void
_fmprb_poly_product_roots(fmprb_ptr poly, fmprb_srcptr xs, long n, long prec)
{
if (n == 0)
{
fmprb_one(poly);
}
else if (n == 1)
{
fmprb_neg(poly, xs);
fmprb_one(poly + 1);
}
else if (n == 2)
{
fmprb_mul(poly, xs + 0, xs + 1, prec);
fmprb_add(poly + 1, xs + 0, xs + 1, prec);
fmprb_neg(poly + 1, poly + 1);
fmprb_one(poly + 2);
}
else
{
const long m = (n + 1) / 2;
fmprb_ptr tmp;
tmp = _fmprb_vec_init(n + 2);
_fmprb_poly_product_roots(tmp, xs, m, prec);
_fmprb_poly_product_roots(tmp + m + 1, xs + m, n - m, prec);
_fmprb_poly_mul_monic(poly, tmp, m + 1, tmp + m + 1, n - m + 1, prec);
_fmprb_vec_clear(tmp, n + 2);
}
}
void
fmprb_mat_det(fmprb_t det, const fmprb_mat_t A, long prec)
{
long n = fmprb_mat_nrows(A);
if (n == 0)
{
fmprb_one(det);
}
else if (n == 1)
{
fmprb_set(det, fmprb_mat_entry(A, 0, 0));
}
else if (n == 2)
{
fmprb_mul(det, fmprb_mat_entry(A, 0, 0), fmprb_mat_entry(A, 1, 1), prec);
fmprb_submul(det, fmprb_mat_entry(A, 0, 1), fmprb_mat_entry(A, 1, 0), prec);
}
else
{
fmprb_mat_t T;
fmprb_mat_init(T, fmprb_mat_nrows(A), fmprb_mat_ncols(A));
fmprb_mat_set(T, A);
fmprb_mat_det_inplace(det, T, prec);
fmprb_mat_clear(T);
}
}
void
bound_K(fmprb_t C, const fmprb_t AN, const fmprb_t B, const fmprb_t T, long wp)
{
if (fmprb_is_zero(B) || fmprb_is_zero(T))
{
fmprb_one(C);
}
else
{
fmprb_div(C, B, AN, wp);
/* TODO: atan is dumb, should also bound by pi/2 */
fmprb_atan(C, C, wp);
fmprb_mul(C, C, T, wp);
if (fmprb_is_nonpositive(C))
fmprb_one(C);
else
fmprb_exp(C, C, wp);
}
}
int
sin_x(fmprb_ptr out, const fmprb_t inp, void * params, long order, long prec)
{
int xlen = FLINT_MIN(2, order);
fmprb_set(out, inp);
if (xlen > 1)
fmprb_one(out + 1);
_fmprb_poly_sin_series(out, out, xlen, order, prec);
eval_count++;
return 0;
}
void
bound_I(fmprb_ptr I, const fmprb_t A, const fmprb_t B, const fmprb_t C, long len, long wp)
{
long k;
fmprb_t D, Dk, L, T, Bm1;
fmprb_init(D);
fmprb_init(Dk);
fmprb_init(Bm1);
fmprb_init(T);
fmprb_init(L);
fmprb_sub_ui(Bm1, B, 1, wp);
fmprb_one(L);
/* T = 1 / (A^Bm1 * Bm1) */
fmprb_inv(T, A, wp);
fmprb_pow(T, T, Bm1, wp);
fmprb_div(T, T, Bm1, wp);
if (len > 1)
{
fmprb_log(D, A, wp);
fmprb_add(D, D, C, wp);
fmprb_mul(D, D, Bm1, wp);
fmprb_set(Dk, D);
}
for (k = 0; k < len; k++)
{
if (k > 0)
{
fmprb_mul_ui(L, L, k, wp);
fmprb_add(L, L, Dk, wp);
fmprb_mul(Dk, Dk, D, wp);
}
fmprb_mul(I + k, L, T, wp);
fmprb_div(T, T, Bm1, wp);
}
fmprb_clear(D);
fmprb_clear(Dk);
fmprb_clear(Bm1);
fmprb_clear(T);
fmprb_clear(L);
}
void
gamma_rising_fmprb_ui_bsplit_eight(fmprb_t y, const fmprb_t x, ulong n, long prec)
