void
arb_root_ui_algebraic(arb_t res, const arb_t x, ulong k, slong prec)
{
mag_t r, msubr, m1k, t;
if (arb_is_exact(x))
{
arb_root_arf(res, arb_midref(x), k, prec);
return;
}
if (!arb_is_nonnegative(x))
{
arb_indeterminate(res);
return;
}
mag_init(r);
mag_init(msubr);
mag_init(m1k);
mag_init(t);
/* x = [m-r, m+r] */
mag_set(r, arb_radref(x));
/* m - r */
arb_get_mag_lower(msubr, x);
/* m^(1/k) */
arb_root_arf(res, arb_midref(x), k, prec);
/* bound for m^(1/k) */
arb_get_mag(m1k, res);
/* C = min(1, log(1+r/(m-r))/k) */
mag_div(t, r, msubr);
mag_log1p(t, t);
mag_div_ui(t, t, k);
if (mag_cmp_2exp_si(t, 0) > 0)
mag_one(t);
/* C m^(1/k) */
mag_mul(t, m1k, t);
mag_add(arb_radref(res), arb_radref(res), t);
mag_clear(r);
mag_clear(msubr);
mag_clear(m1k);
mag_clear(t);
}
void
mag_log_ui(mag_t t, ulong n)
{
if (n <= 1)
{
if (n == 1)
mag_zero(t);
else
mag_inf(t);
}
else
{
mag_set_ui(t, n-1);
mag_log1p(t, t);
}
}
void
mag_root(mag_t y, const mag_t x, ulong n)
{
if (n == 0)
{
mag_inf(y);
}
else if (n == 1 || mag_is_special(x))
{
mag_set(y, x);
}
else if (n == 2)
{
mag_sqrt(y, x);
}
else if (n == 4)
{
mag_sqrt(y, x);
mag_sqrt(y, y);
}
else
{
fmpz_t e, f;
fmpz_init_set_ui(e, MAG_BITS);
fmpz_init(f);
/* We evaluate exp(log(1+2^(kn)x)/n) 2^-k where k is chosen
so that 2^(kn) x ~= 2^30. TODO: this rewriting is probably
unnecessary with the new exp/log functions. */
fmpz_sub(e, e, MAG_EXPREF(x));
fmpz_cdiv_q_ui(e, e, n);
fmpz_mul_ui(f, e, n);
mag_mul_2exp_fmpz(y, x, f);
mag_log1p(y, y);
mag_div_ui(y, y, n);
mag_exp(y, y);
fmpz_neg(e, e);
mag_mul_2exp_fmpz(y, y, e);
fmpz_clear(e);
fmpz_clear(f);
}
}
void
arb_log(arb_t y, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_log_arf(y, arb_midref(x), prec);
}
else
{
/*
Let the input be [a-b, a+b]. We require a > b >= 0 (otherwise the
interval contains zero or a negative number and the logarithm is not
defined). The error is largest at a-b, and we have
log(a) - log(a-b) = log(1 + b/(a-b)).
*/
mag_t err;
mag_init(err);
arb_get_mag_lower_nonnegative(err, x);
if (mag_is_zero(err))
{
mag_inf(err);
}
else
{
mag_div(err, arb_radref(x), err);
mag_log1p(err, err);
}
arb_log_arf(y, arb_midref(x), prec);
mag_add(arb_radref(y), arb_radref(y), err);
mag_clear(err);
}
}
