void
_acb_hypgeom_coulomb_series(acb_ptr F, acb_ptr G,
acb_ptr Hpos, acb_ptr Hneg, const acb_t l, const acb_t eta,
acb_srcptr z, slong zlen, slong len, slong prec)
{
acb_ptr t, v;
if (len == 0)
return;
zlen = FLINT_MIN(zlen, len);
if (zlen == 1)
{
acb_hypgeom_coulomb(F, G, Hpos, Hneg, l, eta, z, prec);
if (F != NULL) _acb_vec_zero(F + 1, len - 1);
if (G != NULL) _acb_vec_zero(G + 1, len - 1);
if (Hpos != NULL) _acb_vec_zero(Hpos + 1, len - 1);
if (Hneg != NULL) _acb_vec_zero(Hneg + 1, len - 1);
return;
}
t = _acb_vec_init(len);
v = _acb_vec_init(zlen);
/* copy nonconstant part first to allow aliasing */
acb_zero(v);
_acb_vec_set(v + 1, z + 1, zlen - 1);
acb_hypgeom_coulomb_jet(F, G, Hpos, Hneg, l, eta, z, len, prec);
if (F != NULL)
{
_acb_vec_set(t, F, len);
_acb_poly_compose_series(F, t, len, v, zlen, len, prec);
}
if (G != NULL)
{
_acb_vec_set(t, G, len);
_acb_poly_compose_series(G, t, len, v, zlen, len, prec);
}
if (Hpos != NULL)
{
_acb_vec_set(t, Hpos, len);
_acb_poly_compose_series(Hpos, t, len, v, zlen, len, prec);
}
if (Hneg != NULL)
{
_acb_vec_set(t, Hneg, len);
_acb_poly_compose_series(Hneg, t, len, v, zlen, len, prec);
}
_acb_vec_clear(t, len);
_acb_vec_clear(v, zlen);
}
void
_acb_poly_compose_series_brent_kung(acb_ptr res,
acb_srcptr poly1, long len1,
acb_srcptr poly2, long len2, long n, long prec)
{
acb_mat_t A, B, C;
acb_ptr t, h;
long i, m;
if (n == 1)
{
acb_set(res, poly1);
return;
}
m = n_sqrt(n) + 1;
acb_mat_init(A, m, n);
acb_mat_init(B, m, m);
acb_mat_init(C, m, n);
h = _acb_vec_init(n);
t = _acb_vec_init(n);
/* Set rows of B to the segments of poly1 */
for (i = 0; i < len1 / m; i++)
_acb_vec_set(B->rows[i], poly1 + i*m, m);
_acb_vec_set(B->rows[i], poly1 + i*m, len1 % m);
/* Set rows of A to powers of poly2 */
acb_set_ui(A->rows[0] + 0, 1UL);
_acb_vec_set(A->rows[1], poly2, len2);
for (i = 2; i < m; i++)
_acb_poly_mullow(A->rows[i], A->rows[(i + 1) / 2], n, A->rows[i / 2], n, n, prec);
acb_mat_mul(C, B, A, prec);
/* Evaluate block composition using the Horner scheme */
_acb_vec_set(res, C->rows[m - 1], n);
_acb_poly_mullow(h, A->rows[m - 1], n, poly2, len2, n, prec);
for (i = m - 2; i >= 0; i--)
{
_acb_poly_mullow(t, res, n, h, n, n, prec);
_acb_poly_add(res, t, n, C->rows[i], n, prec);
}
_acb_vec_clear(h, n);
_acb_vec_clear(t, n);
acb_mat_clear(A);
acb_mat_clear(B);
acb_mat_clear(C);
}
void
acb_poly_add_si(acb_poly_t res, const acb_poly_t x, long y, long prec)
{
long len = x->length;
if (len == 0)
{
acb_poly_set_si(res, y);
}
else
{
acb_poly_fit_length(res, len);
if (y >= 0)
acb_add_ui(res->coeffs, x->coeffs, y, prec);
else
acb_sub_ui(res->coeffs, x->coeffs, -y, prec);
if (res != x)
_acb_vec_set(res->coeffs + 1, x->coeffs + 1, len - 1);
_acb_poly_set_length(res, len);
_acb_poly_normalise(res);
}
}
void
_acb_dirichlet_hardy_z_series(acb_ptr res, acb_srcptr s, slong slen,
const dirichlet_group_t G, const dirichlet_char_t chi,
slong len, slong prec)
{
slen = FLINT_MIN(slen, len);
if (len == 0)
return;
if (slen == 1)
{
acb_dirichlet_hardy_z(res, s, G, chi, 1, prec);
_acb_vec_zero(res + 1, len - 1);
}
else
{
acb_ptr t, u;
