void arf_twobytwo_diag(arf_t u1, arf_t u2, const arf_t a, const arf_t b, const arf_t d, slong prec) {
// Compute the orthogonal matrix that diagonalizes
//
// A = [a b]
// [b d]
//
// This matrix will have the form
//
// U = [cos x , -sin x]
// [sin x, cos x]
//
// where the diagonal matrix is U^t A U.
// We set u1 = cos x, u2 = -sin x.
if(arf_is_zero(b)) {
arf_set_ui(u1, 1);
arf_set_ui(u2, 0);
return;
}
arf_t x; arf_init(x);
arf_mul(u1, b, b, prec, ARF_RND_NEAR); // u1 = b^2
arf_sub(u2, a, d, prec, ARF_RND_NEAR); // u2 = a - d
arf_mul_2exp_si(u2, u2, -1); // u2 = (a - d)/2
arf_mul(u2, u2, u2, prec, ARF_RND_NEAR); // u2 = ( (a - d)/2 )^2
arf_add(u1, u1, u2, prec, ARF_RND_NEAR); // u1 = b^2 + ( (a-d)/2 )^2
arf_sqrt(u1, u1, prec, ARF_RND_NEAR); // u1 = sqrt(above)
arf_mul_2exp_si(u1, u1, 1); // u1 = 2 (sqrt (above) )
arf_add(u1, u1, d, prec, ARF_RND_NEAR); // u1 += d
arf_sub(u1, u1, a, prec, ARF_RND_NEAR); // u1 -= a
arf_mul_2exp_si(u1, u1, -1); // u1 = (d - a)/2 + sqrt(b^2 + ( (a-d)/2 )^2)
arf_mul(x, u1, u1, prec, ARF_RND_NEAR);
arf_addmul(x, b, b, prec, ARF_RND_NEAR); // x = u1^2 + b^2
arf_sqrt(x, x, prec, ARF_RND_NEAR); // x = sqrt(u1^2 + b^2)
arf_div(u2, u1, x, prec, ARF_RND_NEAR);
arf_div(u1, b, x, prec, ARF_RND_NEAR);
arf_neg(u1, u1);
arf_clear(x);
}
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);
}
int
arf_root(arf_ptr z, arf_srcptr x, ulong k, slong prec, arf_rnd_t rnd)
{
mp_size_t xn, zn, val;
mp_srcptr xptr;
mp_ptr tmp, zptr;
mpfr_t xf, zf;
fmpz_t q, r;
int inexact;
if (k == 0)
{
arf_nan(z);
return 0;
}
if (k == 1)
return arf_set_round(z, x, prec, rnd);
if (k == 2)
return arf_sqrt(z, x, prec, rnd);
if (arf_is_special(x))
{
if (arf_is_neg_inf(x))
arf_nan(z);
else
arf_set(z, x);
return 0;
}
if (ARF_SGNBIT(x))
{
arf_nan(z);
return 0;
}
fmpz_init(q);
fmpz_init(r);
/* x = m * 2^e where e = qk + r */
/* x^(1/k) = (m * 2^(qk+r))^(1/k) */
/* x^(1/k) = (m * 2^r)^(1/k) * 2^q */
fmpz_set_ui(r, k);
fmpz_fdiv_qr(q, r, ARF_EXPREF(x), r);
ARF_GET_MPN_READONLY(xptr, xn, x);
zn = (prec + FLINT_BITS - 1) / FLINT_BITS;
zf->_mpfr_d = tmp = flint_malloc(zn * sizeof(mp_limb_t));
zf->_mpfr_prec = prec;
zf->_mpfr_sign = 1;
zf->_mpfr_exp = 0;
xf->_mpfr_d = (mp_ptr) xptr;
xf->_mpfr_prec = xn * FLINT_BITS;
xf->_mpfr_sign = 1;
xf->_mpfr_exp = fmpz_get_ui(r);
inexact = mpfr_root(zf, xf, k, arf_rnd_to_mpfr(rnd));
inexact = (inexact != 0);
val = 0;
while (tmp[val] == 0)
val++;
ARF_GET_MPN_WRITE(zptr, zn - val, z);
flint_mpn_copyi(zptr, tmp + val, zn - val);
fmpz_add_si(ARF_EXPREF(z), q, zf->_mpfr_exp);
flint_free(tmp);
fmpz_clear(q);
fmpz_clear(r);
return inexact;
}