static void
arb_sqrt1pm1_tiny(arb_t r, const arb_t z, slong prec)
{
mag_t b, c;
arb_t t;
mag_init(b);
mag_init(c);
arb_init(t);
/* if |z| < 1, then |(sqrt(1+z)-1) - (z/2-z^2/8)| <= |z|^3/(1-|z|)/16 */
arb_get_mag(b, z);
mag_one(c);
mag_sub_lower(c, c, b);
mag_pow_ui(b, b, 3);
mag_div(b, b, c);
mag_mul_2exp_si(b, b, -4);
arb_mul(t, z, z, prec);
arb_mul_2exp_si(t, t, -2);
arb_sub(r, z, t, prec);
arb_mul_2exp_si(r, r, -1);
if (mag_is_finite(b))
arb_add_error_mag(r, b);
else
arb_indeterminate(r);
mag_clear(b);
mag_clear(c);
arb_clear(t);
}
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);
}
void
arb_div_2expm1_ui(arb_t y, const arb_t x, ulong n, long prec)
{
if (n < FLINT_BITS)
{
arb_div_ui(y, x, (1UL << n) - 1, prec);
}
else if (n < 1024 + prec / 32 || n > LONG_MAX / 4)
{
arb_t t;
fmpz_t e;
arb_init(t);
fmpz_init_set_ui(e, n);
arb_one(t);
arb_mul_2exp_fmpz(t, t, e);
arb_sub_ui(t, t, 1, prec);
arb_div(y, x, t, prec);
arb_clear(t);
fmpz_clear(e);
}
else
{
arb_t s, t;
long i, b;
arb_init(s);
arb_init(t);
/* x / (2^n - 1) = sum_{k>=1} x * 2^(-k*n)*/
arb_mul_2exp_si(s, x, -n);
arb_set(t, s);
b = 1;
for (i = 2; i <= prec / n + 1; i++)
{
arb_mul_2exp_si(t, t, -n);
arb_add(s, s, t, prec);
b = i;
}
/* error bound: sum_{k>b} x * 2^(-k*n) <= x * 2^(-b*n - (n-1)) */
arb_mul_2exp_si(t, x, -b*n - (n-1));
arb_abs(t, t);
arb_add_error(s, t);
arb_set(y, s);
arb_clear(s);
arb_clear(t);
}
}
void
arb_sin_cos_pi(arb_t s, arb_t c, const arb_t x, long prec)
{
arb_t t;
arb_t u;
fmpz_t v;
if (arf_cmpabs_2exp_si(arb_midref(x), FLINT_MAX(65536, (4*prec))) > 0)
{
arf_zero(arb_midref(s));
mag_one(arb_radref(s));
arf_zero(arb_midref(c));
mag_one(arb_radref(c));
return;
}
arb_init(t);
arb_init(u);
fmpz_init(v);
arb_mul_2exp_si(t, x, 1);
arf_get_fmpz(v, arb_midref(t), ARF_RND_NEAR);
arb_sub_fmpz(t, t, v, prec);
arb_const_pi(u, prec);
arb_mul(t, t, u, prec);
arb_mul_2exp_si(t, t, -1);
switch (fmpz_fdiv_ui(v, 4))
{
case 0:
arb_sin_cos(s, c, t, prec);
break;
case 1:
arb_sin_cos(c, s, t, prec);
arb_neg(c, c);
break;
case 2:
arb_sin_cos(s, c, t, prec);
arb_neg(s, s);
arb_neg(c, c);
break;
default:
arb_sin_cos(c, s, t, prec);
arb_neg(s, s);
break;
}
fmpz_clear(v);
arb_clear(t);
arb_clear(u);
}
void
arb_sech(arb_t res, const arb_t x, slong prec)
{
if (arf_cmpabs_2exp_si(arb_midref(x), 0) > 0)
{
arb_t t;
arb_init(t);
if (arf_sgn(arb_midref(x)) > 0)
{
arb_neg(t, x);
arb_exp(t, t, prec + 4);
}
else
{
arb_exp(t, x, prec + 4);
}