t = _acb_vec_init(len);
u = _acb_vec_init(slen);
acb_dirichlet_hardy_z(t, s, G, chi, len, prec);
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, s + 1, slen - 1);
_acb_poly_compose_series(res, t, len, u, slen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, slen);
}
}
void
_acb_poly_revert_series_lagrange_fast(acb_ptr Qinv, acb_srcptr Q, slong Qlen, slong n, slong prec)
{
slong i, j, k, m;
acb_ptr R, S, T, tmp;
acb_t t;
if (n <= 2)
{
if (n >= 1)
acb_zero(Qinv);
if (n == 2)
acb_inv(Qinv + 1, Q + 1, prec);
return;
}
m = n_sqrt(n);
acb_init(t);
R = _acb_vec_init((n - 1) * m);
S = _acb_vec_init(n - 1);
T = _acb_vec_init(n - 1);
acb_zero(Qinv);
acb_inv(Qinv + 1, Q + 1, prec);
_acb_poly_inv_series(Ri(1), Q + 1, FLINT_MIN(Qlen, n) - 1, n - 1, prec);
for (i = 2; i <= m; i++)
_acb_poly_mullow(Ri(i), Ri((i + 1) / 2), n - 1, Ri(i / 2), n - 1, n - 1, prec);
for (i = 2; i < m; i++)
acb_div_ui(Qinv + i, Ri(i) + i - 1, i, prec);
_acb_vec_set(S, Ri(m), n - 1);
for (i = m; i < n; i += m)
{
acb_div_ui(Qinv + i, S + i - 1, i, prec);
for (j = 1; j < m && i + j < n; j++)
{
acb_mul(t, S + 0, Ri(j) + i + j - 1, prec);
for (k = 1; k <= i + j - 1; k++)
acb_addmul(t, S + k, Ri(j) + i + j - 1 - k, prec);
acb_div_ui(Qinv + i + j, t, i + j, prec);
}
if (i + 1 < n)
{
_acb_poly_mullow(T, S, n - 1, Ri(m), n - 1, n - 1, prec);
tmp = S; S = T; T = tmp;
}
}
acb_clear(t);
_acb_vec_clear(R, (n - 1) * m);
_acb_vec_clear(S, n - 1);
_acb_vec_clear(T, n - 1);
}
void
_acb_poly_zeta_series(acb_ptr res, acb_srcptr h, slong hlen, const acb_t a, int deflate, slong len, slong prec)
{
acb_ptr t, u;
hlen = FLINT_MIN(hlen, len);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
_acb_poly_zeta_cpx_reflect(t, h, a, deflate, len, prec);
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, t, len, u, hlen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
int
acb_mat_eig_multiple(acb_ptr E, const acb_mat_t A, acb_srcptr E_approx, const acb_mat_t R_approx, slong prec)
{
slong n;
acb_ptr F;
int success;
n = arb_mat_nrows(A);
F = _acb_vec_init(n);
success = acb_mat_eig_simple_vdhoeven_mourrain(F, NULL, NULL, A, E_approx, R_approx, prec);
if (!success)
success = acb_mat_eig_multiple_rump(F, A, E_approx, R_approx, prec);
_acb_vec_set(E, F, n);
_acb_vec_clear(F, n);
return success;
}
void
_acb_poly_agm1_series(acb_ptr res, acb_srcptr z, long zlen, long len, long prec)
{
acb_ptr t, u;
zlen = FLINT_MIN(zlen, len);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
acb_agm1_cpx(t, z, len, prec);
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, z + 1, zlen - 1);
_acb_poly_compose_series(res, t, len, u, zlen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
void
_acb_poly_powsum_one_series_sieved(acb_ptr z, const acb_t s, slong n, slong len, slong prec)
{
slong * divisors;
slong powers_alloc;
slong i, j, k, ibound, kprev, power_of_two, horner_point;
int critical_line, integer;
acb_ptr powers;
acb_ptr t, u, x;
acb_ptr p1, p2;
arb_t logk, v, w;
critical_line = arb_is_exact(acb_realref(s)) &&
(arf_cmp_2exp_si(arb_midref(acb_realref(s)), -1) == 0);
integer = arb_is_zero(acb_imagref(s)) && arb_is_int(acb_realref(s));
