int
elliptic(acb_ptr out, const acb_t inp, void * params, long order, long prec)
{
acb_ptr t;
t = _acb_vec_init(order);
acb_set(t, inp);
if (order > 1)
acb_one(t + 1);
_acb_poly_sin_series(t, t, FLINT_MIN(2, order), order, prec);
_acb_poly_mullow(out, t, order, t, order, order, prec);
_acb_vec_scalar_mul_2exp_si(t, out, order, -1);
acb_sub_ui(t, t, 1, prec);
_acb_vec_neg(t, t, order);
_acb_poly_rsqrt_series(out, t, order, order, prec);
_acb_vec_clear(t, order);
return 0;
}
void
_acb_poly_zeta_em_sum(acb_ptr z, const acb_t s, const acb_t a, int deflate, ulong N, ulong M, slong d, slong prec)
{
acb_ptr t, u, v, term, sum;
acb_t Na, one;
slong i;
t = _acb_vec_init(d + 1);
u = _acb_vec_init(d);
v = _acb_vec_init(d);
term = _acb_vec_init(d);
sum = _acb_vec_init(d);
acb_init(Na);
acb_init(one);
prec += 2 * (FLINT_BIT_COUNT(N) + FLINT_BIT_COUNT(d));
acb_one(one);
/* sum 1/(k+a)^(s+x) */
if (acb_is_one(a) && d <= 3)
_acb_poly_powsum_one_series_sieved(sum, s, N, d, prec);
else if (N > 50 && flint_get_num_threads() > 1)
_acb_poly_powsum_series_naive_threaded(sum, s, a, one, N, d, prec);
else
_acb_poly_powsum_series_naive(sum, s, a, one, N, d, prec);
/* t = 1/(N+a)^(s+x); we might need one extra term for deflation */
acb_add_ui(Na, a, N, prec);
_acb_poly_acb_invpow_cpx(t, Na, s, d + 1, prec);
/* sum += (N+a) * 1/((s+x)-1) * t */
if (!deflate)
{
/* u = (N+a)^(1-(s+x)) */
acb_sub_ui(v, s, 1, prec);
_acb_poly_acb_invpow_cpx(u, Na, v, d, prec);
/* divide by 1/((s-1) + x) */
acb_sub_ui(v, s, 1, prec);
acb_div(u, u, v, prec);
for (i = 1; i < d; i++)
{
acb_sub(u + i, u + i, u + i - 1, prec);
acb_div(u + i, u + i, v, prec);
}
_acb_vec_add(sum, sum, u, d, prec);
}
/* sum += ((N+a)^(1-(s+x)) - 1) / ((s+x) - 1) */
else
{
/* at s = 1, this becomes (N*t - 1)/x, i.e. just remove one coeff */
if (acb_is_one(s))
{
for (i = 0; i < d; i++)
acb_mul(u + i, t + i + 1, Na, prec);
_acb_vec_add(sum, sum, u, d, prec);
}
else
{
/* TODO: this is numerically unstable for large derivatives,
and divides by zero if s contains 1. We want a good
way to evaluate the power series ((N+a)^y - 1) / y where y has
nonzero constant term, without doing a division.
How is this best done? */
_acb_vec_scalar_mul(t, t, d, Na, prec);
acb_sub_ui(t + 0, t + 0, 1, prec);
acb_sub_ui(u + 0, s, 1, prec);
acb_inv(u + 0, u + 0, prec);
for (i = 1; i < d; i++)
acb_mul(u + i, u + i - 1, u + 0, prec);
for (i = 1; i < d; i += 2)
acb_neg(u + i, u + i);
_acb_poly_mullow(v, u, d, t, d, d, prec);
_acb_vec_add(sum, sum, v, d, prec);
_acb_poly_acb_invpow_cpx(t, Na, s, d, prec);
}
}
/* sum += u = 1/2 * t */
_acb_vec_scalar_mul_2exp_si(u, t, d, -WORD(1));
_acb_vec_add(sum, sum, u, d, prec);
/* Euler-Maclaurin formula tail */
if (d < 5 || d < M / 10)
_acb_poly_zeta_em_tail_naive(u, s, Na, t, M, d, prec);
else
_acb_poly_zeta_em_tail_bsplit(u, s, Na, t, M, d, prec);
_acb_vec_add(z, sum, u, d, prec);
_acb_vec_clear(t, d + 1);
_acb_vec_clear(u, d);
_acb_vec_clear(v, d);
_acb_vec_clear(term, d);
_acb_vec_clear(sum, d);
acb_clear(Na);
acb_clear(one);
}
void
_acb_poly_zeta_em_tail_naive(acb_ptr sum, const acb_t s, const acb_t Na, acb_srcptr Nasx, slong M, slong d, slong prec)
{
acb_ptr u, term;
acb_t Na2, splus, rec;
arb_t x;
fmpz_t c;
int aint;
slong r;
