void
acb_poly_inv_series(acb_poly_t Qinv, const acb_poly_t Q, slong n, slong prec)
{
if (n == 0)
{
acb_poly_zero(Qinv);
return;
}
if (Q->length == 0)
{
acb_poly_fit_length(Qinv, n);
_acb_vec_indeterminate(Qinv->coeffs, n);
_acb_poly_set_length(Qinv, n);
return;
}
if (Qinv == Q)
{
acb_poly_t t;
acb_poly_init(t);
acb_poly_inv_series(t, Q, n, prec);
acb_poly_swap(Qinv, t);
acb_poly_clear(t);
return;
}
acb_poly_fit_length(Qinv, n);
_acb_poly_inv_series(Qinv->coeffs, Q->coeffs, Q->length, n, prec);
_acb_poly_set_length(Qinv, n);
_acb_poly_normalise(Qinv);
}
void
acb_hypgeom_bessel_k_0f1_series(acb_poly_t res,
const acb_poly_t nu, const acb_poly_t z,
slong len, slong prec)
{
acb_poly_t s, u, A, B;
acb_poly_struct b[2];
arb_t c;
slong wlen;
int singular;
acb_poly_init(s);
acb_poly_init(u);
acb_poly_init(A);
acb_poly_init(B);
acb_poly_init(b + 0);
acb_poly_init(b + 1);
arb_init(c);
singular = (nu->length == 0) || acb_is_int(nu->coeffs);
wlen = len + (singular != 0);
/* A = (z/2)^nu, B = 1/A */
acb_poly_scalar_mul_2exp_si(A, z, -1);
acb_poly_pow_series(A, A, nu, wlen, prec);
acb_poly_inv_series(B, A, wlen, prec);
acb_poly_mullow(u, z, z, wlen, prec);
acb_poly_scalar_mul_2exp_si(u, u, -2);
acb_poly_one(b + 1);
acb_poly_add_si(b + 0, nu, 1, prec);
acb_hypgeom_pfq_series_direct(s, NULL, 0, b, 2, u, 1, -1, wlen, prec);
acb_poly_mullow(A, A, s, wlen, prec);
acb_poly_add_si(b + 0, nu, -1, prec);
acb_poly_neg(b + 0, b + 0);
acb_hypgeom_pfq_series_direct(s, NULL, 0, b, 2, u, 1, -1, wlen, prec);
acb_poly_mullow(B, B, s, wlen, prec);
acb_poly_sub(A, B, A, prec);
acb_poly_scalar_mul_2exp_si(A, A, -1);
/* multiply by pi csc(pi nu) */
acb_poly_sin_pi_series(B, nu, wlen, prec);
if (singular)
{
acb_poly_shift_right(A, A, 1);
acb_poly_shift_right(B, B, 1);
}
acb_poly_div_series(res, A, B, len, prec);
arb_const_pi(c, prec);
_acb_vec_scalar_mul_arb(res->coeffs, res->coeffs, res->length, c, prec);
acb_poly_clear(s);
acb_poly_clear(u);
acb_poly_clear(A);
acb_poly_clear(B);
acb_poly_clear(b + 0);
acb_poly_clear(b + 1);
arb_clear(c);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("digamma_series....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++)
{
slong m, n1, n2, rbits1, rbits2, rbits3;
acb_poly_t a, b, c, d;
rbits1 = 2 + n_randint(state, 300);
rbits2 = 2 + n_randint(state, 300);
rbits3 = 2 + n_randint(state, 300);
m = n_randint(state, 25);
n1 = n_randint(state, 25);
n2 = n_randint(state, 25);
acb_poly_init(a);
acb_poly_init(b);
acb_poly_init(c);
acb_poly_init(d);
acb_poly_randtest(a, state, m, rbits1, 10);
acb_poly_digamma_series(b, a, n1, rbits2);
acb_poly_digamma_series(c, a, n2, rbits3);
acb_poly_set(d, b);
acb_poly_truncate(d, FLINT_MIN(n1, n2));
acb_poly_truncate(c, FLINT_MIN(n1, n2));
if (!acb_poly_overlaps(c, d))
{
flint_printf("FAIL\n\n");
flint_printf("n1 = %wd, n2 = %wd, bits2 = %wd, bits3 = %wd\n", n1, n2, rbits2, rbits3);
flint_printf("a = "); acb_poly_printd(a, 50); flint_printf("\n\n");
flint_printf("b = "); acb_poly_printd(b, 50); flint_printf("\n\n");
flint_printf("c = "); acb_poly_printd(c, 50); flint_printf("\n\n");
