void
_arb_poly_sqrt_series(arb_ptr g,
arb_srcptr h, long hlen, long len, long prec)
{
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
arb_sqrt(g, h, prec);
_arb_vec_zero(g + 1, len - 1);
}
else if (len == 2)
{
arb_sqrt(g, h, prec);
arb_div(g + 1, h + 1, h, prec);
arb_mul(g + 1, g + 1, g, prec);
arb_mul_2exp_si(g + 1, g + 1, -1);
}
else
{
arb_ptr t;
t = _arb_vec_init(len);
_arb_poly_rsqrt_series(t, h, hlen, len, prec);
_arb_poly_mullow(g, t, len, h, hlen, len, prec);
_arb_vec_clear(t, len);
}
}
void
arb_root_ui(arb_t res, const arb_t x, ulong k, slong prec)
{
if (k == 0)
{
arb_indeterminate(res);
}
else if (k == 1)
{
arb_set_round(res, x, prec);
}
else if (k == 2)
{
arb_sqrt(res, x, prec);
}
else if (k == 4)
{
arb_sqrt(res, x, prec + 2);
arb_sqrt(res, res, prec);
}
else
{
if (k > 50 || prec < (WORD(1) << ((k / 8) + 8)))
arb_root_ui_exp(res, x, k, prec);
else
arb_root_ui_algebraic(res, x, k, prec);
}
}
void
acb_hypgeom_fresnel(acb_t res1, acb_t res2, const acb_t z, int normalized, slong prec)
{
slong wp;
acb_t w;
arb_t c;
if (!acb_is_finite(z))
{
if (res1 != NULL) acb_indeterminate(res1);
if (res2 != NULL) acb_indeterminate(res2);
return;
}
acb_init(w);
arb_init(c);
wp = prec + 8;
if (normalized)
{
arb_const_pi(c, wp);
arb_sqrt(c, c, wp);
arb_mul_2exp_si(c, c, -1);
acb_mul_arb(w, z, c, wp);
acb_hypgeom_fresnel_erf_error(res1, res2, w, wp);
}
else
{
arb_sqrt_ui(c, 2, wp);
arb_mul_2exp_si(c, c, -1);
acb_mul_arb(w, z, c, wp);
acb_hypgeom_fresnel_erf_error(res1, res2, w, wp);
arb_const_pi(c, wp);
arb_mul_2exp_si(c, c, -1);
arb_sqrt(c, c, wp);
if (res1 != NULL) acb_mul_arb(res1, res1, c, wp);
if (res2 != NULL) acb_mul_arb(res2, res2, c, wp);
}
if (res1 != NULL)
{
acb_mul_2exp_si(res1, res1, -2);
acb_set_round(res1, res1, prec);
}
if (res2 != NULL)
{
acb_mul_2exp_si(res2, res2, -2);
acb_set_round(res2, res2, prec);
}
acb_clear(w);
arb_clear(c);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("sqrt....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000; iter++)
{
arb_t a, b, c;
slong prec = 2 + n_randint(state, 200);
arb_init(a);
arb_init(b);
arb_init(c);
arb_randtest_special(a, state, 1 + n_randint(state, 200), 100);
arb_sqrt(b, a, prec);
arb_mul(c, b, b, prec);
if (!arb_contains(c, a))
{
flint_printf("FAIL: containment\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
abort();
}
arb_sqrt(a, a, prec);
if (!arb_equal(a, b))
{
flint_printf("FAIL: aliasing\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
arb_sqrt1pm1(arb_t r, const arb_t z, slong prec)
{
slong magz, wp;
if (arb_is_zero(z))
{
arb_zero(r);
return;
}
magz = arf_abs_bound_lt_2exp_si(arb_midref(z));
if (magz < -prec)
{
arb_sqrt1pm1_tiny(r, z, prec);
}
else
{
if (magz < 0)
wp = prec + (-magz) + 4;
else
wp = prec + 4;
arb_add_ui(r, z, 1, wp);
arb_sqrt(r, r, wp);