BERNOULLI_ENSURE_CACHED(2 * M);
u = _acb_vec_init(d);
term = _acb_vec_init(d);
acb_init(splus);
acb_init(rec);
acb_init(Na2);
arb_init(x);
fmpz_init(c);
_acb_vec_zero(sum, d);
/* u = 1/2 * Nasx */
_acb_vec_scalar_mul_2exp_si(u, Nasx, d, -WORD(1));
/* term = u * (s+x) / (N+a) */
_acb_poly_mullow_cpx(u, u, d, s, d, prec);
_acb_vec_scalar_div(term, u, d, Na, prec);
/* (N+a)^2 or 1/(N+a)^2 */
acb_mul(Na2, Na, Na, prec);
aint = acb_is_int(Na2);
if (!aint)
acb_inv(Na2, Na2, prec);
for (r = 1; r <= M; r++)
{
/* flint_printf("sum 2: %wd %wd\n", r, M); */
/* sum += bernoulli number * term */
arb_set_round_fmpz(x, fmpq_numref(bernoulli_cache + 2 * r), prec);
arb_div_fmpz(x, x, fmpq_denref(bernoulli_cache + 2 * r), prec);
_acb_vec_scalar_mul_arb(u, term, d, x, prec);
_acb_vec_add(sum, sum, u, d, prec);
/* multiply term by ((s+x)+2r-1)((s+x)+2r) / ((N+a)^2 * (2*r+1)*(2*r+2)) */
acb_set(splus, s);
arb_add_ui(acb_realref(splus), acb_realref(splus), 2*r-1, prec);
_acb_poly_mullow_cpx(term, term, d, splus, d, prec);
arb_add_ui(acb_realref(splus), acb_realref(splus), 1, prec);
_acb_poly_mullow_cpx(term, term, d, splus, d, prec);
/* TODO: combine with previous multiplication? */
if (aint)
{
arb_mul_ui(x, acb_realref(Na2), 2*r+1, prec);
arb_mul_ui(x, x, 2*r+2, prec);
_acb_vec_scalar_div_arb(term, term, d, x, prec);
}
else
{
fmpz_set_ui(c, 2*r+1);
fmpz_mul_ui(c, c, 2*r+2);
acb_div_fmpz(rec, Na2, c, prec);
_acb_vec_scalar_mul(term, term, d, rec, prec);
}
}
_acb_vec_clear(u, d);
_acb_vec_clear(term, d);
acb_clear(splus);
acb_clear(rec);
acb_clear(Na2);
arb_clear(x);
fmpz_clear(c);
}
void
_acb_poly_sin_cos_series_tangent(acb_ptr s, acb_ptr c,
const acb_srcptr h, slong hlen, slong len, slong prec, int times_pi)
{
acb_ptr t, u, v;
acb_t s0, c0;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
if (times_pi)
acb_sin_cos_pi(s, c, h, prec);
else
acb_sin_cos(s, c, h, prec);
_acb_vec_zero(s + 1, len - 1);
_acb_vec_zero(c + 1, len - 1);
return;
}
/*
sin(x) = 2*tan(x/2)/(1+tan(x/2)^2)
cos(x) = (1-tan(x/2)^2)/(1+tan(x/2)^2)
*/
acb_init(s0);
acb_init(c0);
t = _acb_vec_init(3 * len);
u = t + len;
v = u + len;
/* sin, cos of h0 */
if (times_pi)
acb_sin_cos_pi(s0, c0, h, prec);
else
acb_sin_cos(s0, c0, h, prec);
/* t = tan((h-h0)/2) */
acb_zero(u);
_acb_vec_scalar_mul_2exp_si(u + 1, h + 1, hlen - 1, -1);
if (times_pi)
{
acb_const_pi(t, prec);
_acb_vec_scalar_mul(u + 1, u + 1, hlen - 1, t, prec);
}
_acb_poly_tan_series(t, u, hlen, len, prec);
/* v = 1 + t^2 */
_acb_poly_mullow(v, t, len, t, len, len, prec);
acb_add_ui(v, v, 1, prec);
/* u = 1/(1+t^2) */
_acb_poly_inv_series(u, v, len, len, prec);
/* sine */
_acb_poly_mullow(s, t, len, u, len, len, prec);
_acb_vec_scalar_mul_2exp_si(s, s, len, 1);
/* cosine */
acb_sub_ui(v, v, 2, prec);
_acb_vec_neg(v, v, len);
_acb_poly_mullow(c, v, len, u, len, len, prec);
/* sin(h0 + h1) = cos(h0) sin(h1) + sin(h0) cos(h1)
cos(h0 + h1) = cos(h0) cos(h1) - sin(h0) sin(h1) */
if (!acb_is_zero(s0))
{
_acb_vec_scalar_mul(t, s, len, c0, prec);
_acb_vec_scalar_mul(u, c, len, s0, prec);
_acb_vec_scalar_mul(v, s, len, s0, prec);
_acb_vec_add(s, t, u, len, prec);
_acb_vec_scalar_mul(t, c, len, c0, prec);
_acb_vec_sub(c, t, v, len, prec);
}
_acb_vec_clear(t, 3 * len);
acb_clear(s0);
acb_clear(c0);
}
