/* assumes no aliasing of w and p */
void
acb_lambertw_branchpoint_series(acb_t w, const acb_t t, int bound, slong prec)
{
slong i;
static const int coeffs[] = {-130636800,130636800,-43545600,19958400,
-10402560,5813640,-3394560,2042589,-1256320};
acb_zero(w);
for (i = 8; i >= 0; i--)
{
acb_mul(w, w, t, prec);
acb_add_si(w, w, coeffs[i], prec);
}
acb_div_si(w, w, -coeffs[0], prec);
if (bound)
{
mag_t err;
mag_init(err);
acb_get_mag(err, t);
mag_geom_series(err, err, 9);
if (acb_is_real(t))
arb_add_error_mag(acb_realref(w), err);
else
acb_add_error_mag(w, err);
mag_clear(err);
}
}
static void
bsplit(arb_poly_t pol, const arb_t sqrtD,
const slong * qbf, slong a, slong b, slong prec)
{
if (b - a == 0)
{
arb_poly_one(pol);
}
else if (b - a == 1)
{
acb_t z;
acb_init(z);
/* j((-b+sqrt(-D))/(2a)) */
arb_set_si(acb_realref(z), -FLINT_ABS(qbf[3 * a + 1]));
arb_set(acb_imagref(z), sqrtD);
acb_div_si(z, z, 2 * qbf[3 * a], prec);
acb_modular_j(z, z, prec);
if (qbf[3 * a + 1] < 0)
{
/* (x^2 - 2re(j) x + |j|^2) */
arb_poly_fit_length(pol, 3);
arb_mul(pol->coeffs, acb_realref(z), acb_realref(z), prec);
arb_addmul(pol->coeffs, acb_imagref(z), acb_imagref(z), prec);
arb_mul_2exp_si(pol->coeffs + 1, acb_realref(z), 1);
arb_neg(pol->coeffs + 1, pol->coeffs + 1);
arb_one(pol->coeffs + 2);
_arb_poly_set_length(pol, 3);
}
else
{
/* (x-j) */
arb_poly_fit_length(pol, 2);
arb_neg(pol->coeffs, acb_realref(z));
arb_one(pol->coeffs + 1);
_arb_poly_set_length(pol, 2);
}
acb_clear(z);
}
else
{
arb_poly_t tmp;
arb_poly_init(tmp);
bsplit(pol, sqrtD, qbf, a, a + (b - a) / 2, prec);
bsplit(tmp, sqrtD, qbf, a + (b - a) / 2, b, prec);
arb_poly_mul(pol, pol, tmp, prec);
arb_poly_clear(tmp);
}
}
