void
arb_div_arf(arb_t z, const arb_t x, const arf_t y, long prec)
{
mag_t zr, ym;
int inexact;
if (arf_is_zero(y))
{
arb_zero_pm_inf(z);
}
else if (arb_is_exact(x))
{
inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec);
else
mag_zero(arb_radref(z));
}
else if (mag_is_inf(arb_radref(x)))
{
arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
mag_inf(arb_radref(z));
}
else
{
mag_init(ym);
mag_init(zr);
arf_get_mag_lower(ym, y);
mag_div(zr, arb_radref(x), ym);
inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_swap(arb_radref(z), zr);
mag_clear(ym);
mag_clear(zr);
}
}
void
arb_sqrtpos(arb_t z, const arb_t x, long prec)
{
if (!arb_is_finite(x))
{
if (mag_is_zero(arb_radref(x)) && arf_is_pos_inf(arb_midref(x)))
arb_pos_inf(z);
else
arb_zero_pm_inf(z);
}
else if (arb_contains_nonpositive(x))
{
arf_t t;
arf_init(t);
arf_set_mag(t, arb_radref(x));
arf_add(t, arb_midref(x), t, MAG_BITS, ARF_RND_CEIL);
if (arf_sgn(t) <= 0)
{
arb_zero(z);
}
else
{
arf_sqrt(t, t, MAG_BITS, ARF_RND_CEIL);
arf_mul_2exp_si(t, t, -1);
arf_set(arb_midref(z), t);
arf_get_mag(arb_radref(z), t);
}
arf_clear(t);
}
else
{
arb_sqrt(z, x, prec);
}
arb_nonnegative_part(z, z, prec);
}
void
acb_calc_cauchy_bound(arb_t bound, acb_calc_func_t func, void * param,
const acb_t x, const arb_t radius, slong maxdepth, slong prec)
{
slong i, n, depth, wp;
arb_t pi, theta, v, s1, c1, s2, c2, st, ct;
acb_t t, u;
arb_t b;
arb_init(pi);
arb_init(theta);
arb_init(v);
arb_init(s1);
arb_init(c1);
arb_init(s2);
arb_init(c2);
arb_init(st);
arb_init(ct);
acb_init(t);
acb_init(u);
arb_init(b);
wp = prec + 20;
arb_const_pi(pi, wp);
arb_zero_pm_inf(b);
for (depth = 0, n = 16; depth < maxdepth; n *= 2, depth++)
{
arb_zero(b);
/* theta = 2 pi / n */
arb_div_ui(theta, pi, n, wp);
arb_mul_2exp_si(theta, theta, 1);
/* sine and cosine of i*theta and (i+1)*theta */
arb_zero(s1);
arb_one(c1);
arb_sin_cos(st, ct, theta, wp);
arb_set(s2, st);
arb_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 */
arb_union(acb_realref(t), c1, c2, wp);
arb_union(acb_imagref(t), s1, s2, wp);
acb_mul_arb(t, t, radius, wp);
acb_add(t, t, x, prec);
/* next angle */
arb_mul(v, c2, ct, wp);
arb_mul(c1, s2, st, wp);
arb_sub(c1, v, c1, wp);
arb_mul(v, c2, st, wp);
arb_mul(s1, s2, ct, wp);
arb_add(s1, v, s1, wp);
arb_swap(c1, c2);
arb_swap(s1, s2);
func(u, t, param, 1, prec);
acb_abs(v, u, prec);
arb_add(b, b, v, prec);
}
arb_div_ui(b, b, n, prec);
if (arb_is_positive(b))
break;
}
arb_set(bound, b);
arb_clear(pi);
arb_clear(theta);
arb_clear(v);
acb_clear(t);
acb_clear(u);
arb_clear(b);
arb_clear(s1);
arb_clear(c1);
arb_clear(s2);
arb_clear(c2);
arb_clear(st);
arb_clear(ct);
}
slong
_acb_poly_find_roots(acb_ptr roots,
acb_srcptr poly,
acb_srcptr initial, slong len, slong maxiter, slong prec)
{
slong iter, i, deg;
slong rootmag, max_rootmag, correction, max_correction;
deg = len - 1;
if (deg == 0)
{
return 0;
}
else if (acb_contains_zero(poly + len - 1))
{
/* if the leading coefficient contains zero, roots can be anywhere */
for (i = 0; i < deg; i++)
{
arb_zero_pm_inf(acb_realref(roots + i));
arb_zero_pm_inf(acb_imagref(roots + i));
}
return 0;
}
else if (deg == 1)
{
acb_inv(roots + 0, poly + 1, prec);
acb_mul(roots + 0, roots + 0, poly + 0, prec);
acb_neg(roots + 0, roots + 0);
return 1;
}
if (initial == NULL)
_acb_poly_roots_initial_values(roots, deg, prec);
else
_acb_vec_set(roots, initial, deg);
if (maxiter == 0)
maxiter = 2 * deg + n_sqrt(prec);
for (iter = 0; iter < maxiter; iter++)
{
max_rootmag = -ARF_PREC_EXACT;
for (i = 0; i < deg; i++)
{
rootmag = _acb_get_mid_mag(roots + i);
max_rootmag = FLINT_MAX(rootmag, max_rootmag);
}
_acb_poly_refine_roots_durand_kerner(roots, poly, len, prec);
max_correction = -ARF_PREC_EXACT;
for (i = 0; i < deg; i++)
{
correction = _acb_get_rad_mag(roots + i);
max_correction = FLINT_MAX(correction, max_correction);
}
/* estimate the correction relative to the whole set of roots */
max_correction -= max_rootmag;
/* flint_printf("ITER %wd MAX CORRECTION: %wd\n", iter, max_correction); */
if (max_correction < -prec / 2)
maxiter = FLINT_MIN(maxiter, iter + 2);
else if (max_correction < -prec / 3)
maxiter = FLINT_MIN(maxiter, iter + 3);
else if (max_correction < -prec / 4)
maxiter = FLINT_MIN(maxiter, iter + 4);
}
return _acb_poly_validate_roots(roots, poly, len, prec);
}
