */
if (m <= 0 && m < n && m > -10 * prec && (n > 0 || !regularized))
{
acb_hypgeom_m_1f1(res, a, b, z, regularized, prec);
return;
}
/* large */
if (acb_hypgeom_u_use_asymp(z, prec))
{
acb_hypgeom_m_asymp(res, a, b, z, regularized, prec);
return;
}
/* remove singularity */
if (n <= 0 && n > -10 * prec && regularized)
{
acb_t c, d, t, u;
acb_init(c);
acb_init(d);
acb_init(t);
acb_init(u);
acb_sub(c, a, b, prec);
acb_add_ui(c, c, 1, prec);
acb_neg(d, b);
acb_add_ui(d, d, 2, prec);
acb_hypgeom_m_1f1(t, c, d, z, 0, prec);
acb_pow_ui(u, z, 1 - n, prec);
acb_mul(t, t, u, prec);
acb_rising_ui(u, a, 1 - n, prec);
acb_mul(t, t, u, prec);
arb_fac_ui(acb_realref(u), 1 - n, prec);
acb_div_arb(res, t, acb_realref(u), prec);
acb_clear(c);
acb_clear(d);
acb_clear(t);
acb_clear(u);
}
else
{
acb_hypgeom_m_1f1(res, a, b, z, regularized, prec);
}
}
void
_acb_hypgeom_m_1f1(acb_t res, const acb_t a, const acb_t b, const acb_t z,
int regularized, slong prec, slong gamma_prec, int kummer)
{
if (regularized)
{
/* Remove singularity */
if (acb_is_int(b) && arb_is_nonpositive(acb_realref(b)) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(b)), 30) < 0)
{
acb_t c, d, t, u;
slong n;
n = arf_get_si(arb_midref(acb_realref(b)), ARF_RND_DOWN);
acb_init(c);
acb_init(d);
acb_init(t);
acb_init(u);
acb_sub(c, a, b, prec);
acb_add_ui(c, c, 1, prec);
acb_neg(d, b);
acb_add_ui(d, d, 2, prec);
_acb_hypgeom_m_1f1(t, c, d, z, 0, prec, gamma_prec, kummer);
acb_pow_ui(u, z, 1 - n, prec);
acb_mul(t, t, u, prec);
acb_rising_ui(u, a, 1 - n, prec);
acb_mul(t, t, u, prec);
arb_fac_ui(acb_realref(u), 1 - n, prec);
acb_div_arb(res, t, acb_realref(u), prec);
acb_clear(c);
acb_clear(d);
acb_clear(t);
acb_clear(u);
}
else
{
acb_t t;
acb_init(t);
acb_rgamma(t, b, gamma_prec);
_acb_hypgeom_m_1f1(res, a, b, z, 0, prec, gamma_prec, kummer);
acb_mul(res, res, t, prec);
acb_clear(t);
}
return;
}
/* Kummer's transformation */
if (kummer)
{
acb_t u, v;
acb_init(u);
acb_init(v);
acb_sub(u, b, a, prec);
acb_neg(v, z);
_acb_hypgeom_m_1f1(u, u, b, v, regularized, prec, gamma_prec, 0);
acb_exp(v, z, prec);
acb_mul(res, u, v, prec);
acb_clear(u);
acb_clear(v);
return;
}
if (acb_is_one(a))
{
acb_hypgeom_pfq_direct(res, NULL, 0, b, 1, z, -1, prec);
}
else
{
acb_struct c[3];
c[0] = *a;
c[1] = *b;
acb_init(c + 2);
acb_one(c + 2);
acb_hypgeom_pfq_direct(res, c, 1, c + 1, 2, z, -1, prec);
acb_clear(c + 2);
}
}
void
bernoulli_rev_init(bernoulli_rev_t iter, ulong nmax)
{
long j;
fmpz_t t;
arb_t x;
arf_t u;
int round1, round2;
long wp;
nmax -= (nmax % 2);
iter->n = nmax;
iter->alloc = 0;
if (nmax < BERNOULLI_REV_MIN)
return;
iter->prec = wp = bernoulli_global_prec(nmax);
iter->max_power = bernoulli_zeta_terms(nmax, iter->prec);
iter->alloc = iter->max_power + 1;
iter->powers = _fmpz_vec_init(iter->alloc);
fmpz_init(iter->pow_error);
arb_init(iter->prefactor);
arb_init(iter->two_pi_squared);
arb_init(x);
fmpz_init(t);
arf_init(u);
/* precompute powers */
for (j = 3; j <= iter->max_power; j += 2)
{
arb_ui_pow_ui(x, j, nmax, bernoulli_power_prec(j, nmax, wp));
arb_inv(x, x, bernoulli_power_prec(j, nmax, wp));
round1 = arf_get_fmpz_fixed_si(t, arb_midref(x), -wp);
fmpz_set(iter->powers + j, t);
/* error: the radius, plus two roundings */
arf_set_mag(u, arb_radref(x));
round2 = arf_get_fmpz_fixed_si(t, u, -wp);
fmpz_add_ui(t, t, (round1 != 0) + (round2 != 0));
if (fmpz_cmp(iter->pow_error, t) < 0)
fmpz_set(iter->pow_error, t);
}
/* precompute (2pi)^2 and 2*(n!)/(2pi)^n */
arb_fac_ui(iter->prefactor, nmax, wp);
arb_mul_2exp_si(iter->prefactor, iter->prefactor, 1);
arb_const_pi(x, wp);
arb_mul_2exp_si(x, x, 1);
arb_mul(iter->two_pi_squared, x, x, wp);
arb_pow_ui(x, iter->two_pi_squared, nmax / 2, wp);
arb_div(iter->prefactor, iter->prefactor, x, wp);
fmpz_clear(t);
arb_clear(x);
arf_clear(u);
}