arb_sub_ui(r, r, 1, wp);
}
}
void
arb_pow(arb_t z, const arb_t x, const arb_t y, slong prec)
{
if (arb_is_zero(y))
{
arb_one(z);
return;
}
if (arb_is_zero(x))
{
if (arb_is_positive(y))
arb_zero(z);
else
arb_indeterminate(z);
return;
}
if (arb_is_exact(y) && !arf_is_special(arb_midref(x)))
{
const arf_struct * ymid = arb_midref(y);
/* small half-integer or integer */
if (arf_cmpabs_2exp_si(ymid, BINEXP_LIMIT) < 0 &&
arf_is_int_2exp_si(ymid, -1))
{
fmpz_t e;
fmpz_init(e);
if (arf_is_int(ymid))
{
arf_get_fmpz_fixed_si(e, ymid, 0);
arb_pow_fmpz_binexp(z, x, e, prec);
}
else
{
arf_get_fmpz_fixed_si(e, ymid, -1);
arb_sqrt(z, x, prec + fmpz_bits(e));
arb_pow_fmpz_binexp(z, z, e, prec);
}
fmpz_clear(e);
return;
}
else if (arf_is_int(ymid) && arf_sgn(arb_midref(x)) < 0)
{
/* use (-x)^n = (-1)^n * x^n to avoid NaNs
at least at high enough precision */
int odd = !arf_is_int_2exp_si(ymid, 1);
_arb_pow_exp(z, x, 1, y, prec);
if (odd)
arb_neg(z, z);
return;
}
}
_arb_pow_exp(z, x, 0, y, prec);
}
void arb_twobytwo_diag(arb_t u1, arb_t u2, const arb_t a, const arb_t b, const arb_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(arb_contains_zero(b)) {
// this is not quite right (doesn't set error intervals)
arb_set_ui(u1, 1);
arb_set_ui(u2, 0);
return;
}
arb_t x; arb_init(x);
arb_mul(u1, b, b, prec); // u1 = b^2
arb_sub(u2, a, d, prec); // u2 = a - d
arb_mul_2exp_si(u2, u2, -1); // u2 = (a - d)/2
arb_mul(u2, u2, u2, prec); // u2 = ( (a - d)/2 )^2
arb_add(u1, u1, u2, prec); // u1 = b^2 + ( (a-d)/2 )^2
arb_sqrt(u1, u1, prec); // u1 = sqrt(above)
arb_mul_2exp_si(u1, u1, 1); // u1 = 2 (sqrt (above) )
arb_add(u1, u1, d, prec); // u1 += d
arb_sub(u1, u1, a, prec); // u1 -= a
arb_mul_2exp_si(u1, u1, -1); // u1 = (d - a)/2 + sqrt(b^2 + ( (a-d)/2 )^2)
arb_mul(x, u1, u1, prec);
arb_addmul(x, b, b, prec); // x = u1^2 + b^2
arb_sqrt(x, x, prec); // x = sqrt(u1^2 + b^2)
arb_div(u2, u1, x, prec);
arb_div(u1, b, x, prec);
arb_neg(u1, u1);
arb_clear(x);
}
void
arb_agm(arb_t z, const arb_t x, const arb_t y, long prec)
{
arb_t t, u, v, w;
if (arb_contains_negative(x) || arb_contains_negative(y))
{
arb_indeterminate(z);
return;
}
if (arb_is_zero(x) || arb_is_zero(y))
{
arb_zero(z);
return;
}
arb_init(t);
arb_init(u);
arb_init(v);
arb_init(w);
arb_set(t, x);
arb_set(u, y);
while (!arb_overlaps(t, u) &&
!arb_contains_nonpositive(t) &&
!arb_contains_nonpositive(u))
{
arb_add(v, t, u, prec);
arb_mul_2exp_si(v, v, -1);
arb_mul(w, t, u, prec);
arb_sqrt(w, w, prec);
arb_swap(v, t);
arb_swap(w, u);
}
if (!arb_is_finite(t) || !arb_is_finite(u))
{
arb_indeterminate(z);
}
else
{
arb_union(z, t, u, prec);
}
arb_clear(t);
arb_clear(u);
arb_clear(v);
arb_clear(w);
}
/*
Bound for scaled Bessel function: 2/(2 pi x)^(1/2)
Bound for tail of integral: 2 N (k / (pi N))^(k / 2) / (k - 2).