arb_mul(res, t, t, prec + 4);
arb_add_ui(res, res, 1, prec + 4);
arb_div(res, t, res, prec);
arb_mul_2exp_si(res, res, 1);
arb_clear(t);
}
else
{
arb_cosh(res, x, prec + 4);
arb_inv(res, res, prec);
}
}
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_set_interval_arf(arb_t x, const arf_t a, const arf_t b, slong prec)
{
arf_t t;
int inexact;
if (arf_is_inf(a) && arf_equal(a, b))
{
/* [-inf, -inf] or [+inf, +inf] */
arf_set(arb_midref(x), a);
mag_zero(arb_radref(x));
return;
}
arf_init(t);
arf_sub(t, b, a, MAG_BITS, ARF_RND_UP);
if (arf_sgn(t) < 0)
{
flint_printf("exception: arb_set_interval_arf: endpoints not ordered\n");
abort();
}
arf_get_mag(arb_radref(x), t);
inexact = arf_add(arb_midref(x), a, b, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(x), arb_radref(x), arb_midref(x), prec);
arb_mul_2exp_si(x, x, -1);
arf_clear(t);
}
void arb_from_interval(arb_t x, const fmpz_t c, const slong k, const slong prec)
{
/* we build the ball that gives exactly (c 2^k, (c+1) 2^k) */
/* center: (2c+1) 2^(k-1) */
/* radius: 2^(k-1) */
if (prec <= 0 || prec < fmpz_bits(c) + 2)
{
fprintf(stderr, "not enough precision");
abort();
}
arb_set_fmpz(x, c);
arb_mul_2exp_si(x, x, 1);
arb_add_si(x, x, 1, prec);
arb_mul_2exp_si(x, x, k-1);
arb_add_error_2exp_si(x, k-1);
}
void
arb_zeta_ui_borwein_bsplit(arb_t x, ulong s, slong prec)
{
zeta_bsplit_t sum;
mag_t err;
slong wp, n;
/* zeta(0) = -1/2 */
if (s == 0)
{
arb_set_si(x, -1);
arb_mul_2exp_si(x, x, -1);
return;
}
if (s == 1)
{
flint_printf("zeta_ui_borwein_bsplit: zeta(1)");
abort();
}
n = prec / ERROR_B + 2;
wp = prec + 30;
zeta_bsplit_init(sum);
zeta_bsplit(sum, 0, n + 1, n, s, 0, wp);
/* A/Q3 - B/Q3 / (C/Q1) = (A*C - B*Q1) / (Q3*C) */
arb_mul(sum->A, sum->A, sum->C, wp);
arb_mul(sum->B, sum->B, sum->Q1, wp);
arb_sub(sum->A, sum->A, sum->B, wp);
arb_mul(sum->Q3, sum->Q3, sum->C, wp);
arb_div(sum->C, sum->A, sum->Q3, wp);
mag_init(err);
mag_borwein_error(err, n);
mag_add(arb_radref(sum->C), arb_radref(sum->C), err);
mag_clear(err);
/* convert from eta(s) to zeta(s) */
arb_div_2expm1_ui(x, sum->C, s - 1, wp);
arb_mul_2exp_si(x, x, s - 1);
zeta_bsplit_clear(sum);
}
void
arb_bernoulli_fmpz(arb_t res, const fmpz_t n, slong prec)
{
if (fmpz_cmp_ui(n, UWORD_MAX) <= 0)
{
if (fmpz_sgn(n) >= 0)
arb_bernoulli_ui(res, fmpz_get_ui(n), prec);
else
arb_zero(res);
}
else if (fmpz_is_odd(n))
{
arb_zero(res);
}
else
{
arb_t t;
slong wp;
arb_init(t);
wp = prec + 2 * fmpz_bits(n);
