int main()
{
slong iter;
flint_rand_t state;
flint_printf("backlund_s_bound....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 500 * arb_test_multiplier(); iter++)
{
arb_t a, b;
mag_t u, v;
slong aprec, bprec;
slong abits, bbits;
aprec = 2 + n_randint(state, 1000);
bprec = 2 + n_randint(state, 1000);
abits = 2 + n_randint(state, 100);
bbits = 2 + n_randint(state, 100);
arb_init(a);
arb_init(b);
mag_init(u);
mag_init(v);
arb_randtest(a, state, aprec, abits);
arb_randtest(b, state, bprec, bbits);
if (arb_is_nonnegative(a) && arb_is_nonnegative(b))
{
acb_dirichlet_backlund_s_bound(u, a);
acb_dirichlet_backlund_s_bound(v, b);
if ((arb_lt(a, b) && mag_cmp(u, v) > 0) ||
(arb_gt(a, b) && mag_cmp(u, v) < 0))
{
flint_printf("FAIL: increasing on t >= 0\n\n");
flint_printf("a = "); arb_print(a); flint_printf("\n\n");
flint_printf("b = "); arb_print(b); flint_printf("\n\n");
flint_printf("u = "); mag_print(u); flint_printf("\n\n");
flint_printf("v = "); mag_print(v); flint_printf("\n\n");
flint_abort();
}
}
arb_clear(a);
arb_clear(b);
mag_clear(u);
mag_clear(v);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
/* this can be improved */
static int
use_recurrence(const acb_t n, const acb_t a, const acb_t b, slong prec)
{
if (!acb_is_int(n) || !arb_is_nonnegative(acb_realref(n)))
return 0;
if (arf_cmpabs_ui(arb_midref(acb_realref(n)), prec) > 0)
return 0;
if (arb_is_nonnegative(acb_realref(a)) ||
arf_get_d(arb_midref(acb_realref(a)), ARF_RND_DOWN) > -0.9)
return 0;
return 1;
}
void
acb_hypgeom_laguerre_l(acb_t res, const acb_t n, const acb_t m, const acb_t z, slong prec)
{
acb_t t, u, v;
if (use_recurrence(n, m, prec))
{
acb_hypgeom_laguerre_l_ui_recurrence(res,
arf_get_si(arb_midref(acb_realref(n)), ARF_RND_DOWN), m, z, prec);
return;
}
/* todo: should be a test of whether n contains any negative integer */
if (acb_contains_int(n) && !arb_is_nonnegative(acb_realref(n)))
{
acb_indeterminate(res);
return;
}
acb_init(t);
acb_init(u);
acb_init(v);
acb_neg(t, n);
acb_add_ui(u, m, 1, prec);
acb_hypgeom_m(t, t, u, z, 1, prec);
acb_add_ui(u, n, 1, prec);
acb_rising(u, u, m, prec);
acb_mul(res, t, u, prec);
acb_clear(t);
acb_clear(u);
acb_clear(v);
}
/* Extremely close to the branch point at -1/e, use the series expansion directly. */
int
acb_lambertw_try_near_branch_point(acb_t res, const acb_t z,
const acb_t ez1, const fmpz_t k, int flags, slong prec)
{
if (fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z)))
|| (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z))))
{
if (acb_contains_zero(ez1) ||
(arf_cmpabs_2exp_si(arb_midref(acb_realref(ez1)), -prec / 4.5 - 6) < 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(ez1)), -prec / 4.5 - 6) < 0))
{
acb_t t;
acb_init(t);
acb_mul_2exp_si(t, ez1, 1);
acb_sqrt(t, t, prec);
if (!fmpz_is_zero(k))
acb_neg(t, t);
acb_lambertw_branchpoint_series(res, t, 1, prec);
acb_clear(t);
return 1;
}
}
return 0;
}
static void
phase(acb_t res, const arb_t re, const arb_t im)
{
if (arb_is_nonnegative(re) || arb_is_negative(im))
{
acb_one(res);
}
