slong
hypgeom_root_bound(const mag_t z, int r)
{
if (r == 0)
{
return 0;
}
else
{
fmpr_t t;
slong v;
fmpr_init(t);
mag_get_fmpr(t, z);
fmpr_root(t, t, r, MAG_BITS, FMPR_RND_UP);
fmpr_add_ui(t, t, 1, MAG_BITS, FMPR_RND_UP);
v = fmpr_get_si(t, FMPR_RND_UP);
fmpr_clear(t);
return v;
}
}
long
gamma_taylor_bound_mag(long n)
{
fmprb_t t, u;
long v;
fmprb_init(t);
fmprb_init(u);
/* (pi-1) n */
fmprb_const_pi(t, FMPRB_RAD_PREC);
fmprb_sub_ui(t, t, 1, FMPRB_RAD_PREC);
fmprb_mul_ui(t, t, n, FMPRB_RAD_PREC);
/* (3-5n) log(n/6) */
fmprb_set_ui(u, n);
fmprb_div_ui(u, u, 6, FMPRB_RAD_PREC);
fmprb_log(u, u, FMPRB_RAD_PREC);
fmprb_mul_si(u, u, 3 - 5*n, FMPRB_RAD_PREC);
fmprb_add(t, t, u, FMPRB_RAD_PREC);
/* divide by 6 log(2) */
fmprb_log_ui(u, 2, FMPRB_RAD_PREC);
fmprb_mul_ui(u, u, 6, FMPRB_RAD_PREC);
fmprb_div(t, t, u, FMPRB_RAD_PREC);
/* upper bound */
fmpr_add(fmprb_midref(t), fmprb_midref(t), fmprb_radref(t),
FMPRB_RAD_PREC, FMPR_RND_CEIL);
v = fmpr_get_si(fmprb_midref(t), FMPR_RND_CEIL);
fmprb_clear(t);
fmprb_clear(u);
return v;
}
void
_fmprb_poly_rgamma_series(fmprb_ptr res, fmprb_srcptr h, long hlen, long len, long prec)
{
int reflect;
long i, rflen, r, n, wp;
fmprb_ptr t, u, v;
fmprb_struct f[2];
hlen = FLINT_MIN(hlen, len);
wp = prec + FLINT_BIT_COUNT(prec);
t = _fmprb_vec_init(len);
u = _fmprb_vec_init(len);
v = _fmprb_vec_init(len);
fmprb_init(f);
fmprb_init(f + 1);
/* use zeta values at small integers */
if (fmprb_is_int(h) && (fmpr_cmpabs_ui(fmprb_midref(h), prec / 2) < 0))
{
r = fmpr_get_si(fmprb_midref(h), FMPR_RND_DOWN);
gamma_lgamma_series_at_one(u, len, wp);
_fmprb_vec_neg(u, u, len);
_fmprb_poly_exp_series(t, u, len, len, wp);
if (r == 1)
{
_fmprb_vec_swap(v, t, len);
}
else if (r <= 0)
{
fmprb_set(f, h);
fmprb_one(f + 1);
rflen = FLINT_MIN(len, 2 - r);
_fmprb_poly_rising_ui_series(u, f, FLINT_MIN(2, len), 1 - r, rflen, wp);
_fmprb_poly_mullow(v, t, len, u, rflen, len, wp);
}
else
{
fmprb_one(f);
fmprb_one(f + 1);
rflen = FLINT_MIN(len, r);
_fmprb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r - 1, rflen, wp);
/* TODO: use div_series? */
_fmprb_poly_inv_series(u, v, rflen, len, wp);
_fmprb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* otherwise use Stirling series */
gamma_stirling_choose_param_fmprb(&reflect, &r, &n, h, 1, 0, wp);
/* rgamma(h) = (gamma(1-h+r) sin(pi h)) / (rf(1-h, r) * pi), h = h0 + t*/
if (reflect)
{
/* u = gamma(r+1-h) */
fmprb_sub_ui(f, h, r + 1, wp);
fmprb_neg(f, f);
gamma_stirling_eval_fmprb_series(t, f, n, len, wp);
_fmprb_poly_exp_series(u, t, len, len, wp);
for (i = 1; i < len; i += 2)
fmprb_neg(u + i, u + i);
/* v = sin(pi x) */
fmprb_const_pi(f + 1, wp);
fmprb_mul(f, h, f + 1, wp);
_fmprb_poly_sin_series(v, f, 2, len, wp);
_fmprb_poly_mullow(t, u, len, v, len, len, wp);
/* rf(1-h,r) * pi */
if (r == 0)
{
fmprb_const_pi(u, wp);
_fmprb_vec_scalar_div(v, t, len, u, wp);
}
else
{
fmprb_sub_ui(f, h, 1, wp);
fmprb_neg(f, f);
fmprb_set_si(f + 1, -1);
rflen = FLINT_MIN(len, r + 1);
_fmprb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
fmprb_const_pi(u, wp);
_fmprb_vec_scalar_mul(v, v, rflen, u, wp);
/* divide by rising factorial */
/* TODO: might better to use div_series, when it has a good basecase */
_fmprb_poly_inv_series(u, v, rflen, len, wp);
_fmprb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* rgamma(h) = rgamma(h+r) rf(h,r) */
if (r == 0)
{
fmprb_add_ui(f, h, r, wp);
gamma_stirling_eval_fmprb_series(t, f, n, len, wp);
_fmprb_vec_neg(t, t, len);
_fmprb_poly_exp_series(v, t, len, len, wp);
}
else
{
fmprb_set(f, h);
fmprb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_fmprb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);
fmprb_add_ui(f, h, r, wp);
gamma_stirling_eval_fmprb_series(v, f, n, len, wp);
_fmprb_vec_neg(v, v, len);
_fmprb_poly_exp_series(u, v, len, len, wp);
_fmprb_poly_mullow(v, u, len, t, rflen, len, wp);
}
}
}
/* compose with nonconstant part */
fmprb_zero(t);
_fmprb_vec_set(t + 1, h + 1, hlen - 1);
_fmprb_poly_compose_series(res, v, len, t, hlen, len, prec);
fmprb_clear(f);
fmprb_clear(f + 1);
_fmprb_vec_clear(t, len);
_fmprb_vec_clear(u, len);
_fmprb_vec_clear(v, len);
}