divisors = flint_calloc(n / 2 + 1, sizeof(slong));
powers_alloc = (n / 6 + 1) * len;
powers = _acb_vec_init(powers_alloc);
ibound = n_sqrt(n);
for (i = 3; i <= ibound; i += 2)
if (DIVISOR(i) == 0)
for (j = i * i; j <= n; j += 2 * i)
DIVISOR(j) = i;
t = _acb_vec_init(len);
u = _acb_vec_init(len);
x = _acb_vec_init(len);
arb_init(logk);
arb_init(v);
arb_init(w);
power_of_two = 1;
while (power_of_two * 2 <= n)
power_of_two *= 2;
horner_point = n / power_of_two;
_acb_vec_zero(z, len);
kprev = 0;
COMPUTE_POWER(x, 2, kprev);
for (k = 1; k <= n; k += 2)
{
/* t = k^(-s) */
if (DIVISOR(k) == 0)
{
COMPUTE_POWER(t, k, kprev);
}
else
{
p1 = POWER(DIVISOR(k));
p2 = POWER(k / DIVISOR(k));
if (len == 1)
acb_mul(t, p1, p2, prec);
else
_acb_poly_mullow(t, p1, len, p2, len, len, prec);
}
if (k * 3 <= n)
_acb_vec_set(POWER(k), t, len);
_acb_vec_add(u, u, t, len, prec);
while (k == horner_point && power_of_two != 1)
{
_acb_poly_mullow(t, z, len, x, len, len, prec);
_acb_vec_add(z, t, u, len, prec);
power_of_two /= 2;
horner_point = n / power_of_two;
horner_point -= (horner_point % 2 == 0);
}
}
_acb_poly_mullow(t, z, len, x, len, len, prec);
_acb_vec_add(z, t, u, len, prec);
flint_free(divisors);
_acb_vec_clear(powers, powers_alloc);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
_acb_vec_clear(x, len);
arb_clear(logk);
arb_clear(v);
arb_clear(w);
}
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_dirichlet_hurwitz_precomp_init(acb_dirichlet_hurwitz_precomp_t pre,
const acb_t s, int deflate, slong A, slong K, slong N, slong prec)
{
slong i, k;
if (A < 1 || K < 1 || N < 1)
abort();
pre->deflate = deflate;
pre->A = A;
pre->K = K;
pre->N = N;
pre->coeffs = _acb_vec_init(N * K);
mag_init(&pre->err);
acb_init(&pre->s);
acb_set(&pre->s, s);
acb_dirichlet_hurwitz_precomp_bound(&pre->err, s, A, K, N);
if (mag_is_finite(&pre->err))
{
acb_t t, a;
acb_init(t);
acb_init(a);
/* (-1)^k (s)_k / k! */
acb_one(pre->coeffs + 0);
for (k = 1; k < K; k++)
{
acb_add_ui(pre->coeffs + k, s, k - 1, prec);
acb_mul(pre->coeffs + k, pre->coeffs + k, pre->coeffs + k - 1, prec);
acb_div_ui(pre->coeffs + k, pre->coeffs + k, k, prec);
acb_neg(pre->coeffs + k, pre->coeffs + k);
}
for (i = 1; i < N; i++)
_acb_vec_set(pre->coeffs + i * K, pre->coeffs, K);
/* zeta(s+k,a) where a = A + (2*i+1)/(2*N) */
for (i = 0; i < N; i++)
{
acb_set_ui(a, 2 * i + 1);
acb_div_ui(a, a, 2 * N, prec);
acb_add_ui(a, a, A, prec);
for (k = 0; k < K; k++)
{
acb_add_ui(t, s, k, prec);
if (deflate && k == 0)
_acb_poly_zeta_cpx_series(t, t, a, 1, 1, prec);
else
acb_hurwitz_zeta(t, t, a, prec);
acb_mul(pre->coeffs + i * K + k,
pre->coeffs + i * K + k, t, prec);
}
}
acb_clear(t);
acb_clear(a);
}
}
void
_acb_poly_zeta_series(acb_ptr res, acb_srcptr h, long hlen, const acb_t a, int deflate, long len, long prec)
{
long i;
acb_ptr t, u;
hlen = FLINT_MIN(hlen, len);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
/* use reflection formula */
if (arf_sgn(arb_midref(acb_realref(h))) < 0 && acb_is_one(a))
{
/* zeta(s) = (2*pi)**s * sin(pi*s/2) / pi * gamma(1-s) * zeta(1-s) */
acb_t pi;
acb_ptr f, s1, s2, s3, s4;