else if (arb_is_negative(re) && arb_is_nonnegative(im))
{
acb_set_si(res, -3);
}
else
{
arb_zero(acb_imagref(res));
/* -1 +/- 2 */
arf_set_si(arb_midref(acb_realref(res)), -1);
mag_one(arb_radref(acb_realref(res)));
mag_mul_2exp_si(arb_radref(acb_realref(res)), arb_radref(acb_realref(res)), 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;
}
/* this can be improved */
static int
use_recurrence(const acb_t n, const acb_t m, slong prec)
{
if (!acb_is_int(n) || !arb_is_nonnegative(acb_realref(n)))
return 0;
if (arf_cmpabs_ui(arb_midref(acb_realref(n)), prec) > 0)
return 0;
if (arf_cmpabs(arb_midref(acb_realref(n)), arb_midref(acb_realref(m))) >= 0)
return 0;
return 1;
}
/* assumes no aliasing */
slong
acb_lambertw_initial(acb_t res, const acb_t z, const acb_t ez1, const fmpz_t k, slong prec)
{
/* Handle z very close to 0 on the principal branch. */
if (fmpz_is_zero(k) &&
(arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), -20) <= 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), -20) <= 0))
{
acb_set(res, z);
acb_submul(res, res, res, prec);
return 40; /* could be tightened... */
}
/* For moderate input not close to the branch point, compute a double
approximation as the initial value. */
if (fmpz_is_zero(k) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), 400) < 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 400) < 0 &&
(arf_cmp_d(arb_midref(acb_realref(z)), -0.37) < 0 ||
arf_cmp_d(arb_midref(acb_realref(z)), -0.36) > 0 ||
arf_cmpabs_d(arb_midref(acb_imagref(z)), 0.01) > 0))
{
acb_lambertw_principal_d(res, z);
return 48;
}
/* Check if we are close to the branch point at -1/e. */
if ((fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z)))
|| (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z))))
&& ((arf_cmpabs_2exp_si(arb_midref(acb_realref(ez1)), -2) <= 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(ez1)), -2) <= 0)))
{
acb_t t;
acb_init(t);
acb_mul_2exp_si(t, ez1, 1);
mag_zero(arb_radref(acb_realref(t)));
mag_zero(arb_radref(acb_imagref(t)));
acb_mul_ui(t, t, 3, prec);
acb_sqrt(t, t, prec);
if (!fmpz_is_zero(k))
acb_neg(t, t);
acb_lambertw_branchpoint_series(res, t, 0, prec);
acb_clear(t);
return 1; /* todo: estimate */
}
acb_lambertw_initial_asymp(res, z, k, prec);
return 1; /* todo: estimate */
}
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
arb_hypgeom_li(arb_t res, const arb_t z, int offset, slong prec)
{
if (!arb_is_finite(z) || !arb_is_nonnegative(z))
{
arb_indeterminate(res);
}
else
{
acb_t t;
acb_init(t);
arb_set(acb_realref(t), z);
acb_hypgeom_li(t, t, offset, prec);
arb_swap(res, acb_realref(t));
acb_clear(t);
}
}
/* Check if z crosses a branch cut. */
int
acb_lambertw_branch_crossing(const acb_t z, const acb_t ez1, const fmpz_t k)
{
if (arb_contains_zero(acb_imagref(z)) && !arb_is_nonnegative(acb_imagref(z)))
{
if (fmpz_is_zero(k))
{
if (!arb_is_positive(acb_realref(ez1)))
{
return 1;
}
}
else if (!arb_is_positive(acb_realref(z)))
{
return 1;
}
}
return 0;
}
void
acb_hypgeom_chebyshev_t(acb_t res, const acb_t n, const acb_t z, slong prec)
{
acb_t t;