/* zeta(n) ~= 1 */
arf_one(arb_midref(res));
mag_one(arb_radref(res));
mag_mul_2exp_si(arb_radref(res), arb_radref(res), WORD_MIN);
/* |B_n| = 2 * n! / (2*pi)^n * zeta(n) */
arb_gamma_fmpz(t, n, wp);
arb_mul_fmpz(t, t, n, wp);
arb_mul(res, res, t, wp);
arb_const_pi(t, wp);
arb_mul_2exp_si(t, t, 1);
arb_pow_fmpz(t, t, n, wp);
arb_div(res, res, t, prec);
arb_mul_2exp_si(res, res, 1);
if (fmpz_fdiv_ui(n, 4) == 0)
arb_neg(res, res);
arb_clear(t);
}
}
slong renf_set_embeddings_fmpz_poly(renf * nf, fmpz_poly_t pol, slong lim, slong prec)
{
slong i, n, n_exact, n_interval;
fmpq_poly_t p2;
arb_t a;
fmpz * c;
slong * k;
n = fmpz_poly_num_real_roots_upper_bound(pol);
c = _fmpz_vec_init(n);
k = (slong *) flint_malloc(n * sizeof(slong));
fmpz_poly_isolate_real_roots(NULL, &n_exact, c, k, &n_interval, pol);
if (n_exact)
{
fprintf(stderr, "ERROR (fmpz_poly_real_embeddings): rational roots\n");
abort();
}
arb_init(a);
fmpq_poly_init(p2);
fmpz_one(fmpq_poly_denref(p2));
fmpq_poly_fit_length(p2, pol->length);
_fmpz_vec_set(p2->coeffs, pol->coeffs, pol->length);
p2->length = pol->length;
for (i = 0; i < FLINT_MIN(lim, n_interval); i++)
{
arb_set_fmpz(a, c + i);
arb_mul_2exp_si(a, a, 1);
arb_add_si(a, a, 1, prec);
mag_one(arb_radref(a));
arb_mul_2exp_si(a, a, k[i] - 1);
renf_init(nf + i, p2, a, prec);
}
arb_clear(a);
fmpq_poly_clear(p2);
_fmpz_vec_clear(c, n);
flint_free(k);
return n_interval;
}
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);
}
int
f_aj(arb_t m, const arb_t t, params_t * p, slong prec)
{
slong k;
acb_t z, zu;
arb_t abs;
arb_init(abs);
acb_init(z);
acb_init(zu);
arb_const_pi(abs, prec);
arb_mul_2exp_si(abs, abs, -2); /* Pi/4 */
arb_set(acb_realref(z), t);
arb_set(acb_imagref(z), abs);
acb_sinh(z, z, prec);
arb_mul_2exp_si(abs, abs, 1); /* Pi/2 */
acb_mul_arb(z, z, abs, prec);
acb_tanh(z, z, prec);
arb_one(m);
for (k = 0; k < p->len; k++)
{
acb_sub(zu, z, p->z + k, prec);
if (acb_contains_zero(zu))
{
arb_clear(abs);
acb_clear(zu);
acb_clear(z);
return 0;
}
acb_abs(abs, zu, prec);
arb_mul(m, m, abs, prec);
}
arb_inv(m, m, prec);
arb_clear(abs);
acb_clear(zu);
acb_clear(z);
return 1;
}
int
acb_modular_is_in_fundamental_domain(const acb_t z, const arf_t tol, long prec)
{
arb_t t;
arb_init(t);
/* require re(w) + 1/2 >= 0 */
arb_set_ui(t, 1);
arb_mul_2exp_si(t, t, -1);
arb_add(t, t, acb_realref(z), prec);
arb_add_arf(t, t, tol, prec);
if (!arb_is_nonnegative(t))
{
arb_clear(t);
return 0;
}