acb_init(pi);
f = _acb_vec_init(2);
s1 = _acb_vec_init(len);
s2 = _acb_vec_init(len);
s3 = _acb_vec_init(len);
s4 = _acb_vec_init(len);
acb_const_pi(pi, prec);
/* s1 = (2*pi)**s */
acb_mul_2exp_si(pi, pi, 1);
_acb_poly_pow_cpx(s1, pi, h, len, prec);
acb_mul_2exp_si(pi, pi, -1);
/* s2 = sin(pi*s/2) / pi */
acb_set(f, h);
acb_one(f + 1);
acb_mul_2exp_si(f, f, -1);
acb_mul_2exp_si(f + 1, f + 1, -1);
_acb_poly_sin_pi_series(s2, f, 2, len, prec);
_acb_vec_scalar_div(s2, s2, len, pi, prec);
/* s3 = gamma(1-s) */
acb_sub_ui(f, h, 1, prec);
acb_neg(f, f);
acb_set_si(f + 1, -1);
_acb_poly_gamma_series(s3, f, 2, len, prec);
/* s4 = zeta(1-s) */
acb_sub_ui(f, h, 1, prec);
acb_neg(f, f);
_acb_poly_zeta_cpx_series(s4, f, a, 0, len, prec);
for (i = 1; i < len; i += 2)
acb_neg(s4 + i, s4 + i);
_acb_poly_mullow(u, s1, len, s2, len, len, prec);
_acb_poly_mullow(s1, s3, len, s4, len, len, prec);
_acb_poly_mullow(t, u, len, s1, len, len, prec);
/* add 1/(1-(s+t)) = 1/(1-s) + t/(1-s)^2 + ... */
if (deflate)
{
acb_sub_ui(u, h, 1, prec);
acb_neg(u, u);
acb_inv(u, u, prec);
for (i = 1; i < len; i++)
acb_mul(u + i, u + i - 1, u, prec);
_acb_vec_add(t, t, u, len, prec);
}
acb_clear(pi);
_acb_vec_clear(f, 2);
_acb_vec_clear(s1, len);
_acb_vec_clear(s2, len);
_acb_vec_clear(s3, len);
_acb_vec_clear(s4, len);
}
else
{
_acb_poly_zeta_cpx_series(t, h, a, deflate, len, prec);
}
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, t, len, u, hlen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
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);
}
slong
_acb_poly_find_roots(acb_ptr roots,
acb_srcptr poly,
acb_srcptr initial, slong len, slong maxiter, slong prec)
{
slong iter, i, deg;
slong rootmag, max_rootmag, correction, max_correction;
deg = len - 1;
if (deg == 0)
{
return 0;
}
else if (acb_contains_zero(poly + len - 1))
{
/* if the leading coefficient contains zero, roots can be anywhere */
for (i = 0; i < deg; i++)
{
arb_zero_pm_inf(acb_realref(roots + i));
arb_zero_pm_inf(acb_imagref(roots + i));
}
return 0;
}
else if (deg == 1)
{
acb_inv(roots + 0, poly + 1, prec);
acb_mul(roots + 0, roots + 0, poly + 0, prec);
acb_neg(roots + 0, roots + 0);
return 1;
}
if (initial == NULL)
_acb_poly_roots_initial_values(roots, deg, prec);
else
_acb_vec_set(roots, initial, deg);
if (maxiter == 0)
maxiter = 2 * deg + n_sqrt(prec);
for (iter = 0; iter < maxiter; iter++)
{
max_rootmag = -ARF_PREC_EXACT;
for (i = 0; i < deg; i++)
{
rootmag = _acb_get_mid_mag(roots + i);
max_rootmag = FLINT_MAX(rootmag, max_rootmag);
}
_acb_poly_refine_roots_durand_kerner(roots, poly, len, prec);
max_correction = -ARF_PREC_EXACT;
for (i = 0; i < deg; i++)
{
correction = _acb_get_rad_mag(roots + i);
max_correction = FLINT_MAX(correction, max_correction);
}
/* estimate the correction relative to the whole set of roots */
max_correction -= max_rootmag;
/* flint_printf("ITER %wd MAX CORRECTION: %wd\n", iter, max_correction); */
if (max_correction < -prec / 2)
maxiter = FLINT_MIN(maxiter, iter + 2);
else if (max_correction < -prec / 3)
maxiter = FLINT_MIN(maxiter, iter + 3);
else if (max_correction < -prec / 4)