{
if (n == 0)
{
fmprb_one(y);
}
else if (n == 1)
{
fmprb_set_round(y, x, prec);
}
else
{
ulong k, a;
long wp;
fmprb_t t, u;
wp = FMPR_PREC_ADD(prec, FLINT_BIT_COUNT(n));
fmprb_init(t);
fmprb_init(u);
if (n >= 8)
{
bsplit(t, x, 0, (n / 8) * 8, wp);
a = (n / 8) * 8;
}
else
{
fmprb_set(t, x);
a = 1;
}
for (k = a; k < n; k++)
{
fmprb_add_ui(u, x, k, wp);
fmprb_mul(t, t, u, wp);
}
fmprb_set_round(y, t, prec);
fmprb_clear(t);
fmprb_clear(u);
}
}
int
z_function(fmprb_ptr out, const fmprb_t inp, void * params, long order, long prec)
{
fmprb_struct x[2];
fmprb_init(x);
fmprb_init(x + 1);
fmprb_set(x, inp);
fmprb_one(x + 1);
_fmprb_poly_riemann_siegel_z_series(out, x, FLINT_MIN(2, order), order, prec);
fmprb_clear(x);
fmprb_clear(x + 1);
eval_count++;
return 0;
}
int
sin_1x(fmprb_ptr out, const fmprb_t inp, void * params, long order, long prec)
{
fmprb_ptr x;
int xlen = FLINT_MIN(2, order);
x = _fmprb_vec_init(xlen);
fmprb_set(x, inp);
if (xlen > 1)
fmprb_one(x + 1);
_fmprb_poly_inv_series(out, x, xlen, order, prec);
_fmprb_poly_sin_series(out, out, order, order, prec);
_fmprb_vec_clear(x, xlen);
eval_count++;
return 0;
}
void
bound_rfac(fmprb_ptr F, const fmpcb_t s, ulong n, long len, long wp)
{
if (len == 1)
{
fmpcb_rfac_abs_ubound2(fmprb_midref(F + 0), s, n, wp);
fmpr_zero(fmprb_radref(F + 0));
}
else
{
fmprb_struct sx[2];
fmprb_init(sx + 0);
fmprb_init(sx + 1);
fmpcb_abs(sx + 0, s, wp);
fmprb_one(sx + 1);
_fmprb_vec_zero(F, len);
_fmprb_poly_rising_ui_series(F, sx, 2, n, len, wp);
fmprb_clear(sx + 0);
fmprb_clear(sx + 1);
}
}
void
fmprb_hypgeom_sum(fmprb_t P, fmprb_t Q, const hypgeom_t hyp, long n, long prec)
{
if (n < 1)
{
fmprb_zero(P);
fmprb_one(Q);
}
else
{
fmprb_t B, T;
fmprb_init(B);
fmprb_init(T);
bsplit_recursive_fmprb(P, Q, B, T, hyp, 0, n, 0, prec);
if (!fmprb_is_one(B))
fmprb_mul(Q, Q, B, prec);
fmprb_swap(P, T);
fmprb_clear(B);
fmprb_clear(T);
}
}
void
_fmprb_poly_riemann_siegel_theta_series(fmprb_ptr res,
fmprb_srcptr h, long hlen, long len, long prec)
{
fmpcb_ptr s;
fmprb_t u;
long i;
hlen = FLINT_MIN(hlen, len);
s = _fmpcb_vec_init(len);
fmprb_init(u);
/* s = 1/4 + (1/2) i h */
for (i = 0; i < hlen; i++)
fmprb_mul_2exp_si(fmpcb_imagref(s + i), h + i, -1);
fmprb_one(u);
fmprb_mul_2exp_si(u, u, -2);
fmprb_add(fmpcb_realref(s), fmpcb_realref(s), u, prec);
/* log gamma */
_fmpcb_poly_lgamma_series(s, s, hlen, len, prec);
/* imaginary part */
for (i = 0; i < len; i++)
fmprb_set(res + i, fmpcb_imagref(s + i));
/* subtract log(pi)/2 * h */
fmprb_const_pi(u, prec);
fmprb_log(u, u, prec);