flint_abort();
}
/* check psi(a) + 1/a = psi(a+1) */
acb_poly_inv_series(c, a, n1, rbits2);
acb_poly_add(c, b, c, rbits2);
acb_poly_add_si(d, a, 1, rbits2);
acb_poly_digamma_series(d, d, n1, rbits2);
if (!acb_poly_overlaps(c, d))
{
flint_printf("FAIL (functional equation)\n\n");
flint_printf("a = "); acb_poly_printd(a, 15); flint_printf("\n\n");
flint_printf("b = "); acb_poly_printd(b, 15); flint_printf("\n\n");
flint_printf("c = "); acb_poly_printd(c, 15); flint_printf("\n\n");
flint_printf("d = "); acb_poly_printd(d, 15); flint_printf("\n\n");
flint_abort();
}
acb_poly_digamma_series(a, a, n1, rbits2);
if (!acb_poly_overlaps(a, b))
{
flint_printf("FAIL (aliasing)\n\n");
flint_abort();
}
acb_poly_clear(a);
acb_poly_clear(b);
acb_poly_clear(c);
acb_poly_clear(d);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("rsqrt_series....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 5000; iter++)
{
slong m, n, qbits, rbits1, rbits2;
fmpq_poly_t A;
acb_poly_t a, b, c;
qbits = 2 + n_randint(state, 200);
rbits1 = 2 + n_randint(state, 200);
rbits2 = 2 + n_randint(state, 200);
m = 1 + n_randint(state, 30);
n = 1 + n_randint(state, 30);
fmpq_poly_init(A);
acb_poly_init(a);
acb_poly_init(b);
acb_poly_init(c);
fmpq_poly_randtest_not_zero(A, state, m, qbits);
acb_poly_set_fmpq_poly(a, A, rbits1);
acb_poly_randtest(b, state, 1 + n_randint(state, 30), rbits1, 5);
acb_poly_rsqrt_series(b, a, n, rbits2);
/* Check 1/((1/sqrt(a))^2) = a */
acb_poly_mullow(c, b, b, n, rbits2);
acb_poly_inv_series(c, c, n, rbits2);
fmpq_poly_truncate(A, n);
if (!acb_poly_contains_fmpq_poly(c, A))
{
flint_printf("FAIL\n\n");
flint_printf("bits2 = %wd\n", rbits2);
flint_printf("A = "); fmpq_poly_print(A); flint_printf("\n\n");
flint_printf("a = "); acb_poly_printd(a, 15); flint_printf("\n\n");
flint_printf("b = "); acb_poly_printd(b, 15); flint_printf("\n\n");
flint_printf("c = "); acb_poly_printd(c, 15); flint_printf("\n\n");
abort();
}
acb_poly_rsqrt_series(a, a, n, rbits2);
if (!acb_poly_equal(a, b))
{
flint_printf("FAIL (aliasing)\n\n");
abort();
}
fmpq_poly_clear(A);
acb_poly_clear(a);
acb_poly_clear(b);
acb_poly_clear(c);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
_acb_hypgeom_2f1_transform_limit(acb_t res, const acb_poly_t a, const acb_poly_t b,
const acb_poly_t c, const acb_poly_t z, int which, slong prec)
{
acb_poly_t ba, ca, cb, cab, ac1, bc1, ab1, ba1, w, t, u, v, s;
acb_t tt;
acb_poly_init(ba);
acb_poly_init(ca); acb_poly_init(cb); acb_poly_init(cab);
acb_poly_init(ac1); acb_poly_init(bc1);
acb_poly_init(ab1); acb_poly_init(ba1);
acb_poly_init(w); acb_poly_init(t);
acb_poly_init(u); acb_poly_init(v);
acb_poly_init(s);
acb_init(tt);
acb_poly_add_si(s, z, -1, prec); /* s = 1 - z */
acb_poly_neg(s, s);
acb_poly_sub(ba, b, a, prec); /* ba = b - a */
acb_poly_sub(ca, c, a, prec); /* ca = c - a */
acb_poly_sub(cb, c, b, prec); /* cb = c - b */
acb_poly_sub(cab, ca, b, prec); /* cab = c - a - b */
acb_poly_add_si(ac1, ca, -1, prec); acb_poly_neg(ac1, ac1); /* ac1 = a - c + 1 */
