int
bessel(acb_ptr out, const acb_t inp, void * params, long order, long prec)
{
acb_ptr t;
acb_t z;
ulong n;
t = _acb_vec_init(order);
acb_init(z);
acb_set(t, inp);
if (order > 1)
acb_one(t + 1);
n = 10;
arb_set_si(acb_realref(z), 20);
arb_set_si(acb_imagref(z), 10);
/* z sin(t) */
_acb_poly_sin_series(out, t, FLINT_MIN(2, order), order, prec);
_acb_vec_scalar_mul(out, out, order, z, prec);
/* t n */
_acb_vec_scalar_mul_ui(t, t, FLINT_MIN(2, order), n, prec);
_acb_poly_sub(out, t, FLINT_MIN(2, order), out, order, prec);
_acb_poly_cos_series(out, out, order, order, prec);
_acb_vec_clear(t, order);
acb_clear(z);
return 0;
}
void
_acb_poly_mullow_classical(acb_ptr res,
acb_srcptr poly1, slong len1,
acb_srcptr poly2, slong len2, slong n, slong prec)
{
len1 = FLINT_MIN(len1, n);
len2 = FLINT_MIN(len2, n);
if (n == 1)
{
acb_mul(res, poly1, poly2, prec);
}
else if (poly1 == poly2 && len1 == len2)
{
slong i;
_acb_vec_scalar_mul(res, poly1, FLINT_MIN(len1, n), poly1, prec);
_acb_vec_scalar_mul(res + len1, poly1 + 1, n - len1, poly1 + len1 - 1, prec);
for (i = 1; i < len1 - 1; i++)
_acb_vec_scalar_addmul(res + i + 1, poly1 + 1,
FLINT_MIN(i - 1, n - (i + 1)), poly1 + i, prec);
for (i = 1; i < FLINT_MIN(2 * len1 - 2, n); i++)
acb_mul_2exp_si(res + i, res + i, 1);
for (i = 1; i < FLINT_MIN(len1 - 1, (n + 1) / 2); i++)
acb_addmul(res + 2 * i, poly1 + i, poly1 + i, prec);
}
else
{
slong i;
_acb_vec_scalar_mul(res, poly1, FLINT_MIN(len1, n), poly2, prec);
if (n > len1)
_acb_vec_scalar_mul(res + len1, poly2 + 1, n - len1,
poly1 + len1 - 1, prec);
for (i = 0; i < FLINT_MIN(len1, n) - 1; i++)
_acb_vec_scalar_addmul(res + i + 1, poly2 + 1,
FLINT_MIN(len2, n - i) - 1,
poly1 + i, prec);
}
}
void
integrals_edge_factors_gc(acb_ptr res, const acb_t cab, const acb_t ba2, sec_t c, slong prec)
{
slong i;
acb_t cj, ci;
acb_init(cj);
acb_init(ci);
/* polynomial shift */
acb_vec_polynomial_shift(res, cab, c.g, prec);
/* constants cj, j = 1 */
/* c_1 = (1-zeta^-1) ba2^(-d/2) (-I)^i
* = 2 / ba2^(d/2) */
acb_pow_ui(cj, ba2, c.d / 2, prec);
if (c.d % 2)
{
acb_t t;
acb_init(t);
acb_sqrt(t, ba2, prec);
acb_mul(cj, cj, t, prec);
acb_clear(t);
}
acb_inv(cj, cj, prec);
acb_mul_2exp_si(cj, cj, 1);
_acb_vec_scalar_mul(res, res, c.g, cj, prec);
/* constant ci = -I * ba2*/
acb_one(ci);
for (i = 1; i < c.g; i++)
{
acb_mul(ci, ci, ba2, prec);
acb_div_onei(ci, ci);
acb_mul(res + i, res + i, ci, prec);
}
acb_clear(ci);
acb_clear(cj);
}
void
_acb_poly_compose_series_horner(acb_ptr res, acb_srcptr poly1, slong len1,
acb_srcptr poly2, slong len2, slong n, slong prec)
{
if (n == 1)
{
acb_set(res, poly1);
}
else
{
slong i = len1 - 1;
slong lenr;
acb_ptr t = _acb_vec_init(n);
lenr = len2;
_acb_vec_scalar_mul(res, poly2, len2, poly1 + i, prec);
i--;
acb_add(res, res, poly1 + i, prec);
while (i > 0)
{
i--;
if (lenr + len2 - 1 < n)
{
_acb_poly_mul(t, res, lenr, poly2, len2, prec);
lenr = lenr + len2 - 1;
}
else
{
_acb_poly_mullow(t, res, lenr, poly2, len2, n, prec);
lenr = n;
}
_acb_poly_add(res, t, lenr, poly1 + i, 1, prec);
}
_acb_vec_zero(res + lenr, n - lenr);
_acb_vec_clear(t, n);
}
}
void
_acb_poly_rgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong i, rflen, r, n, wp;
acb_ptr t, u, v;
acb_struct f[2];
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
acb_rgamma(res, h, prec);
_acb_vec_zero(res + 1, len - 1);
return;
