void
acb_hypgeom_bessel_i(acb_t res, const acb_t nu, const acb_t z, long prec)
{
mag_t zmag;
mag_init(zmag);
acb_get_mag(zmag, z);
if (mag_cmp_2exp_si(zmag, 4) < 0 ||
(mag_cmp_2exp_si(zmag, 64) < 0 && 2 * mag_get_d(zmag) < prec))
acb_hypgeom_bessel_i_0f1(res, nu, z, prec);
else
acb_hypgeom_bessel_i_asymp(res, nu, z, prec);
mag_clear(zmag);
}
void
acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, long prec)
{
acb_struct b[2];
acb_t w, c, t;
if (acb_is_int(nu) && arb_is_negative(acb_realref(nu)))
{
acb_init(t);
acb_neg(t, nu);
acb_hypgeom_bessel_i_0f1(res, t, z, prec);
acb_clear(t);
return;
}
acb_init(b + 0);
acb_init(b + 1);
acb_init(w);
acb_init(c);
acb_init(t);
acb_add_ui(b + 0, nu, 1, prec);
acb_one(b + 1);
/* (z/2)^nu / gamma(nu+1) */
acb_mul_2exp_si(c, z, -1);
acb_pow(c, c, nu, prec);
acb_rgamma(t, b + 0, prec);
acb_mul(c, t, c, prec);
/* z^2/4 */
acb_mul(w, z, z, prec);
acb_mul_2exp_si(w, w, -2);
acb_hypgeom_pfq_direct(t, NULL, 0, b, 2, w, -1, prec);
acb_mul(res, t, c, prec);
acb_clear(b + 0);
acb_clear(b + 1);
acb_clear(w);
acb_clear(c);
acb_clear(t);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("bessel_i....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 2000; iter++)
{
acb_t nu, z, jv, iv, t;
slong prec;
acb_init(nu);
acb_init(z);
acb_init(jv);
acb_init(iv);
acb_init(t);
prec = 2 + n_randint(state, 500);
acb_randtest_param(nu, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 10));
acb_randtest(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(iv, nu, z, prec);
break;
case 1:
acb_hypgeom_bessel_i_0f1(iv, nu, z, prec);
break;
default:
acb_hypgeom_bessel_i(iv, nu, z, prec);
}
acb_mul_onei(t, z);
acb_hypgeom_bessel_j(jv, nu, t, prec);
acb_pow(t, z, nu, prec);
acb_mul(jv, jv, t, prec);
acb_mul_onei(t, z);
acb_pow(t, t, nu, prec);
acb_div(jv, jv, t, prec);
if (!acb_overlaps(iv, jv))
{
flint_printf("FAIL: consistency with bessel_j\n\n");
flint_printf("nu = "); acb_printd(nu, 30); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("iv = "); acb_printd(iv, 30); flint_printf("\n\n");
flint_printf("jv = "); acb_printd(jv, 30); flint_printf("\n\n");
abort();
}
acb_clear(nu);
acb_clear(z);
acb_clear(jv);
acb_clear(iv);
acb_clear(t);
}
for (iter = 0; iter < 2000; iter++)
{
acb_t nu0, nu1, nu2, z, w0, w1, w2, t, u;
slong prec0, prec1, prec2;
acb_init(nu0);
acb_init(nu1);
acb_init(nu2);
acb_init(z);
acb_init(w0);
acb_init(w1);
acb_init(w2);
acb_init(t);
acb_init(u);
prec0 = 2 + n_randint(state, 1000);
prec1 = 2 + n_randint(state, 1000);
prec2 = 2 + n_randint(state, 1000);
acb_randtest_param(nu0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest(w0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest(w1, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest(w2, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_sub_ui(nu1, nu0, 1, prec0);
acb_sub_ui(nu2, nu0, 2, prec0);
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w0, nu0, z, prec0);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w0, nu0, z, prec0);
break;
default:
acb_hypgeom_bessel_i(w0, nu0, z, prec0);
}
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w1, nu0, z, prec1);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w1, nu0, z, prec1);
break;
default:
acb_hypgeom_bessel_i(w1, nu0, z, prec1);
}
if (!acb_overlaps(w0, w1))
{
flint_printf("FAIL: consistency\n\n");
flint_printf("nu = "); acb_printd(nu0, 30); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("w0 = "); acb_printd(w0, 30); flint_printf("\n\n");
flint_printf("w1 = "); acb_printd(w1, 30); flint_printf("\n\n");
abort();
}
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w1, nu1, z, prec1);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w1, nu1, z, prec1);
break;
default:
acb_hypgeom_bessel_i(w1, nu1, z, prec1);
}
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w2, nu2, z, prec2);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w2, nu2, z, prec2);
break;
default:
acb_hypgeom_bessel_i(w2, nu2, z, prec2);
}
acb_mul(t, w1, nu1, prec0);
acb_mul_2exp_si(t, t, 1);
acb_submul(t, w2, z, prec0);
acb_addmul(t, w0, z, prec0);
if (!acb_contains_zero(t))
{
flint_printf("FAIL: contiguous relation\n\n");
flint_printf("nu = "); acb_printd(nu0, 30); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("w0 = "); acb_printd(w0, 30); flint_printf("\n\n");
flint_printf("w1 = "); acb_printd(w1, 30); flint_printf("\n\n");
flint_printf("w2 = "); acb_printd(w2, 30); flint_printf("\n\n");
flint_printf("t = "); acb_printd(t, 30); flint_printf("\n\n");
abort();
}
acb_neg(t, nu0);
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w2, t, z, prec2);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w2, t, z, prec2);
break;
default:
acb_hypgeom_bessel_i(w2, t, z, prec2);
}
acb_mul(w1, w1, w2, prec2);
acb_neg(t, nu1);
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w2, t, z, prec2);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w2, t, z, prec2);
break;
default:
acb_hypgeom_bessel_i(w2, t, z, prec2);
}
acb_mul(w0, w0, w2, prec2);
acb_sub(w0, w1, w0, prec2);
acb_sin_pi(t, nu0, prec2);
acb_const_pi(u, prec2);
acb_mul(u, u, z, prec2);
acb_div(t, t, u, prec2);
acb_mul_2exp_si(t, t, 1);
if (!acb_overlaps(w0, t))
{
flint_printf("FAIL: wronskian\n\n");
flint_printf("nu = "); acb_printd(nu0, 30); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("w0 = "); acb_printd(w0, 30); flint_printf("\n\n");
flint_printf("t = "); acb_printd(t, 30); flint_printf("\n\n");
abort();
}
acb_clear(nu0);
acb_clear(nu1);
acb_clear(nu2);
acb_clear(z);
acb_clear(w0);
acb_clear(w1);
acb_clear(w2);
acb_clear(t);
acb_clear(u);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