maxiter = FLINT_MIN(maxiter, iter + 4);
}
return _acb_poly_validate_roots(roots, poly, len, prec);
}
void
fmpz_poly_complex_roots_squarefree(const fmpz_poly_t poly,
slong initial_prec,
slong target_prec,
slong print_digits)
{
slong i, j, prec, deg, deg_deflated, isolated, maxiter, deflation;
acb_poly_t cpoly, cpoly_deflated;
fmpz_poly_t poly_deflated;
acb_ptr roots, roots_deflated;
int removed_zero;
if (fmpz_poly_degree(poly) < 1)
return;
fmpz_poly_init(poly_deflated);
acb_poly_init(cpoly);
acb_poly_init(cpoly_deflated);
/* try to write poly as poly_deflated(x^deflation), possibly multiplied by x */
removed_zero = fmpz_is_zero(poly->coeffs);
if (removed_zero)
fmpz_poly_shift_right(poly_deflated, poly, 1);
else
fmpz_poly_set(poly_deflated, poly);
deflation = fmpz_poly_deflation(poly_deflated);
fmpz_poly_deflate(poly_deflated, poly_deflated, deflation);
deg = fmpz_poly_degree(poly);
deg_deflated = fmpz_poly_degree(poly_deflated);
flint_printf("searching for %wd roots, %wd deflated\n", deg, deg_deflated);
roots = _acb_vec_init(deg);
roots_deflated = _acb_vec_init(deg_deflated);
for (prec = initial_prec; ; prec *= 2)
{
acb_poly_set_fmpz_poly(cpoly_deflated, poly_deflated, prec);
maxiter = FLINT_MIN(FLINT_MAX(deg_deflated, 32), prec);
TIMEIT_ONCE_START
flint_printf("prec=%wd: ", prec);
isolated = acb_poly_find_roots(roots_deflated, cpoly_deflated,
prec == initial_prec ? NULL : roots_deflated, maxiter, prec);
flint_printf("%wd isolated roots | ", isolated);
TIMEIT_ONCE_STOP
if (isolated == deg_deflated)
{
if (!check_accuracy(roots_deflated, deg_deflated, target_prec))
continue;
if (deflation == 1)
{
_acb_vec_set(roots, roots_deflated, deg_deflated);
}
else /* compute all nth roots */
{
acb_t w, w2;
acb_init(w);
acb_init(w2);
acb_unit_root(w, deflation, prec);
acb_unit_root(w2, 2 * deflation, prec);
for (i = 0; i < deg_deflated; i++)
{
if (arf_sgn(arb_midref(acb_realref(roots_deflated + i))) > 0)
{
acb_root_ui(roots + i * deflation,
roots_deflated + i, deflation, prec);
}
else
{
acb_neg(roots + i * deflation, roots_deflated + i);
acb_root_ui(roots + i * deflation,
roots + i * deflation, deflation, prec);
acb_mul(roots + i * deflation,
roots + i * deflation, w2, prec);
}
for (j = 1; j < deflation; j++)
{
acb_mul(roots + i * deflation + j,
roots + i * deflation + j - 1, w, prec);
}
}
acb_clear(w);
acb_clear(w2);
}
/* by assumption that poly is squarefree, must be just one */
if (removed_zero)
acb_zero(roots + deg_deflated * deflation);
if (!check_accuracy(roots, deg, target_prec))
continue;
acb_poly_set_fmpz_poly(cpoly, poly, prec);
if (!acb_poly_validate_real_roots(roots, cpoly, prec))
continue;
for (i = 0; i < deg; i++)
{
if (arb_contains_zero(acb_imagref(roots + i)))
arb_zero(acb_imagref(roots + i));
}
flint_printf("done!\n");
break;
}
}
if (print_digits != 0)
{
_acb_vec_sort_pretty(roots, deg);
for (i = 0; i < deg; i++)
{
acb_printn(roots + i, print_digits, 0);
flint_printf("\n");
}
}
fmpz_poly_clear(poly_deflated);
acb_poly_clear(cpoly);
acb_poly_clear(cpoly_deflated);
_acb_vec_clear(roots, deg);
_acb_vec_clear(roots_deflated, deg_deflated);
}