}
/* use real code for real input */
if (_acb_vec_is_real(h, hlen))
{
arb_ptr tmp = _arb_vec_init(len);
for (i = 0; i < hlen; i++)
arb_set(tmp + i, acb_realref(h + i));
_arb_poly_rgamma_series(tmp, tmp, hlen, len, prec);
for (i = 0; i < len; i++)
acb_set_arb(res + i, tmp + i);
_arb_vec_clear(tmp, len);
return;
}
wp = prec + FLINT_BIT_COUNT(prec);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
v = _acb_vec_init(len);
acb_init(f);
acb_init(f + 1);
/* otherwise use Stirling series */
acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);
/* rgamma(h) = (gamma(1-h+r) sin(pi h)) / (rf(1-h, r) * pi), h = h0 + t*/
if (reflect)
{
/* u = gamma(r+1-h) */
acb_sub_ui(f, h, r + 1, wp);
acb_neg(f, f);
_acb_poly_gamma_stirling_eval(t, f, n, len, wp);
_acb_poly_exp_series(u, t, len, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(u + i, u + i);
/* v = sin(pi x) */
acb_set(f, h);
acb_one(f + 1);
_acb_poly_sin_pi_series(v, f, 2, len, wp);
_acb_poly_mullow(t, u, len, v, len, len, wp);
/* rf(1-h,r) * pi */
if (r == 0)
{
acb_const_pi(u, wp);
_acb_vec_scalar_div(v, t, len, u, wp);
}
else
{
acb_sub_ui(f, h, 1, wp);
acb_neg(f, f);
acb_set_si(f + 1, -1);
rflen = FLINT_MIN(len, r + 1);
_acb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
acb_const_pi(u, wp);
_acb_vec_scalar_mul(v, v, rflen, u, wp);
/* divide by rising factorial */
/* TODO: might better to use div_series, when it has a good basecase */
_acb_poly_inv_series(u, v, rflen, len, wp);
_acb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* rgamma(h) = rgamma(h+r) rf(h,r) */
if (r == 0)
{
acb_add_ui(f, h, r, wp);
_acb_poly_gamma_stirling_eval(t, f, n, len, wp);
_acb_vec_neg(t, t, len);
_acb_poly_exp_series(v, t, len, len, wp);
}
else
{
acb_set(f, h);
acb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_acb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);
acb_add_ui(f, h, r, wp);
_acb_poly_gamma_stirling_eval(v, f, n, len, wp);
_acb_vec_neg(v, v, len);
_acb_poly_exp_series(u, v, len, len, wp);
_acb_poly_mullow(v, u, len, t, rflen, len, wp);
}
}
/* compose with nonconstant part */
acb_zero(t);
_acb_vec_set(t + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, v, len, t, hlen, len, prec);
acb_clear(f);
acb_clear(f + 1);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
_acb_vec_clear(v, len);
}
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_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_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);
}
void
_acb_poly_lgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong i, r, n, wp;
acb_t zr;
acb_ptr t, u;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
acb_lgamma(res, h, prec);
if (acb_is_finite(res))
_acb_vec_zero(res + 1, len - 1);
else
_acb_vec_indeterminate(res + 1, len - 1);
return;
}
if (len == 2)
{
acb_t v;
acb_init(v);
acb_set(v, h + 1);
acb_digamma(res + 1, h, prec);
acb_lgamma(res, h, prec);
acb_mul(res + 1, res + 1, v, prec);
acb_clear(v);
return;
}
/* use real code for real input and output */
if (_acb_vec_is_real(h, hlen) && arb_is_positive(acb_realref(h)))
{
arb_ptr tmp = _arb_vec_init(len);
for (i = 0; i < hlen; i++)
arb_set(tmp + i, acb_realref(h + i));
_arb_poly_lgamma_series(tmp, tmp, hlen, len, prec);
for (i = 0; i < len; i++)
acb_set_arb(res + i, tmp + i);
_arb_vec_clear(tmp, len);
return;
}