fmprb_mul_2exp_si(u, u, -1);
fmprb_neg(u, u);
_fmprb_vec_scalar_addmul(res, h, hlen, u, prec);
_fmpcb_vec_clear(s, len);
fmprb_clear(u);
}
static __inline__ void
zeta_coeff_k(zeta_bsplit_t S, long k, long n, long s)
{
fmprb_set_si(S->D, 2 * (n + k));
fmprb_mul_si(S->D, S->D, n - k, FMPR_PREC_EXACT);
fmprb_set_si(S->Q1, k + 1);
fmprb_mul_si(S->Q1, S->Q1, 2*k + 1, FMPR_PREC_EXACT);
if (k == 0)
{
fmprb_zero(S->A);
fmprb_one(S->Q2);
}
else
{
fmprb_set_si(S->A, k % 2 ? 1 : -1);
fmprb_mul(S->A, S->A, S->Q1, FMPR_PREC_EXACT);
fmprb_ui_pow_ui(S->Q2, k, s, FMPR_PREC_EXACT);
}
fmprb_mul(S->Q3, S->Q1, S->Q2, FMPR_PREC_EXACT);
fmprb_zero(S->B);
fmprb_set(S->C, S->Q1);
}
void
gamma_stirling_eval_fmprb(fmprb_t s, const fmprb_t z, long nterms, int digamma, long prec)
{
fmprb_t b, t, logz, zinv, zinv2;
fmpr_t err;
long k, term_prec;
double z_mag, term_mag;
fmprb_init(b);
fmprb_init(t);
fmprb_init(logz);
fmprb_init(zinv);
fmprb_init(zinv2);
fmprb_log(logz, z, prec);
fmprb_ui_div(zinv, 1UL, z, prec);
nterms = FLINT_MAX(nterms, 1);
fmprb_zero(s);
if (nterms > 1)
{
fmprb_mul(zinv2, zinv, zinv, prec);
z_mag = fmpr_get_d(fmprb_midref(logz), FMPR_RND_UP) * 1.44269504088896;
for (k = nterms - 1; k >= 1; k--)
{
term_mag = bernoulli_bound_2exp_si(2 * k);
term_mag -= (2 * k - 1) * z_mag;
term_prec = prec + term_mag;
term_prec = FLINT_MIN(term_prec, prec);
term_prec = FLINT_MAX(term_prec, 10);
if (prec > 2000)
{
fmprb_set_round(t, zinv2, term_prec);
fmprb_mul(s, s, t, term_prec);
}
else
fmprb_mul(s, s, zinv2, term_prec);
gamma_stirling_coeff(b, k, digamma, term_prec);
fmprb_add(s, s, b, term_prec);
}
if (digamma)
fmprb_mul(s, s, zinv2, prec);
else
fmprb_mul(s, s, zinv, prec);
}
/* remainder bound */
fmpr_init(err);
gamma_stirling_bound_fmprb(err, z, digamma ? 1 : 0, 1, nterms);
fmprb_add_error_fmpr(s, err);
fmpr_clear(err);
if (digamma)
{
fmprb_neg(s, s);
fmprb_mul_2exp_si(zinv, zinv, -1);
fmprb_sub(s, s, zinv, prec);
fmprb_add(s, s, logz, prec);
}
else
{
/* (z-0.5)*log(z) - z + log(2*pi)/2 */
fmprb_one(t);
fmprb_mul_2exp_si(t, t, -1);
fmprb_sub(t, z, t, prec);
fmprb_mul(t, logz, t, prec);
fmprb_add(s, s, t, prec);
fmprb_sub(s, s, z, prec);
fmprb_const_log_sqrt2pi(t, prec);
fmprb_add(s, s, t, prec);
}
fmprb_clear(t);
fmprb_clear(b);
fmprb_clear(zinv);
fmprb_clear(zinv2);
fmprb_clear(logz);
}
void
fmpcb_calc_cauchy_bound(fmprb_t bound, fmpcb_calc_func_t func, void * param,
const fmpcb_t x, const fmprb_t radius, long maxdepth, long prec)
{
long i, n, depth, wp;
fmprb_t pi, theta, v, s1, c1, s2, c2, st, ct;
fmpcb_t t, u;
fmprb_t b;
fmprb_init(pi);