/* require re(w) - 1/2 <= 0 */
arb_set_ui(t, 1);
arb_mul_2exp_si(t, t, -1);
arb_sub(t, acb_realref(z), t, prec);
arb_sub_arf(t, t, tol, prec);
if (!arb_is_nonpositive(t))
{
arb_clear(t);
return 0;
}
/* require |w| >= 1 - tol, i.e. |w| - 1 + tol >= 0 */
acb_abs(t, z, prec);
arb_sub_ui(t, t, 1, prec);
arb_add_arf(t, t, tol, prec);
if (!arb_is_nonnegative(t))
{
arb_clear(t);
return 0;
}
arb_clear(t);
return 1;
}
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);
}
/* 0.5*(B/AN)^2 + |B|/AN */
static void
bound_C(arb_t C, const arb_t AN, const arb_t B, slong wp)
{
arb_t t;
arb_init(t);
arb_abs(t, B);
arb_div(t, t, AN, wp);
arb_mul_2exp_si(C, t, -1);
arb_add_ui(C, C, 1, wp);
arb_mul(C, C, t, wp);
arb_clear(t);
}
void
scaled_bessel_select_N(arb_t N, ulong k, slong prec)
{
slong e;
double f = log(k/3.14159265358979)/log(2);
e = 1;
while ((k / 2.0) * (e - f) - e < prec + 5)
e++;
arb_one(N);
arb_mul_2exp_si(N, N, e);
}
void
arb_atanh(arb_t z, const arb_t x, slong prec)
{
if (arb_is_zero(x))
{
arb_zero(z);
}
else
{
arb_t t;
arb_init(t);
arb_sub_ui(t, x, 1, prec + 4);
arb_div(t, x, t, prec + 4);
arb_mul_2exp_si(t, t, 1);
arb_neg(t, t);
arb_log1p(z, t, prec);
arb_mul_2exp_si(z, z, -1);
arb_clear(t);
}
}
static void
bsplit(arb_poly_t pol, const arb_t sqrtD,
const slong * qbf, slong a, slong b, slong prec)
{
if (b - a == 0)
{
arb_poly_one(pol);
}
else if (b - a == 1)
{
acb_t z;
acb_init(z);
/* j((-b+sqrt(-D))/(2a)) */
arb_set_si(acb_realref(z), -FLINT_ABS(qbf[3 * a + 1]));
arb_set(acb_imagref(z), sqrtD);
acb_div_si(z, z, 2 * qbf[3 * a], prec);
acb_modular_j(z, z, prec);
if (qbf[3 * a + 1] < 0)
{
/* (x^2 - 2re(j) x + |j|^2) */
arb_poly_fit_length(pol, 3);
arb_mul(pol->coeffs, acb_realref(z), acb_realref(z), prec);
arb_addmul(pol->coeffs, acb_imagref(z), acb_imagref(z), prec);
arb_mul_2exp_si(pol->coeffs + 1, acb_realref(z), 1);
arb_neg(pol->coeffs + 1, pol->coeffs + 1);
arb_one(pol->coeffs + 2);
_arb_poly_set_length(pol, 3);
}
else
{
/* (x-j) */
arb_poly_fit_length(pol, 2);
arb_neg(pol->coeffs, acb_realref(z));
arb_one(pol->coeffs + 1);
_arb_poly_set_length(pol, 2);
}
acb_clear(z);
}
else
{
arb_poly_t tmp;
arb_poly_init(tmp);
bsplit(pol, sqrtD, qbf, a, a + (b - a) / 2, prec);
bsplit(tmp, sqrtD, qbf, a + (b - a) / 2, b, prec);
arb_poly_mul(pol, pol, tmp, prec);
arb_poly_clear(tmp);
}
}
/* The floor+vec method *requires* n <= 1498 for floor(p(n)/2^64)
to be equal to floor(T/2^64). It is faster up to n ~= 1200.