wp = prec + FLINT_BIT_COUNT(prec);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
acb_init(zr);
/* use Stirling series */
acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);
if (reflect)
{
/* log gamma(h+x) = log rf(1-(h+x), r) - log gamma(1-(h+x)+r) - log sin(pi (h+x)) + log(pi) */
if (r != 0) /* otherwise t = 0 */
{
acb_sub_ui(u, h, 1, wp);
acb_neg(u, u);
_log_rising_ui_series(t, u, r, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(t + i, t + i);
}
acb_sub_ui(u, h, 1, wp);
acb_neg(u, u);
acb_add_ui(zr, u, r, wp);
_acb_poly_gamma_stirling_eval(u, zr, n, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(u + i, u + i);
_acb_vec_sub(t, t, u, len, wp);
/* log(sin) is unstable with large imaginary parts;
cot_pi is implemented in a numerically stable way */
acb_set(u, h);
acb_one(u + 1);
_acb_poly_cot_pi_series(u, u, 2, len - 1, wp);
_acb_poly_integral(u, u, len, wp);
acb_const_pi(u, wp);
_acb_vec_scalar_mul(u + 1, u + 1, len - 1, u, wp);
acb_log_sin_pi(u, h, wp);
_acb_vec_sub(u, t, u, len, wp);
acb_const_pi(t, wp); /* todo: constant for log pi */
acb_log(t, t, wp);
acb_add(u, u, t, wp);
}
else
{
/* log gamma(x) = log gamma(x+r) - log rf(x,r) */
acb_add_ui(zr, h, r, wp);
_acb_poly_gamma_stirling_eval(u, zr, n, len, wp);
if (r != 0)
{
_log_rising_ui_series(t, h, r, len, wp);
_acb_vec_sub(u, u, t, len, wp);
}
}
/* compose with nonconstant part */
acb_zero(t);
_acb_vec_set(t + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, u, len, t, hlen, len, prec);
acb_clear(zr);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
void
_acb_poly_digamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong i, r, n, rflen, wp;
acb_t zr;
acb_ptr t, u, v;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
acb_digamma(res, h, prec);
if (acb_is_finite(res))
_acb_vec_zero(res + 1, len - 1);
else
_acb_vec_indeterminate(res + 1, len - 1);
return;
}
/* use real code for real input */
if (_acb_vec_is_real(h, hlen))
{
arb_ptr tmp = _arb_vec_init(len);
for (i = 0; i < hlen; i++)
arb_set(tmp + i, acb_realref(h + i));
_arb_poly_digamma_series(tmp, tmp, hlen, len, prec);
for (i = 0; i < len; i++)
acb_set_arb(res + i, tmp + i);
_arb_vec_clear(tmp, len);
return;
}
wp = prec + FLINT_BIT_COUNT(prec);
t = _acb_vec_init(len + 1);
u = _acb_vec_init(len + 1);
v = _acb_vec_init(len + 1);
acb_init(zr);
/* use Stirling series */
acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 1, wp);
/* psi(x) = psi((1-x)+r) - h(1-x,r) - pi*cot(pi*x) */
if (reflect)
{
if (r != 0) /* otherwise t = 0 */
{
acb_sub_ui(v, h, 1, wp);
acb_neg(v, v);
acb_one(v + 1);
rflen = FLINT_MIN(len + 1, r + 1);
_acb_poly_rising_ui_series(u, v, 2, r, rflen, wp);
_acb_poly_derivative(v, u, rflen, wp);
_acb_poly_div_series(t, v, rflen - 1, u, rflen, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(t + i, t + i);
}
acb_sub_ui(zr, h, r + 1, wp);
acb_neg(zr, zr);
_acb_poly_gamma_stirling_eval2(u, zr, n, len + 1, 1, wp);
for (i = 1; i < len; i += 2)
acb_neg(u + i, u + i);
_acb_vec_sub(u, u, t, len, wp);
acb_set(t, h);
acb_one(t + 1);
_acb_poly_cot_pi_series(t, t, 2, len, wp);
acb_const_pi(v, wp);
_acb_vec_scalar_mul(t, t, len, v, wp);
_acb_vec_sub(u, u, t, len, wp);
}
else
{
if (r == 0)
{
acb_add_ui(zr, h, r, wp);
_acb_poly_gamma_stirling_eval2(u, zr, n, len + 1, 1, wp);
}
else
{
acb_set(v, h);
acb_one(v + 1);
rflen = FLINT_MIN(len + 1, r + 1);