*/
void
scaled_bessel_tail_bound(arb_t b, ulong k, const arb_t N, slong prec)
{
arb_const_pi(b, prec);
arb_mul(b, b, N, prec);
arb_ui_div(b, k, b, prec);
arb_sqrt(b, b, prec);
arb_pow_ui(b, b, k, prec);
arb_mul(b, b, N, prec);
arb_mul_ui(b, b, 2, prec);
arb_div_ui(b, b, k - 2, prec);
}
void
_arb_poly_sqrt_series(arb_ptr g,
arb_srcptr h, slong hlen, slong len, slong prec)
{
hlen = FLINT_MIN(hlen, len);
while (hlen > 0 && arb_is_zero(h + hlen - 1))
hlen--;
if (hlen <= 1)
{
arb_sqrt(g, h, prec);
_arb_vec_zero(g + 1, len - 1);
}
else if (len == 2)
{
arb_sqrt(g, h, prec);
arb_div(g + 1, h + 1, h, prec);
arb_mul(g + 1, g + 1, g, prec);
arb_mul_2exp_si(g + 1, g + 1, -1);
}
else if (_arb_vec_is_zero(h + 1, hlen - 2))
{
arb_t t;
arb_init(t);
arf_set_si_2exp_si(arb_midref(t), 1, -1);
_arb_poly_binomial_pow_arb_series(g, h, hlen, t, len, prec);
arb_clear(t);
}
else
{
arb_ptr t;
t = _arb_vec_init(len);
_arb_poly_rsqrt_series(t, h, hlen, len, prec);
_arb_poly_mullow(g, t, len, h, hlen, len, prec);
_arb_vec_clear(t, len);
}
}
void arb_mat_cholesky(arb_mat_t out, const arb_mat_t in, slong prec) {
int nrows = arb_mat_nrows(in);
for(int j = 0; j < nrows; j++) {
for(int i = j; i < nrows; i++) {
arb_set(arb_mat_entry(out, i, j), arb_mat_entry(in, i, j));
for(int k = 0; k < j; k++) {
arb_submul(arb_mat_entry(out, i, j), arb_mat_entry(out, i, k), arb_mat_entry(out, j, k), prec);
}
if(i == j) {
arb_sqrt(arb_mat_entry(out, i, j), arb_mat_entry(out, i, j), prec);
}
else {
arb_div(arb_mat_entry(out, i, j), arb_mat_entry(out, i, j), arb_mat_entry(out, j, j), prec);
}
}
}
}
void
acb_dirichlet_vec_mellin_arb(acb_ptr res, const dirichlet_group_t G, const dirichlet_char_t chi, slong len, const arb_t t, slong n, slong prec)
{
slong k;
arb_t tk, xt, stk, st;
acb_ptr a;
mag_t e;
a = _acb_vec_init(len);
acb_dirichlet_chi_vec(a, G, chi, len, prec);
if (dirichlet_parity_char(G, chi))
{
for (k = 2; k < len; k++)
acb_mul_si(a + k, a + k, k, prec);
}
arb_init(tk);
arb_init(xt);
arb_init(st);
arb_init(stk);
mag_init(e);
arb_sqrt(st, t, prec);
arb_one(tk);
arb_one(stk);
for (k = 0; k < n; k++)
{
_acb_dirichlet_theta_argument_at_arb(xt, G->q, tk, prec);
mag_tail_kexpk2_arb(e, xt, len);
arb_neg(xt, xt);
arb_exp(xt, xt, prec);
/* TODO: reduce len */
acb_dirichlet_qseries_arb(res + k, a, xt, len, prec);
acb_add_error_mag(res + k, e);
acb_mul_arb(res + k, res + k, stk, prec);
arb_mul(tk, tk, t, prec);
arb_mul(stk, stk, st, prec);
}
mag_clear(e);
arb_clear(xt);
arb_clear(tk);
arb_clear(stk);
arb_clear(st);
_acb_vec_clear(a, len);
}
void
arb_acosh(arb_t z, const arb_t x, slong prec)
{
if (arb_is_one(x))
{
arb_zero(z);
}
else
{
arb_t t;
arb_init(t);
arb_mul(t, x, x, prec + 4);
arb_sub_ui(t, t, 1, prec + 4);
arb_sqrt(t, t, prec + 4);
arb_add(t, t, x, prec + 4);
arb_log(z, t, prec);
arb_clear(t);
}
}
int
_acb_modular_hilbert_class_poly(fmpz_poly_t res, slong D,
const slong * qbf, slong qbf_len, slong prec)
{
arb_t sqrtD;
arb_poly_t pol;
int success;
arb_init(sqrtD);
arb_poly_init(pol);
arb_set_si(sqrtD, -D);
arb_sqrt(sqrtD, sqrtD, prec);
bsplit(pol, sqrtD, qbf, 0, qbf_len, prec);
success = arb_poly_get_unique_fmpz_poly(res, pol);
arb_clear(sqrtD);
arb_poly_clear(pol);
return success;
}
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 main()
