void
acb_dirichlet_zeta_rs_mid(acb_t res, const acb_t s, slong K, slong prec)
{
acb_t R1, R2, X, t;
slong wp;
if (arf_sgn(arb_midref(acb_imagref(s))) < 0)
{
acb_init(t);
acb_conj(t, s);
acb_dirichlet_zeta_rs(res, t, K, prec);
acb_conj(res, res);
acb_clear(t);
return;
}
acb_init(R1);
acb_init(R2);
acb_init(X);
acb_init(t);
/* rs_r increases the precision internally */
wp = prec;
acb_dirichlet_zeta_rs_r(R1, s, K, wp);
if (arb_is_exact(acb_realref(s)) &&
(arf_cmp_2exp_si(arb_midref(acb_realref(s)), -1) == 0))
{
acb_conj(R2, R1);
}
else
{
/* conj(R(conj(1-s))) */
arb_sub_ui(acb_realref(t), acb_realref(s), 1, 10 * wp);
arb_neg(acb_realref(t), acb_realref(t));
arb_set(acb_imagref(t), acb_imagref(s));
acb_dirichlet_zeta_rs_r(R2, t, K, wp);
acb_conj(R2, R2);
}
if (acb_is_finite(R1) && acb_is_finite(R2))
{
wp += 10 + arf_abs_bound_lt_2exp_si(arb_midref(acb_imagref(s)));
wp = FLINT_MAX(wp, 10);
/* X = pi^(s-1/2) gamma((1-s)/2) rgamma(s/2)
= (2 pi)^s rgamma(s) / (2 cos(pi s / 2)) */
acb_rgamma(X, s, wp);
acb_const_pi(t, wp);
acb_mul_2exp_si(t, t, 1);
acb_pow(t, t, s, wp);
acb_mul(X, X, t, wp);
acb_mul_2exp_si(t, s, -1);
acb_cos_pi(t, t, wp);
acb_mul_2exp_si(t, t, 1);
acb_div(X, X, t, wp);
acb_mul(R2, R2, X, wp);
}
/* R1 + X * R2 */
acb_add(res, R1, R2, prec);
acb_clear(R1);
acb_clear(R2);
acb_clear(X);
acb_clear(t);
}
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);
}
int main()
{
long iter;
flint_rand_t state;
printf("legendre_q....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 1000; iter++)
{
acb_t n, m, z, res1, res2;
long prec1, prec2, ebits;
acb_init(n);
acb_init(m);
acb_init(z);
acb_init(res1);
acb_init(res2);
prec1 = 2 + n_randint(state, 300);
prec2 = 2 + n_randint(state, 300);
ebits = 1 + n_randint(state, 10);
if (n_randint(state, 2))
{
acb_set_si(m, n_randint(state, 20) - 10);
acb_set_si(n, n_randint(state, 20) - 10);
}
else
{
acb_randtest_param(n, state, 1 + n_randint(state, 400), ebits);
acb_randtest_param(m, state, 1 + n_randint(state, 400), ebits);
}
acb_randtest_param(z, state, 1 + n_randint(state, 400), ebits);
_acb_hypgeom_legendre_q_single(res1, n, m, z, prec1);
_acb_hypgeom_legendre_q_double(res2, n, m, z, prec2);
if (!acb_overlaps(res1, res2))
{
printf("FAIL: consistency 1\n\n");
printf("iter = %ld, prec1 = %ld, prec2 = %ld\n\n", iter, prec1, prec2);
printf("m = "); acb_printd(m, 30); printf("\n\n");
printf("n = "); acb_printd(n, 30); printf("\n\n");
printf("z = "); acb_printd(z, 30); printf("\n\n");
printf("res1 = "); acb_printd(res1, 30); printf("\n\n");
printf("res2 = "); acb_printd(res2, 30); printf("\n\n");
abort();
}
acb_clear(n);
acb_clear(m);
acb_clear(z);
acb_clear(res1);
acb_clear(res2);
}
for (iter = 0; iter < 2000; iter++)
{
acb_t n, m, z, res1, res2, t, u;
long prec1, prec2, ebits;
int type;
acb_init(n);
acb_init(m);
acb_init(z);
acb_init(res1);
acb_init(res2);
acb_init(t);
acb_init(u);
prec1 = 2 + n_randint(state, 300);
prec2 = 2 + n_randint(state, 300);
ebits = 1 + n_randint(state, 10);
if (n_randint(state, 2))
{
acb_set_si(m, n_randint(state, 20) - 10);
acb_set_si(n, n_randint(state, 20) - 10);
}
else
{
acb_randtest_param(n, state, 1 + n_randint(state, 400), ebits);
acb_randtest_param(m, state, 1 + n_randint(state, 400), ebits);
}
acb_randtest_param(z, state, 1 + n_randint(state, 400), ebits);
type = n_randint(state, 2);
acb_hypgeom_legendre_q(res1, n, m, z, type, prec1);
acb_neg(t, m);
acb_hypgeom_legendre_p(res2, n, t, z, type, prec2);
acb_add(u, m, n, prec2);
acb_add_ui(u, u, 1, prec2);
acb_gamma(u, u, prec2);
acb_mul(res2, res2, u, prec2);
acb_sub(u, n, m, prec2);
acb_add_ui(u, u, 1, prec2);
acb_rgamma(u, u, prec2);
acb_mul(res2, res2, u, prec2);
acb_hypgeom_legendre_p(t, n, m, z, type, prec2);
if (type == 0)
{
acb_cos_pi(u, m, prec2);
acb_mul(t, t, u, prec2);
}
acb_sub(res2, t, res2, prec2);
if (type == 1)
{
acb_exp_pi_i(t, m, prec2);
acb_mul(res2, res2, t, prec2);
}
acb_sin_pi(t, m, prec2);
if (acb_contains_zero(t))
acb_indeterminate(res2);
else
acb_div(res2, res2, t, prec2);
acb_const_pi(t, prec2);
acb_mul(res2, res2, t, prec2);
acb_mul_2exp_si(res2, res2, -1);
if (!acb_overlaps(res1, res2))
{
printf("FAIL: consistency 2\n\n");
printf("iter = %ld, prec1 = %ld, prec2 = %ld\n\n", iter, prec1, prec2);
printf("type = %d\n\n", type);
printf("m = "); acb_printd(m, 30); printf("\n\n");
printf("n = "); acb_printd(n, 30); printf("\n\n");
printf("z = "); acb_printd(z, 30); printf("\n\n");
printf("res1 = "); acb_printd(res1, 30); printf("\n\n");
printf("res2 = "); acb_printd(res2, 30); printf("\n\n");
abort();
}
acb_clear(n);
acb_clear(m);
acb_clear(z);
acb_clear(res1);
acb_clear(res2);
acb_clear(t);
acb_clear(u);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}