if (acb_is_int(n) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(n)), FLINT_BITS - 1) < 0)
{
slong k = arf_get_si(arb_midref(acb_realref(n)), ARF_RND_DOWN);
acb_chebyshev_t_ui(res, FLINT_ABS(k), z, prec);
return;
}
if (acb_is_zero(z))
{
acb_mul_2exp_si(res, n, -1);
acb_cos_pi(res, res, prec);
return;
}
if (acb_is_one(z))
{
acb_one(res);
return;
}
acb_init(t);
acb_set_si(t, -1);
if (acb_equal(t, z))
{
acb_cos_pi(res, n, prec);
}
else
{
acb_sub_ui(t, z, 1, prec);
if (arf_cmpabs_2exp_si(arb_midref(acb_realref(t)), -2 - prec / 10) < 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(t)), -2 - prec / 10) < 0)
{
acb_t a, c;
acb_init(a);
acb_init(c);
acb_neg(a, n);
acb_one(c);
acb_mul_2exp_si(c, c, -1);
acb_neg(t, t);
acb_mul_2exp_si(t, t, -1);
acb_hypgeom_2f1(res, a, n, c, t, 0, prec);
acb_clear(a);
acb_clear(c);
}
else if (arb_is_nonnegative(acb_realref(t)))
{
acb_acosh(t, z, prec);
acb_mul(t, t, n, prec);
acb_cosh(res, t, prec);
}
else
{
acb_acos(t, z, prec);
acb_mul(t, t, n, prec);
acb_cos(res, t, prec);
}
}
acb_clear(t);
}
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);
}
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
}
/* computes the factors that are independent of n (all are upper bounds) */
void
acb_hypgeom_u_asymp_bound_factors(int * R, mag_t alpha,
mag_t nu, mag_t sigma, mag_t rho, mag_t zinv,
const acb_t a, const acb_t b, const acb_t z)
{
mag_t r, u, zre, zim, zlo, sigma_prime;
acb_t t;
mag_init(r);
mag_init(u);
mag_init(zre);
mag_init(zim);
mag_init(zlo);
mag_init(sigma_prime);
acb_init(t);
/* lower bounds for |re(z)|, |im(z)|, |z| */
arb_get_mag_lower(zre, acb_realref(z));
arb_get_mag_lower(zim, acb_imagref(z));
acb_get_mag_lower(zlo, z); /* todo: hypot */
/* upper bound for 1/|z| */
mag_one(u);
mag_div(zinv, u, zlo);
/* upper bound for r = |b - 2a| */
acb_mul_2exp_si(t, a, 1);
acb_sub(t, b, t, MAG_BITS);
acb_get_mag(r, t);
/* determine region */
*R = 0;
if (mag_cmp(zlo, r) >= 0)
{
int znonneg = arb_is_nonnegative(acb_realref(z));
if (znonneg && mag_cmp(zre, r) >= 0)
{
*R = 1;
}
else if (mag_cmp(zim, r) >= 0 || znonneg)
{
*R = 2;
}
else
{
mag_mul_2exp_si(u, r, 1);
if (mag_cmp(zlo, u) >= 0)
*R = 3;
}
}
if (R == 0)
{
mag_inf(alpha);
mag_inf(nu);
mag_inf(sigma);
mag_inf(rho);
}
else
{
/* sigma = |(b-2a)/z| */
mag_mul(sigma, r, zinv);
/* nu = (1/2 + 1/2 sqrt(1-4 sigma^2))^(-1/2) <= 1 + 2 sigma^2 */
if (mag_cmp_2exp_si(sigma, -1) <= 0)
{
mag_mul(nu, sigma, sigma);
mag_mul_2exp_si(nu, nu, 1);
mag_one(u);
mag_add(nu, nu, u);
}
else
{
mag_inf(nu);
}
/* modified sigma for alpha, beta, rho when in R3 */
if (*R == 3)
mag_mul(sigma_prime, sigma, nu);
else
mag_set(sigma_prime, sigma);
/* alpha = 1/(1-sigma') */
mag_one(alpha);
mag_sub_lower(alpha, alpha, sigma_prime);
mag_one(u);
mag_div(alpha, u, alpha);
/* rho = |2a^2-2ab+b|/2 + sigma'*(1+sigma'/4)/(1-sigma')^2 */
mag_mul_2exp_si(rho, sigma_prime, -2);
mag_one(u);
mag_add(rho, rho, u);
mag_mul(rho, rho, sigma_prime);
mag_mul(rho, rho, alpha);
mag_mul(rho, rho, alpha);
acb_sub(t, a, b, MAG_BITS);
acb_mul(t, t, a, MAG_BITS);
acb_mul_2exp_si(t, t, 1);