With doubles, it is faster up to n ~= 500. */
void
_partitions_fmpz_ui(fmpz_t res, ulong n, int use_doubles)
{
if (n < NUMBER_OF_SMALL_PARTITIONS)
{
fmpz_set_ui(res, partitions_lookup[n]);
}
else if (FLINT_BITS == 64 && (n < 500 || (!use_doubles && n < 1200)))
{
mp_ptr tmp = flint_malloc((n + 1) * sizeof(mp_limb_t));
if (n < 417) /* p(n) < 2^64 */
{
partitions_vec(tmp, n + 1);
fmpz_set_ui(res, tmp[n]);
}
else
{
arb_t x;
arb_init(x);
fmpz_set_ui(res, n);
partitions_leading_fmpz(x, res, 4 * sqrt(n) - 50);
arb_mul_2exp_si(x, x, -64);
arb_floor(x, x, 4 * sqrt(n) - 50);
if (arb_get_unique_fmpz(res, x))
{
fmpz_mul_2exp(res, res, 64);
partitions_vec(tmp, n + 1);
fmpz_add_ui(res, res, tmp[n]);
}
else
{
flint_printf("warning: failed at %wu\n", n);
fmpz_set_ui(res, n);
partitions_fmpz_fmpz_hrr(res, res, use_doubles);
}
arb_clear(x);
}
flint_free(tmp);
}
else
{
fmpz_set_ui(res, n);
partitions_fmpz_fmpz_hrr(res, res, use_doubles);
}
}
void
_arb_poly_mullow_classical(arb_ptr res,
arb_srcptr poly1, long len1,
arb_srcptr poly2, long len2, long n, long prec)
{
len1 = FLINT_MIN(len1, n);
len2 = FLINT_MIN(len2, n);
if (n == 1)
{
arb_mul(res, poly1, poly2, prec);
}
else if (poly1 == poly2 && len1 == len2)
{
long i;
_arb_vec_scalar_mul(res, poly1, FLINT_MIN(len1, n), poly1, prec);
_arb_vec_scalar_mul(res + len1, poly1 + 1, n - len1, poly1 + len1 - 1, prec);
for (i = 1; i < len1 - 1; i++)
_arb_vec_scalar_addmul(res + i + 1, poly1 + 1,
FLINT_MIN(i - 1, n - (i + 1)), poly1 + i, prec);
for (i = 1; i < FLINT_MIN(2 * len1 - 2, n); i++)
arb_mul_2exp_si(res + i, res + i, 1);
for (i = 1; i < FLINT_MIN(len1 - 1, (n + 1) / 2); i++)
arb_addmul(res + 2 * i, poly1 + i, poly1 + i, prec);
}
else
{
long i;
_arb_vec_scalar_mul(res, poly1, FLINT_MIN(len1, n), poly2, prec);
if (n > len1)
_arb_vec_scalar_mul(res + len1, poly2 + 1, n - len1,
poly1 + len1 - 1, prec);
for (i = 0; i < FLINT_MIN(len1, n) - 1; i++)
_arb_vec_scalar_addmul(res + i + 1, poly2 + 1,
FLINT_MIN(len2, n - i) - 1,
poly1 + i, prec);
}
}
void
arb_fac2_ui(arb_t res, ulong n, long prec)
{
if (n % 2 == 0)
{
arb_fac_ui(res, n / 2, prec);
arb_mul_2exp_si(res, res, n / 2);
}
else
{
arb_t t;
arb_init(t);
arb_fac2_ui(t, n - 1, prec + 5);
arb_fac_ui(res, n, prec + 5);
arb_div(res, res, t, prec);
arb_clear(t);
}
}
/* sin((pi/2)x) */
static int
sin_pi2_x(arb_ptr out, const arb_t inp, void * params, slong order, slong prec)
{
arb_ptr x;
x = _arb_vec_init(2);
arb_set(x, inp);
arb_one(x + 1);
arb_const_pi(out, prec);
arb_mul_2exp_si(out, out, -1);
_arb_vec_scalar_mul(x, x, 2, out, prec);
_arb_poly_sin_series(out, x, order, order, prec);
_arb_vec_clear(x, 2);
return 0;
}
void arb_calc_newton_conv_factor(arf_t conv_factor,
arb_calc_func_t func, void * param,
const arb_t conv_region, long prec)
{
arb_struct t[3];
arb_init(t);
arb_init(t + 1);
arb_init(t + 2);
func(t, conv_region, param, 3, prec);
arb_div(t, t + 2, t + 1, prec);
arb_mul_2exp_si(t, t, -1);
arb_get_abs_ubound_arf(conv_factor, t, prec);
arb_clear(t);
arb_clear(t + 1);
arb_clear(t + 2);
}
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_const_log2_hypgeom_eval(arb_t s, slong prec)
