/* Differential equation for F(a,b,c,y+z):
(y+z)(y-1+z) F''(z) + ((y+z)(a+b+1) - c) F'(z) + a b F(z) = 0
Coefficients in the Taylor series are bounded by
A * binomial(N+k, k) * nu^k
using the Cauchy-Kovalevskaya majorant method.
See J. van der Hoeven, "Fast evaluation of holonomic functions near
and in regular singularities"
*/
static void
bound(mag_t A, mag_t nu, mag_t N,
const acb_t a, const acb_t b, const acb_t c, const acb_t y,
const acb_t f0, const acb_t f1)
{
mag_t M0, M1, t, u;
acb_t d;
acb_init(d);
mag_init(M0);
mag_init(M1);
mag_init(t);
mag_init(u);
/* nu = max(1/|y-1|, 1/|y|) = 1/min(|y-1|, |y|) */
acb_get_mag_lower(t, y);
acb_sub_ui(d, y, 1, MAG_BITS);
acb_get_mag_lower(u, d);
mag_min(t, t, u);
mag_one(u);
mag_div(nu, u, t);
/* M0 = 2 nu |ab| */
acb_get_mag(t, a);
acb_get_mag(u, b);
mag_mul(M0, t, u);
mag_mul(M0, M0, nu);
mag_mul_2exp_si(M0, M0, 1);
/* M1 = 2 nu |(a+b+1)y-c| + 2|a+b+1| */
acb_add(d, a, b, MAG_BITS);
acb_add_ui(d, d, 1, MAG_BITS);
acb_get_mag(t, d);
acb_mul(d, d, y, MAG_BITS);
acb_sub(d, d, c, MAG_BITS);
acb_get_mag(u, d);
mag_mul(u, u, nu);
mag_add(M1, t, u);
mag_mul_2exp_si(M1, M1, 1);
/* N = max(sqrt(2 M0), 2 M1) / nu */
mag_mul_2exp_si(M0, M0, 1);
mag_sqrt(M0, M0);
mag_mul_2exp_si(M1, M1, 1);
mag_max(N, M0, M1);
mag_div(N, N, nu);
/* A = max(|f0|, |f1| / (nu (N+1)) */
acb_get_mag(t, f0);
acb_get_mag(u, f1);
mag_div(u, u, nu);
mag_div(u, u, N); /* upper bound for dividing by N+1 */
mag_max(A, t, u);
acb_clear(d);
mag_clear(M0);
mag_clear(M1);
mag_clear(t);
mag_clear(u);
}
void
acb_sinc_pi(acb_t res, const acb_t x, slong prec)
{
mag_t m;
acb_t t;
if (acb_is_zero(x))
{
acb_one(res);
return;
}
mag_init(m);
acb_init(t);
acb_get_mag_lower(m, x);
if (mag_cmp_2exp_si(m, -1) > 0)
{
acb_const_pi(t, prec + 4);
acb_mul(t, t, x, prec + 4);
acb_sin_pi(res, x, prec + 4);
acb_div(res, res, t, prec);
}
else
{
acb_const_pi(t, prec + 4);
acb_mul(t, t, x, prec + 4);
acb_sinc(res, t, prec);
}
mag_clear(m);
acb_clear(t);
}
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);
}
/* computes the factors that are independent of n (all are upper bounds) */
void
acb_hypgeom_u_asymp_bound_factors(int * R, mag_t alpha,
mag_t nu, mag_t sigma, mag_t rho, mag_t zinv,
const acb_t a, const acb_t b, const acb_t z)
{
mag_t r, u, zre, zim, zlo, sigma_prime;
acb_t t;
mag_init(r);
mag_init(u);
mag_init(zre);
mag_init(zim);
mag_init(zlo);
mag_init(sigma_prime);
acb_init(t);
/* lower bounds for |re(z)|, |im(z)|, |z| */
arb_get_mag_lower(zre, acb_realref(z));
arb_get_mag_lower(zim, acb_imagref(z));
acb_get_mag_lower(zlo, z); /* todo: hypot */
/* upper bound for 1/|z| */
mag_one(u);
mag_div(zinv, u, zlo);
/* upper bound for r = |b - 2a| */
acb_mul_2exp_si(t, a, 1);
acb_sub(t, b, t, MAG_BITS);
acb_get_mag(r, t);
/* determine region */
*R = 0;
if (mag_cmp(zlo, r) >= 0)
{
int znonneg = arb_is_nonnegative(acb_realref(z));
if (znonneg && mag_cmp(zre, r) >= 0)
{
*R = 1;
}
else if (mag_cmp(zim, r) >= 0 || znonneg)
{
*R = 2;
}
else
{
mag_mul_2exp_si(u, r, 1);
if (mag_cmp(zlo, u) >= 0)
*R = 3;
}
}
if (R == 0)
{
mag_inf(alpha);
mag_inf(nu);
mag_inf(sigma);
mag_inf(rho);
}
else
{
/* sigma = |(b-2a)/z| */
mag_mul(sigma, r, zinv);
/* nu = (1/2 + 1/2 sqrt(1-4 sigma^2))^(-1/2) <= 1 + 2 sigma^2 */
if (mag_cmp_2exp_si(sigma, -1) <= 0)
{
mag_mul(nu, sigma, sigma);
mag_mul_2exp_si(nu, nu, 1);
mag_one(u);
mag_add(nu, nu, u);
}
else
{
mag_inf(nu);
}
/* modified sigma for alpha, beta, rho when in R3 */
if (*R == 3)
mag_mul(sigma_prime, sigma, nu);
else
mag_set(sigma_prime, sigma);
/* alpha = 1/(1-sigma') */
mag_one(alpha);
mag_sub_lower(alpha, alpha, sigma_prime);
mag_one(u);
mag_div(alpha, u, alpha);
/* rho = |2a^2-2ab+b|/2 + sigma'*(1+sigma'/4)/(1-sigma')^2 */
mag_mul_2exp_si(rho, sigma_prime, -2);
mag_one(u);
mag_add(rho, rho, u);
mag_mul(rho, rho, sigma_prime);
mag_mul(rho, rho, alpha);
mag_mul(rho, rho, alpha);
acb_sub(t, a, b, MAG_BITS);
acb_mul(t, t, a, MAG_BITS);
acb_mul_2exp_si(t, t, 1);
acb_add(t, t, b, MAG_BITS);
acb_get_mag(u, t);
mag_mul_2exp_si(u, u, -1);
mag_add(rho, rho, u);
}
mag_clear(r);
mag_clear(u);
mag_clear(zre);
mag_clear(zim);
mag_clear(zlo);
mag_clear(sigma_prime);
acb_clear(t);
}