{
slong iter;
flint_rand_t state;
flint_printf("sqrt1pm1....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 20000; iter++)
{
arb_t a, b, c, d;
slong prec0, prec1, prec2;
if (iter % 10 == 0)
prec0 = 10000;
else
prec0 = 1000;
prec1 = 2 + n_randint(state, prec0);
prec2 = 2 + n_randint(state, prec0);
arb_init(a);
arb_init(b);
arb_init(c);
arb_init(d);
arb_randtest_special(a, state, 1 + n_randint(state, prec0), 100);
arb_randtest_special(b, state, 1 + n_randint(state, prec0), 100);
arb_randtest_special(c, state, 1 + n_randint(state, prec0), 100);
arb_sqrt1pm1(b, a, prec1);
arb_sqrt1pm1(c, a, prec2);
if (!arb_overlaps(b, c))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
abort();
}
/* compare with sqrt */
arb_add_ui(d, a, 1, prec2);
arb_sqrt(d, d, prec2);
arb_sub_ui(d, d, 1, prec2);
if (!arb_overlaps(c, d))
{
flint_printf("FAIL: comparison with log\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("c = "); arb_print(c); flint_printf("\n\n");
flint_printf("d = "); arb_print(d); flint_printf("\n\n");
abort();
}
arb_sqrt1pm1(a, a, prec1);
if (!arb_overlaps(a, b))
{
flint_printf("FAIL: aliasing\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
arb_clear(d);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
acb_sqrt(acb_t y, const acb_t x, slong prec)
{
arb_t r, t, u;
slong wp;
#define a acb_realref(x)
#define b acb_imagref(x)
#define c acb_realref(y)
#define d acb_imagref(y)
if (arb_is_zero(b))
{
if (arb_is_nonnegative(a))
{
arb_sqrt(c, a, prec);
arb_zero(d);
return;
}
else if (arb_is_nonpositive(a))
{
arb_neg(d, a);
arb_sqrt(d, d, prec);
arb_zero(c);
return;
}
}
if (arb_is_zero(a))
{
if (arb_is_nonnegative(b))
{
arb_mul_2exp_si(c, b, -1);
arb_sqrt(c, c, prec);
arb_set(d, c);
return;
}
else if (arb_is_nonpositive(b))
{
arb_mul_2exp_si(c, b, -1);
arb_neg(c, c);
arb_sqrt(c, c, prec);
arb_neg(d, c);
return;
}
}
wp = prec + 4;
arb_init(r);
arb_init(t);
arb_init(u);
acb_abs(r, x, wp);
arb_add(t, r, a, wp);
if (arb_rel_accuracy_bits(t) > 8)
{
/* sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i, r = |a+bi| */
arb_mul_2exp_si(u, t, 1);
arb_sqrt(u, u, wp);
arb_div(d, b, u, prec);
arb_set_round(c, u, prec);
arb_mul_2exp_si(c, c, -1);
}
else
{
/*
sqrt(a+bi) = sqrt((r+a)/2) + (b/|b|)*sqrt((r-a)/2)*i
(sign)
*/
arb_mul_2exp_si(t, t, -1);
arb_sub(u, r, a, wp);
arb_mul_2exp_si(u, u, -1);
arb_sqrtpos(c, t, prec);
if (arb_is_nonnegative(b))
{
arb_sqrtpos(d, u, prec);
}
else if (arb_is_nonpositive(b))
{
arb_sqrtpos(d, u, prec);
arb_neg(d, d);
}
else
{
arb_sqrtpos(t, u, wp);
arb_neg(u, t);
arb_union(d, t, u, prec);
}
}
arb_clear(r);
arb_clear(t);
arb_clear(u);
#undef a
#undef b
#undef c
#undef d
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("frobenius_norm....");
fflush(stdout);
flint_randinit(state);
/* compare to the exact rational norm */
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
fmpq_mat_t Q;
fmpq_t q;
arb_mat_t A;
slong n, qbits, prec;
n = n_randint(state, 8);
qbits = 1 + n_randint(state, 100);
prec = 2 + n_randint(state, 200);
fmpq_mat_init(Q, n, n);
fmpq_init(q);
arb_mat_init(A, n, n);
fmpq_mat_randtest(Q, state, qbits);
_fmpq_mat_sum_of_squares(q, Q);
arb_mat_set_fmpq_mat(A, Q, prec);
/* check that the arb interval contains the exact value */