{
hypgeom_t series;
arb_t t;
arb_init(t);
hypgeom_init(series);
fmpz_poly_set_str(series->A, "1 1");
fmpz_poly_set_str(series->B, "1 1");
fmpz_poly_set_str(series->P, "2 0 -1");
fmpz_poly_set_str(series->Q, "2 4 8");
prec += FLINT_CLOG2(prec);
arb_hypgeom_infsum(s, t, series, prec, prec);
arb_mul_ui(s, s, 3, prec);
arb_mul_2exp_si(t, t, 2);
arb_div(s, s, t, prec);
hypgeom_clear(series);
arb_clear(t);
}
/* atan(x) = pi/2 - eps, eps < 1/x <= 2^(1-mag) */
void
arb_atan_inf_eps(arb_t z, const arf_t x, slong prec)
{
fmpz_t mag;
fmpz_init(mag);
fmpz_neg(mag, ARF_EXPREF(x));
fmpz_add_ui(mag, mag, 1);
if (arf_sgn(x) > 0)
{
arb_const_pi(z, prec);
}
else
{
arb_const_pi(z, prec);
arb_neg(z, z);
}
arb_mul_2exp_si(z, z, -1);
arb_add_error_2exp_fmpz(z, mag);
fmpz_clear(mag);
}
int main()
{
long iter;
flint_rand_t state;
printf("div_2expm1_ui....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000; iter++)
{
arb_t a, b, c;
ulong n;
long prec, acc1, acc2;
fmpz_t t;
arb_init(a);
arb_init(b);
arb_init(c);
fmpz_init(t);
prec = 2 + n_randint(state, 10000);
arb_randtest(a, state, 1 + n_randint(state, 10000), 10);
if (n_randint(state, 2))
n = 1 + (n_randtest(state) % (10 * prec));
else
n = n_randtest(state);
arb_div_2expm1_ui(b, a, n, prec);
arb_one(c);
if (n >= (1UL << (FLINT_BITS-1)))
{
arb_mul_2exp_si(c, c, (1UL << (FLINT_BITS-2)));
arb_mul_2exp_si(c, c, (1UL << (FLINT_BITS-2)));
arb_mul_2exp_si(c, c, n - (1UL << (FLINT_BITS-1)));
}
else
{
arb_mul_2exp_si(c, c, n);
}
arb_sub_ui(c, c, 1, prec);
arb_div(c, a, c, prec);
acc1 = arb_rel_accuracy_bits(a);
acc2 = arb_rel_accuracy_bits(b);
if (!arb_overlaps(b, c))
{
printf("FAIL: containment\n\n");
printf("n = %lu\n", n);
printf("a = "); arb_print(a); printf("\n\n");
printf("b = "); arb_print(b); printf("\n\n");
printf("c = "); arb_print(c); printf("\n\n");
abort();
}
if (n > 0 && (acc2 < FLINT_MIN(prec, acc1) - 10) &&
!(acc1 == -ARF_PREC_EXACT && acc2 == -ARF_PREC_EXACT))
{
printf("FAIL: poor accuracy\n\n");
printf("prec=%ld, acc1=%ld, acc2=%ld\n\n", prec, acc1, acc2);
printf("n = %lu\n\n", n);
printf("a = "); arb_print(a); printf("\n\n");
printf("b = "); arb_print(b); printf("\n\n");
printf("c = "); arb_print(c); printf("\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
fmpz_clear(t);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("isolate_roots....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 40; iter++)
{
slong m, r, a, b, maxdepth, maxeval, maxfound, prec, i, j, num;
arf_interval_ptr blocks;
int * info;
arf_interval_t interval;
arb_t t;
fmpz_t nn;
prec = 2 + n_randint(state, 50);
m = n_randint(state, 80);
r = 1 + n_randint(state, 80);
a = m - r;
b = m + r;
maxdepth = 1 + n_randint(state, 60);
maxeval = 1 + n_randint(state, 5000);
maxfound = 1 + n_randint(state, 100);
arf_interval_init(interval);
arb_init(t);
fmpz_init(nn);
arf_set_si(&interval->a, a);
arf_set_si(&interval->b, b);
num = arb_calc_isolate_roots(&blocks, &info, sin_pi2_x, NULL,
interval, maxdepth, maxeval, maxfound, prec);
/* check that all roots are accounted for */
for (i = a; i <= b; i++)