_acb_poly_rising_ui_series(u, v, 2, r, rflen, wp);
_acb_poly_derivative(v, u, rflen, wp);
_acb_poly_div_series(t, v, rflen - 1, u, rflen, len, wp);
acb_add_ui(zr, h, r, wp);
_acb_poly_gamma_stirling_eval2(u, zr, n, len + 1, 1, wp);
_acb_vec_sub(u, u, t, len, wp);
}
}
/* compose with nonconstant part */
acb_zero(t);
_acb_vec_set(t + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, u, len, t, hlen, len, prec);
acb_clear(zr);
_acb_vec_clear(t, len + 1);
_acb_vec_clear(u, len + 1);
_acb_vec_clear(v, len + 1);
}
void
acb_elliptic_p_jet(acb_ptr r, const acb_t z, const acb_t tau, slong len, slong prec)
{
acb_t t01, t02, t03, t04;
acb_ptr tz1, tz2, tz3, tz4;
acb_t t;
int real;
slong k;
if (len < 1)
return;
if (len == 1)
{
acb_elliptic_p(r, z, tau, prec);
return;
}
real = acb_is_real(z) && arb_is_int_2exp_si(acb_realref(tau), -1) &&
arb_is_positive(acb_imagref(tau));
acb_init(t);
acb_init(t01);
acb_init(t02);
acb_init(t03);
acb_init(t04);
tz1 = _acb_vec_init(len);
tz2 = _acb_vec_init(len);
tz3 = _acb_vec_init(len);
tz4 = _acb_vec_init(len);
acb_modular_theta_jet(tz1, tz2, tz3, tz4, z, tau, len, prec);
/* [theta_4(z) / theta_1(z)]^2 */
_acb_poly_div_series(tz2, tz4, len, tz1, len, len, prec);
_acb_poly_mullow(tz1, tz2, len, tz2, len, len, prec);
acb_zero(t);
acb_modular_theta(t01, t02, t03, t04, t, tau, prec);
/* [theta_2(0) * theta_3(0)] ^2 */
acb_mul(t, t02, t03, prec);
acb_mul(t, t, t, prec);
_acb_vec_scalar_mul(tz1, tz1, len, t, prec);
/* - [theta_2(0)^4 + theta_3(0)^4] / 3 */
acb_pow_ui(t02, t02, 4, prec);
acb_pow_ui(t03, t03, 4, prec);
acb_add(t, t02, t03, prec);
acb_div_ui(t, t, 3, prec);
acb_sub(tz1, tz1, t, prec);
/* times pi^2 */
acb_const_pi(t, prec);
acb_mul(t, t, t, prec);
_acb_vec_scalar_mul(r, tz1, len, t, prec);
if (real)
{
for (k = 0; k < len; k++)
arb_zero(acb_imagref(r + k));
}
acb_clear(t);
acb_clear(t01);
acb_clear(t02);
acb_clear(t03);
acb_clear(t04);
_acb_vec_clear(tz1, len);
_acb_vec_clear(tz2, len);
_acb_vec_clear(tz3, len);
_acb_vec_clear(tz4, len);
}
int main()
{
long iter;
flint_rand_t state;
printf("root_bound_fujiwara....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000; iter++)
{
acb_poly_t a;
acb_ptr roots;
acb_t t;
mag_t mag1, mag2;
long i, deg, prec;
prec = 10 + n_randint(state, 400);
deg = n_randint(state, 10);
acb_init(t);
acb_poly_init(a);
mag_init(mag1);
mag_init(mag2);
roots = _acb_vec_init(deg);
for (i = 0; i < deg; i++)
acb_randtest(roots + i, state, prec, 1 + n_randint(state, 20));
acb_poly_product_roots(a, roots, deg, prec);
acb_randtest(t, state, prec, 1 + n_randint(state, 20));
_acb_vec_scalar_mul(a->coeffs, a->coeffs, a->length, t, prec);
acb_poly_root_bound_fujiwara(mag1, a);
for (i = 0; i < deg; i++)
{
acb_get_mag(mag2, roots + i);
/* acb_get_mag gives an upper bound which due to rounding
could be larger than mag1, so we pick a slightly
smaller number */
mag_mul_ui(mag2, mag2, 10000);
mag_div_ui(mag2, mag2, 10001);
if (mag_cmp(mag2, mag1) > 0)
{
printf("FAIL\n");
printf("a = "); acb_poly_printd(a, 15); printf("\n\n");
printf("root = "); acb_printd(roots + i, 15); printf("\n\n");
printf("mag1 = "); mag_printd(mag1, 10); printf("\n\n");
printf("mag2 = "); mag_printd(mag2, 10); printf("\n\n");
abort();
}
}
_acb_vec_clear(roots, deg);
acb_clear(t);
acb_poly_clear(a);
mag_clear(mag1);
mag_clear(mag2);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