{
arb_t a;
arb_init(a);
arb_mat_frobenius_norm(a, A, prec);
arb_mul(a, a, a, prec);
if (!arb_contains_fmpq(a, q))
{
flint_printf("FAIL (containment, iter = %wd)\n", iter);
flint_printf("n = %wd, prec = %wd\n", n, prec);
flint_printf("\n");
flint_printf("Q = \n");
fmpq_mat_print(Q);
flint_printf("\n\n");
flint_printf("frobenius_norm(Q)^2 = \n");
fmpq_print(q);
flint_printf("\n\n");
flint_printf("A = \n");
arb_mat_printd(A, 15);
flint_printf("\n\n");
flint_printf("frobenius_norm(A)^2 = \n");
arb_printd(a, 15);
flint_printf("\n\n");
flint_printf("frobenius_norm(A)^2 = \n");
arb_print(a);
flint_printf("\n\n");
abort();
}
arb_clear(a);
}
/* check that the upper bound is not less than the exact value */
{
mag_t b;
fmpq_t y;
mag_init(b);
fmpq_init(y);
arb_mat_bound_frobenius_norm(b, A);
mag_mul(b, b, b);
mag_get_fmpq(y, b);
if (fmpq_cmp(q, y) > 0)
{
flint_printf("FAIL (bound, iter = %wd)\n", iter);
flint_printf("n = %wd, prec = %wd\n", n, prec);
flint_printf("\n");
flint_printf("Q = \n");
fmpq_mat_print(Q);
flint_printf("\n\n");
flint_printf("frobenius_norm(Q)^2 = \n");
fmpq_print(q);
flint_printf("\n\n");
flint_printf("A = \n");
arb_mat_printd(A, 15);
flint_printf("\n\n");
flint_printf("bound_frobenius_norm(A)^2 = \n");
mag_printd(b, 15);
flint_printf("\n\n");
flint_printf("bound_frobenius_norm(A)^2 = \n");
mag_print(b);
flint_printf("\n\n");
abort();
}
mag_clear(b);
fmpq_clear(y);
}
fmpq_mat_clear(Q);
fmpq_clear(q);
arb_mat_clear(A);
}
/* check trace(A^T A) = frobenius_norm(A)^2 */
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
slong m, n, prec;
arb_mat_t A, AT, ATA;
arb_t t;
prec = 2 + n_randint(state, 200);
m = n_randint(state, 10);
n = n_randint(state, 10);
arb_mat_init(A, m, n);
arb_mat_init(AT, n, m);
arb_mat_init(ATA, n, n);
arb_init(t);
arb_mat_randtest(A, state, 2 + n_randint(state, 100), 10);
arb_mat_transpose(AT, A);
arb_mat_mul(ATA, AT, A, prec);
arb_mat_trace(t, ATA, prec);
arb_sqrt(t, t, prec);
/* check the norm bound */
{
mag_t low, frobenius;
mag_init(low);
arb_get_mag_lower(low, t);
mag_init(frobenius);
arb_mat_bound_frobenius_norm(frobenius, A);
if (mag_cmp(low, frobenius) > 0)
{
flint_printf("FAIL (bound)\n", iter);
flint_printf("m = %wd, n = %wd, prec = %wd\n", m, n, prec);
flint_printf("\n");
flint_printf("A = \n");
arb_mat_printd(A, 15);
flint_printf("\n\n");
flint_printf("lower(sqrt(trace(A^T A))) = \n");
mag_printd(low, 15);
flint_printf("\n\n");
flint_printf("bound_frobenius_norm(A) = \n");
mag_printd(frobenius, 15);
flint_printf("\n\n");
abort();
}
mag_clear(low);
mag_clear(frobenius);
}
/* check the norm interval */
{
arb_t frobenius;
arb_init(frobenius);
arb_mat_frobenius_norm(frobenius, A, prec);
if (!arb_overlaps(t, frobenius))
{
flint_printf("FAIL (overlap)\n", iter);
flint_printf("m = %wd, n = %wd, prec = %wd\n", m, n, prec);
flint_printf("\n");
flint_printf("A = \n");
arb_mat_printd(A, 15);
flint_printf("\n\n");
flint_printf("sqrt(trace(A^T A)) = \n");
arb_printd(t, 15);
flint_printf("\n\n");
flint_printf("frobenius_norm(A) = \n");
arb_printd(frobenius, 15);
flint_printf("\n\n");
abort();
}
arb_clear(frobenius);
}
arb_mat_clear(A);
arb_mat_clear(AT);
arb_mat_clear(ATA);
arb_clear(t);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