{
if (i % 2 == 0)
{
int found = 0;
for (j = 0; j < num; j++)
{
arf_interval_get_arb(t, blocks + j, ARF_PREC_EXACT);
if (arb_contains_si(t, i))
{
found = 1;
break;
}
}
if (!found)
{
flint_printf("FAIL: missing root %wd\n", i);
flint_printf("a = %wd, b = %wd, maxdepth = %wd, maxeval = %wd, maxfound = %wd, prec = %wd\n",
a, b, maxdepth, maxeval, maxfound, prec);
for (j = 0; j < num; j++)
{
arf_interval_printd(blocks + j, 15);
flint_printf(" %d \n", info[i]);
}
abort();
}
}
}
/* check that all reported single roots are good */
for (i = 0; i < num; i++)
{
if (info[i] == 1)
{
/* b contains unique 2n -> b/2 contains unique n */
arf_interval_get_arb(t, blocks + i, ARF_PREC_EXACT);
arb_mul_2exp_si(t, t, -1);
if (!arb_get_unique_fmpz(nn, t))
{
flint_printf("FAIL: bad root %wd\n", i);
flint_printf("a = %wd, b = %wd, maxdepth = %wd, maxeval = %wd, maxfound = %wd, prec = %wd\n",
a, b, maxdepth, maxeval, maxfound, prec);
for (j = 0; j < num; j++)
{
arf_interval_printd(blocks + j, 15);
flint_printf(" %d \n", info[i]);
}
abort();
}
}
}
_arf_interval_vec_clear(blocks, num);
flint_free(info);
arf_interval_clear(interval);
arb_clear(t);
fmpz_clear(nn);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
acb_hypgeom_ci_asymp(acb_t res, const acb_t z, slong prec)
{
acb_t t, u, w, v, one;
acb_init(t);
acb_init(u);
acb_init(w);
acb_init(v);
acb_init(one);
acb_one(one);
acb_mul_onei(w, z);
/* u = U(1,1,iz) */
acb_hypgeom_u_asymp(u, one, one, w, -1, prec);
/* v = e^(-iz) */
acb_neg(v, w);
acb_exp(v, v, prec);
acb_mul(t, u, v, prec);
if (acb_is_real(z))
{
arb_div(acb_realref(t), acb_imagref(t), acb_realref(z), prec);
arb_zero(acb_imagref(t));
acb_neg(t, t);
}
else
{
/* u = U(1,1,-iz) */
acb_neg(w, w);
acb_hypgeom_u_asymp(u, one, one, w, -1, prec);
acb_inv(v, v, prec);
acb_submul(t, u, v, prec);
acb_div(t, t, w, prec);
acb_mul_2exp_si(t, t, -1);
}
if (arb_is_zero(acb_realref(z)))
{
if (arb_is_positive(acb_imagref(z)))
{
arb_const_pi(acb_imagref(t), prec);
arb_mul_2exp_si(acb_imagref(t), acb_imagref(t), -1);
}
else if (arb_is_negative(acb_imagref(z)))
{
arb_const_pi(acb_imagref(t), prec);
arb_mul_2exp_si(acb_imagref(t), acb_imagref(t), -1);
arb_neg(acb_imagref(t), acb_imagref(t));
}
else
{
acb_const_pi(u, prec);
acb_mul_2exp_si(u, u, -1);
arb_zero(acb_imagref(t));
arb_add_error(acb_imagref(t), acb_realref(u));
}
}
else
{
/* 0 if positive or positive imaginary
pi if upper left quadrant (including negative real axis)
-pi if lower left quadrant (including negative imaginary axis) */
if (arb_is_positive(acb_realref(z)))
{
/* do nothing */
}
else if (arb_is_negative(acb_realref(z)) && arb_is_nonnegative(acb_imagref(z)))
{
acb_const_pi(u, prec);
arb_add(acb_imagref(t), acb_imagref(t), acb_realref(u), prec);
}
else if (arb_is_nonpositive(acb_realref(z)) && arb_is_negative(acb_imagref(z)))
{
acb_const_pi(u, prec);
arb_sub(acb_imagref(t), acb_imagref(t), acb_realref(u), prec);
}
else
{
/* add [-pi,pi] */
acb_const_pi(u, prec);
arb_add_error(acb_imagref(t), acb_realref(u));
}
}
acb_swap(res, t);
acb_clear(t);
acb_clear(u);
acb_clear(w);
acb_clear(v);
acb_clear(one);
}