acb_poly_add_si(bc1, cb, -1, prec); acb_poly_neg(bc1, bc1); /* bc1 = b - c + 1 */
acb_poly_add_si(ab1, ba, -1, prec); acb_poly_neg(ab1, ab1); /* ab1 = a - b + 1 */
acb_poly_add_si(ba1, ba, 1, prec); /* ba1 = b - a + 1 */
/* t = left term, u = right term (DLMF 15.8.1 - 15.8.5) */
if (which == 2)
{
acb_poly_inv_series(w, z, 2, prec); /* w = 1/z */
acb_hypgeom_2f1_series_direct(t, a, ac1, ab1, w, 1, 2, prec);
acb_hypgeom_2f1_series_direct(u, b, bc1, ba1, w, 1, 2, prec);
}
else if (which == 3)
{
acb_poly_inv_series(w, s, 2, prec); /* w = 1/(1-z) */
acb_hypgeom_2f1_series_direct(t, a, cb, ab1, w, 1, 2, prec);
acb_hypgeom_2f1_series_direct(u, b, ca, ba1, w, 1, 2, prec);
}
else if (which == 4)
{
acb_poly_set(w, s); /* w = 1-z */
acb_poly_add(v, ac1, b, prec); /* v = a+b-c+1 */
acb_hypgeom_2f1_series_direct(t, a, b, v, w, 1, 2, prec);
acb_poly_add_si(v, cab, 1, prec); /* v = c-a-b+1 */
acb_hypgeom_2f1_series_direct(u, ca, cb, v, w, 1, 2, prec);
}
else if (which == 5)
{
acb_poly_inv_series(w, z, 2, prec); /* w = 1-1/z */
acb_poly_neg(w, w);
acb_poly_add_si(w, w, 1, prec);
acb_poly_add(v, ac1, b, prec); /* v = a+b-c+1 */
acb_hypgeom_2f1_series_direct(t, a, ac1, v, w, 1, 2, prec);
acb_poly_add_si(v, cab, 1, prec); /* v = c-a-b+1 */
acb_poly_add_si(u, a, -1, prec); /* u = 1-a */
acb_poly_neg(u, u);
acb_hypgeom_2f1_series_direct(u, ca, u, v, w, 1, 2, prec);
}
else
{
flint_printf("invalid transformation!\n");
flint_abort();
}
/* gamma factors */
acb_poly_rgamma_series(v, a, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
acb_poly_rgamma_series(v, ca, 2, prec);
acb_poly_mullow(t, t, v, 2, prec);
acb_poly_rgamma_series(v, b, 2, prec);
if (which == 2 || which == 3)
acb_poly_mullow(t, t, v, 2, prec);
else
acb_poly_mullow(u, u, v, 2, prec);
acb_poly_rgamma_series(v, cb, 2, prec);
if (which == 2 || which == 3)
acb_poly_mullow(u, u, v, 2, prec);
else
acb_poly_mullow(t, t, v, 2, prec);
if (which == 2 || which == 3)
{
if (which == 2)
acb_poly_neg(s, z); /* -z, otherwise 1-z since before */
acb_poly_neg(v, a);
acb_poly_pow_series(v, s, v, 2, prec);
acb_poly_mullow(t, t, v, 2, prec);
acb_poly_neg(v, b);
acb_poly_pow_series(v, s, v, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
}
else
{
acb_poly_pow_series(v, s, cab, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
if (which == 5)
{
acb_poly_neg(v, a);
acb_poly_pow_series(v, z, v, 2, prec);
acb_poly_mullow(t, t, v, 2, prec);
acb_poly_neg(v, ca);
acb_poly_pow_series(v, z, v, 2, prec);
acb_poly_mullow(u, u, v, 2, prec);
}
}
acb_poly_sub(t, t, u, prec);
if (which == 2 || which == 3)
acb_poly_sin_pi_series(v, ba, 2, prec);
else
acb_poly_sin_pi_series(v, cab, 2, prec);
acb_poly_get_coeff_acb(tt, t, 1);
acb_poly_get_coeff_acb(res, v, 1);
acb_div(res, tt, res, prec);
acb_const_pi(tt, prec);
acb_mul(res, res, tt, prec);
acb_poly_clear(ba);
acb_poly_clear(ca); acb_poly_clear(cb); acb_poly_clear(cab);
acb_poly_clear(ac1); acb_poly_clear(bc1);
acb_poly_clear(ab1); acb_poly_clear(ba1);
acb_poly_clear(w); acb_poly_clear(t);
acb_poly_clear(u); acb_poly_clear(v);
acb_poly_clear(s);
acb_clear(tt);
}