acb_add(t, t, b, MAG_BITS);
acb_get_mag(u, t);
mag_mul_2exp_si(u, u, -1);
mag_add(rho, rho, u);
}
mag_clear(r);
mag_clear(u);
mag_clear(zre);
mag_clear(zim);
mag_clear(zlo);
mag_clear(sigma_prime);
acb_clear(t);
}
/* note: z should be exact here */
void acb_lambertw_main(acb_t res, const acb_t z,
const acb_t ez1, const fmpz_t k, int flags, slong prec)
{
acb_t w, t, oldw, ew;
mag_t err;
slong i, wp, accuracy, ebits, kbits, mbits, wp_initial, extraprec;
int have_ew;
acb_init(t);
acb_init(w);
acb_init(oldw);
acb_init(ew);
mag_init(err);
/* We need higher precision for large k, large exponents, or very close
to the branch point at -1/e. todo: we should be recomputing
ez1 to higher precision when close... */
acb_get_mag(err, z);
if (fmpz_is_zero(k) && mag_cmp_2exp_si(err, 0) < 0)
ebits = 0;
else
ebits = fmpz_bits(MAG_EXPREF(err));
if (fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z)))
|| (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z))))
{
acb_get_mag(err, ez1);
mbits = -MAG_EXP(err);
mbits = FLINT_MAX(mbits, 0);
mbits = FLINT_MIN(mbits, prec);
}
else
{
mbits = 0;
}
kbits = fmpz_bits(k);
extraprec = FLINT_MAX(ebits, kbits);
extraprec = FLINT_MAX(extraprec, mbits);
wp = wp_initial = 40 + extraprec;
accuracy = acb_lambertw_initial(w, z, ez1, k, wp_initial);
mag_zero(arb_radref(acb_realref(w)));
mag_zero(arb_radref(acb_imagref(w)));
/* We should be able to compute e^w for the final certification
during the Halley iteration. */
have_ew = 0;
for (i = 0; i < 5 + FLINT_BIT_COUNT(prec + extraprec); i++)
{
/* todo: should we restart? */
if (!acb_is_finite(w))
break;
wp = FLINT_MIN(3 * accuracy, 1.1 * prec + 10);
wp = FLINT_MAX(wp, 40);
wp += extraprec;
acb_set(oldw, w);
acb_lambertw_halley_step(t, ew, z, w, wp);
/* estimate the error (conservatively) */
acb_sub(w, w, t, wp);
acb_get_mag(err, w);
acb_set(w, t);
acb_add_error_mag(t, err);
accuracy = acb_rel_accuracy_bits(t);
if (accuracy > 2 * extraprec)
accuracy *= 2.9; /* less conservatively */
accuracy = FLINT_MIN(accuracy, wp);
accuracy = FLINT_MAX(accuracy, 0);
if (accuracy > prec + extraprec)
{
/* e^w = e^oldw * e^(w-oldw) */
acb_sub(t, w, oldw, wp);
acb_exp(t, t, wp);
acb_mul(ew, ew, t, wp);
have_ew = 1;
break;
}
mag_zero(arb_radref(acb_realref(w)));
mag_zero(arb_radref(acb_imagref(w)));
}
wp = FLINT_MIN(3 * accuracy, 1.1 * prec + 10);
wp = FLINT_MAX(wp, 40);
wp += extraprec;
if (acb_lambertw_check_branch(w, k, wp))
{
acb_t u, r, eu1;
mag_t err, rad;
acb_init(u);
acb_init(r);
acb_init(eu1);
mag_init(err);
mag_init(rad);
if (have_ew)
acb_set(t, ew);
else
acb_exp(t, w, wp);
/* t = w e^w */
acb_mul(t, t, w, wp);
acb_sub(r, t, z, wp);
/* Bound W' on the straight line path between t and z */
acb_union(u, t, z, wp);
arb_const_e(acb_realref(eu1), wp);
arb_zero(acb_imagref(eu1));
acb_mul(eu1, eu1, u, wp);
acb_add_ui(eu1, eu1, 1, wp);
if (acb_lambertw_branch_crossing(u, eu1, k))
{
mag_inf(err);
}
else
{
acb_lambertw_bound_deriv(err, u, eu1, k);
acb_get_mag(rad, r);
mag_mul(err, err, rad);
}
acb_add_error_mag(w, err);
acb_set(res, w);
