void
fmprb_sqrt(fmprb_t z, const fmprb_t x, slong prec)
{
slong r;
if (fmprb_contains_negative(x))
{
fmpr_nan(fmprb_midref(z));
fmpr_pos_inf(fmprb_radref(z));
return;
}
if (fmprb_is_exact(x))
{
r = fmpr_sqrt(fmprb_midref(z), fmprb_midref(x), prec, FMPR_RND_DOWN);
fmpr_set_error_result(fmprb_radref(z), fmprb_midref(z), r);
}
else
{
fmpr_t err;
fmpr_init(err);
fmpr_sub(err, fmprb_midref(x), fmprb_radref(x), FMPRB_RAD_PREC, FMPR_RND_DOWN);
fmpr_rsqrt(err, err, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul(err, fmprb_radref(x), err, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(err, err, -1);
r = fmpr_sqrt(fmprb_midref(z), fmprb_midref(x), prec, FMPR_RND_DOWN);
fmpr_add_error_result(fmprb_radref(z), err, fmprb_midref(z), r,
FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_clear(err);
}
fmprb_adjust(z);
}
/* Absolute value of rising factorial (could speed up once complex gamma is available). */
void
fmpcb_rfac_abs_ubound2(fmpr_t bound, const fmpcb_t s, ulong n, long prec)
{
fmpr_t term, t;
ulong k;
/* M(k) = (a+k)^2 + b^2
M(0) = a^2 + b^2
M(k+1) = M(k) + 2*a + (2*k+1)
*/
fmpr_init(t);
fmpr_init(term);
fmpr_one(bound);
/* M(0) = a^2 + b^2 */
fmprb_get_abs_ubound_fmpr(t, fmpcb_realref(s), prec);
fmpr_mul(term, t, t, prec, FMPR_RND_UP);
fmprb_get_abs_ubound_fmpr(t, fmpcb_imagref(s), prec);
fmpr_mul(t, t, t, prec, FMPR_RND_UP);
fmpr_add(term, term, t, prec, FMPR_RND_UP);
/* we add t = 2*a to each term. note that this can be signed;
we always want the most positive value */
fmpr_add(t, fmprb_midref(fmpcb_realref(s)),
fmprb_radref(fmpcb_realref(s)), prec, FMPR_RND_CEIL);
fmpr_mul_2exp_si(t, t, 1);
for (k = 0; k < n; k++)
{
fmpr_mul(bound, bound, term, prec, FMPR_RND_UP);
fmpr_add_ui(term, term, 2 * k + 1, prec, FMPR_RND_UP);
fmpr_add(term, term, t, prec, FMPR_RND_UP);
}
fmpr_sqrt(bound, bound, prec, FMPR_RND_UP);
fmpr_clear(t);
fmpr_clear(term);
}
void
fmprb_mat_det_inplace(fmprb_t det, fmprb_mat_t A, long prec)
{
long i, n, sign, rank;
n = fmprb_mat_nrows(A);
rank = fmprb_mat_gauss_partial(A, prec);
sign = (rank < 0) ? -1 : 1;
rank = FLINT_ABS(rank);
fmprb_set_si(det, sign);
for (i = 0; i < rank; i++)
fmprb_mul(det, det, fmprb_mat_entry(A, i, i), prec);
/* bound unreduced part using Hadamard's inequality */
if (rank < n)
{
fmpr_t t;
fmprb_t d;
fmpr_init(t);
fmprb_init(d);
fmpr_one(fmprb_radref(d));
for (i = rank; i < n; i++)
{
fmprb_vec_get_fmpr_2norm_squared_bound(t, A->rows[i] + rank,
n - rank, FMPRB_RAD_PREC);
fmpr_mul(fmprb_radref(d), fmprb_radref(d), t, FMPRB_RAD_PREC, FMPR_RND_UP);
}
fmpr_sqrt(fmprb_radref(d), fmprb_radref(d), FMPRB_RAD_PREC, FMPR_RND_UP);
fmprb_mul(det, det, d, prec);
fmprb_clear(d);
fmpr_clear(t);
}
}
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);
}
