void
fmprb_sqrtpos(fmprb_t z, const fmprb_t x, slong prec)
{
if (!fmprb_is_finite(x))
{
if (fmpr_is_zero(fmprb_radref(x)) && fmpr_is_pos_inf(fmprb_midref(x)))
fmprb_pos_inf(z);
else
fmprb_zero_pm_inf(z);
}
else if (fmprb_contains_nonpositive(x))
{
fmpr_t t;
fmpr_init(t);
fmpr_add(t, fmprb_midref(x), fmprb_radref(x),
FMPRB_RAD_PREC, FMPR_RND_CEIL);
if (fmpr_sgn(t) <= 0)
{
fmprb_zero(z);
}
else
{
fmpr_sqrt(t, t, FMPRB_RAD_PREC, FMPR_RND_CEIL);
fmpr_mul_2exp_si(t, t, -1);
fmpr_set(fmprb_midref(z), t);
fmpr_set(fmprb_radref(z), t);
}
fmpr_clear(t);
}
else
{
fmprb_sqrt(z, x, prec);
}
fmprb_nonnegative_part(z, z, prec);
}
void
fmpcb_calc_cauchy_bound(fmprb_t bound, fmpcb_calc_func_t func, void * param,
const fmpcb_t x, const fmprb_t radius, long maxdepth, long prec)
{
long i, n, depth, wp;
fmprb_t pi, theta, v, s1, c1, s2, c2, st, ct;
fmpcb_t t, u;
fmprb_t b;
fmprb_init(pi);
fmprb_init(theta);
fmprb_init(v);
fmprb_init(s1);
fmprb_init(c1);
fmprb_init(s2);
fmprb_init(c2);
fmprb_init(st);
fmprb_init(ct);
fmpcb_init(t);
fmpcb_init(u);
fmprb_init(b);
wp = prec + 20;
fmprb_const_pi(pi, wp);
fmprb_zero_pm_inf(b);
for (depth = 0, n = 16; depth < maxdepth; n *= 2, depth++)
{
fmprb_zero(b);
/* theta = 2 pi / n */
fmprb_div_ui(theta, pi, n, wp);
fmprb_mul_2exp_si(theta, theta, 1);
/* sine and cosine of i*theta and (i+1)*theta */
fmprb_zero(s1);
fmprb_one(c1);
fmprb_sin_cos(st, ct, theta, wp);
fmprb_set(s2, st);
fmprb_set(c2, ct);
for (i = 0; i < n; i++)
{
/* sine and cosine of 2 pi ([i,i+1]/n) */
/* since we use power of two subdivision points, the
sine and cosine are monotone on each subinterval */
fmprb_union(fmpcb_realref(t), c1, c2, wp);
fmprb_union(fmpcb_imagref(t), s1, s2, wp);
fmpcb_mul_fmprb(t, t, radius, wp);
fmpcb_add(t, t, x, prec);
/* next angle */
fmprb_mul(v, c2, ct, wp);
fmprb_mul(c1, s2, st, wp);
fmprb_sub(c1, v, c1, wp);
fmprb_mul(v, c2, st, wp);
fmprb_mul(s1, s2, ct, wp);
fmprb_add(s1, v, s1, wp);
fmprb_swap(c1, c2);
fmprb_swap(s1, s2);
func(u, t, param, 1, prec);
fmpcb_abs(v, u, prec);
fmprb_add(b, b, v, prec);
}
fmprb_div_ui(b, b, n, prec);
if (fmprb_is_exact(b) || fmpr_cmp(fmprb_radref(b), fmprb_midref(b)) < 0)
break;
}
fmprb_set(bound, b);
fmprb_clear(pi);
fmprb_clear(theta);
fmprb_clear(v);
fmpcb_clear(t);
fmpcb_clear(u);
fmprb_clear(b);
fmprb_clear(s1);
fmprb_clear(c1);
fmprb_clear(s2);
fmprb_clear(c2);
fmprb_clear(st);
fmprb_clear(ct);
}