acb_clear(u);
acb_clear(r);
acb_clear(eu1);
mag_clear(err);
mag_clear(rad);
}
else
{
acb_indeterminate(res);
}
acb_clear(t);
acb_clear(w);
acb_clear(oldw);
acb_clear(ew);
mag_clear(err);
}
void
acb_hypgeom_2f1_transform_limit(acb_t res, const acb_t a, const acb_t b,
const acb_t c, const acb_t z, int regularized, int which, slong prec)
{
acb_poly_t aa, bb, cc, zz;
acb_t t;
if (acb_contains_zero(z) || !acb_is_finite(z))
{
acb_indeterminate(res);
return;
}
if (arb_contains_si(acb_realref(z), 1) && arb_contains_zero(acb_imagref(z)))
{
acb_indeterminate(res);
return;
}
if (!regularized)
{
acb_init(t);
acb_gamma(t, c, prec);
acb_hypgeom_2f1_transform_limit(res, a, b, c, z, 1, which, prec);
acb_mul(res, res, t, prec);
acb_clear(t);
return;
}
acb_poly_init(aa);
acb_poly_init(bb);
acb_poly_init(cc);
acb_poly_init(zz);
acb_init(t);
acb_poly_set_acb(aa, a);
acb_poly_set_acb(bb, b);
acb_poly_set_acb(cc, c);
acb_poly_set_acb(zz, z);
if (which == 2 || which == 3)
{
acb_sub(t, b, a, prec);
acb_poly_set_coeff_si(aa, 1, 1);
/* prefer b-a nonnegative (either is correct) to avoid
expensive operations in the hypergeometric series */
if (arb_is_nonnegative(acb_realref(t)))
_acb_hypgeom_2f1_transform_limit(res, aa, bb, cc, zz, which, prec);
else
_acb_hypgeom_2f1_transform_limit(res, bb, aa, cc, zz, which, prec);
}
else
{
acb_poly_set_coeff_si(aa, 1, 1);
_acb_hypgeom_2f1_transform_limit(res, aa, bb, cc, zz, which, prec);
}
acb_poly_clear(aa);
acb_poly_clear(bb);
acb_poly_clear(cc);
acb_poly_clear(zz);
acb_clear(t);
}
void
acb_lambertw_middle(acb_t res, const acb_t z, slong prec)
{
fmpz_t k;
if (acb_contains_zero(z))
{
acb_indeterminate(res);
return;
}
fmpz_init(k);
fmpz_set_si(k, -1);
if (arb_is_positive(acb_imagref(z)))
{
acb_lambertw(res, z, k, 0, prec);
}
else if (arb_is_negative(acb_imagref(z)))
{
acb_conj(res, z);
acb_lambertw(res, res, k, 0, prec);
acb_conj(res, res);
}
else if (arb_is_negative(acb_realref(z)))
{
if (arb_is_nonnegative(acb_imagref(z)))
{
acb_lambertw(res, z, k, 0, prec);
}
else if (arb_is_negative(acb_imagref(z)))
{
acb_conj(res, z);
acb_lambertw(res, res, k, 0, prec);
acb_conj(res, res);
}
else
{
acb_t za, zb;
acb_init(za);
acb_init(zb);
acb_set(za, z);
acb_conj(zb, z);
arb_nonnegative_part(acb_imagref(za), acb_imagref(za));
arb_nonnegative_part(acb_imagref(zb), acb_imagref(zb));
acb_lambertw(za, za, k, 0, prec);
acb_lambertw(zb, zb, k, 0, prec);
acb_conj(zb, zb);
acb_union(res, za, zb, prec);
acb_clear(za);
acb_clear(zb);
}
}
else /* re is positive */
{
if (arb_is_positive(acb_imagref(z)))
{
acb_lambertw(res, z, k, 0, prec);
}
else if (arb_is_nonpositive(acb_imagref(z)))
{
acb_conj(res, z);
acb_lambertw(res, res, k, 0, prec);
acb_conj(res, res);
}
else
{
acb_t za, zb;
acb_init(za);
acb_init(zb);
acb_set(za, z);
acb_conj(zb, z);
arb_nonnegative_part(acb_imagref(za), acb_imagref(za));
arb_nonnegative_part(acb_imagref(zb), acb_imagref(zb));
acb_lambertw(za, za, k, 0, prec);
acb_lambertw(zb, zb, k, 0, prec);
acb_conj(zb, zb);
acb_union(res, za, zb, prec);
acb_clear(za);
acb_clear(zb);
}
}
fmpz_clear(k);
}
