void
acb_hypgeom_mag_chi(mag_t chi, ulong n)
{
mag_t p, q;
ulong k;
mag_init(p);
mag_init(q);
if (n % 2 == 0)
{
mag_one(p);
mag_one(q);
}
else
{
/* upper bound for pi/2 */
mag_set_ui_2exp_si(p, 843314857, -28);
mag_one(q);
}
for (k = n; k >= 2; k -= 2)
{
mag_mul_ui(p, p, k);
mag_mul_ui_lower(q, q, k - 1);
}
mag_div(chi, p, q);
mag_clear(p);
mag_clear(q);
}
slong
hypgeom_bound(mag_t error, int r,
slong A, slong B, slong K, const mag_t TK, const mag_t z, slong tol_2exp)
{
mag_t Tn, t, u, one, tol, num, den;
slong n, m;
mag_init(Tn);
mag_init(t);
mag_init(u);
mag_init(one);
mag_init(tol);
mag_init(num);
mag_init(den);
mag_one(one);
mag_set_ui_2exp_si(tol, UWORD(1), -tol_2exp);
/* approximate number of needed terms */
n = hypgeom_estimate_terms(z, r, tol_2exp);
/* required for 1 + O(1/k) part to be decreasing */
n = FLINT_MAX(n, K + 1);
/* required for z^k / (k!)^r to be decreasing */
m = hypgeom_root_bound(z, r);
n = FLINT_MAX(n, m);
/* We now have |R(k)| <= G(k) where G(k) is monotonically decreasing,
and can bound the tail using a geometric series as soon
as soon as G(k) < 1. */
/* bound T(n-1) */
hypgeom_term_bound(Tn, TK, K, A, B, r, z, n-1);
while (1)
{
/* bound R(n) */
mag_mul_ui(num, z, n);
mag_mul_ui(num, num, n - B);
mag_set_ui_lower(den, n - A);
mag_mul_ui_lower(den, den, n - 2*B);
if (r != 0)
{
mag_set_ui_lower(u, n);
mag_pow_ui_lower(u, u, r);
mag_mul_lower(den, den, u);
}
mag_div(t, num, den);
/* multiply bound for T(n-1) by bound for R(n) to bound T(n) */
mag_mul(Tn, Tn, t);
/* geometric series termination check */
/* u = max(1-t, 0), rounding down [lower bound] */
mag_sub_lower(u, one, t);
if (!mag_is_zero(u))
{
mag_div(u, Tn, u);
if (mag_cmp(u, tol) < 0)
{
mag_set(error, u);
break;
}
}
/* move on to next term */
n++;
}
mag_clear(Tn);
mag_clear(t);
mag_clear(u);
mag_clear(one);
mag_clear(tol);
mag_clear(num);
mag_clear(den);
return n;
}