fmprb_init(theta);
fmprb_init(v);
fmprb_init(s1);
fmprb_init(c1);
fmprb_init(s2);
fmprb_init(c2);
fmprb_init(st);
fmprb_init(ct);
fmpcb_init(t);
fmpcb_init(u);
fmprb_init(b);
wp = prec + 20;
fmprb_const_pi(pi, wp);
fmprb_zero_pm_inf(b);
for (depth = 0, n = 16; depth < maxdepth; n *= 2, depth++)
{
fmprb_zero(b);
/* theta = 2 pi / n */
fmprb_div_ui(theta, pi, n, wp);
fmprb_mul_2exp_si(theta, theta, 1);
/* sine and cosine of i*theta and (i+1)*theta */
fmprb_zero(s1);
fmprb_one(c1);
fmprb_sin_cos(st, ct, theta, wp);
fmprb_set(s2, st);
fmprb_set(c2, ct);
for (i = 0; i < n; i++)
{
/* sine and cosine of 2 pi ([i,i+1]/n) */
/* since we use power of two subdivision points, the
sine and cosine are monotone on each subinterval */
fmprb_union(fmpcb_realref(t), c1, c2, wp);
fmprb_union(fmpcb_imagref(t), s1, s2, wp);
fmpcb_mul_fmprb(t, t, radius, wp);
fmpcb_add(t, t, x, prec);
/* next angle */
fmprb_mul(v, c2, ct, wp);
fmprb_mul(c1, s2, st, wp);
fmprb_sub(c1, v, c1, wp);
fmprb_mul(v, c2, st, wp);
fmprb_mul(s1, s2, ct, wp);
fmprb_add(s1, v, s1, wp);
fmprb_swap(c1, c2);
fmprb_swap(s1, s2);
func(u, t, param, 1, prec);
fmpcb_abs(v, u, prec);
fmprb_add(b, b, v, prec);
}
fmprb_div_ui(b, b, n, prec);
if (fmprb_is_exact(b) || fmpr_cmp(fmprb_radref(b), fmprb_midref(b)) < 0)
break;
}
fmprb_set(bound, b);
fmprb_clear(pi);
fmprb_clear(theta);
fmprb_clear(v);
fmpcb_clear(t);
fmpcb_clear(u);
fmprb_clear(b);
fmprb_clear(s1);
fmprb_clear(c1);
fmprb_clear(s2);
fmprb_clear(c2);
fmprb_clear(st);
fmprb_clear(ct);
}
void
gamma_rising_fmprb_ui_delta(fmprb_t y, const fmprb_t x, ulong n, ulong m, long prec)
{
fmprb_ptr xs;
fmprb_t t, u, v;
ulong i, k, rem;
fmpz_t c, h;
fmpz *s, *d;
long wp;
if (n == 0)
{
fmprb_one(y);
return;
}
if (n == 1)
{
fmprb_set_round(y, x, prec);
return;
}
wp = FMPR_PREC_ADD(prec, FLINT_BIT_COUNT(n));
fmprb_init(t);
fmprb_init(u);
fmprb_init(v);
fmpz_init(c);
fmpz_init(h);
if (m == 0)
{
ulong m1, m2;
m1 = 0.2 * pow(wp, 0.4);
m2 = n_sqrt(n);
m = FLINT_MIN(m1, m2);
}
m = FLINT_MIN(m, n);
m = FLINT_MAX(m, 1);
xs = _fmprb_vec_init(m + 1);
d = _fmpz_vec_init(m * m);
s = _fmpz_vec_init(m + 1);
_fmprb_vec_set_powers(xs, x, m + 1, wp);
rising_difference_polynomial(s, d, m);
/* tail */
rem = m;
while (rem + m <= n)
rem += m;
fmprb_one(y);
for (k = rem; k < n; k++)
{
fmprb_add_ui(t, xs + 1, k, wp);
fmprb_mul(y, y, t, wp);
}
/* initial rising factorial */
fmprb_zero(t);
for (i = 1; i <= m; i++)
fmprb_addmul_fmpz(t, xs + i, s + i, wp);
fmprb_mul(y, y, t, wp);
/* the leading coefficient is always the same */
fmprb_mul_fmpz(xs + m - 1, xs + m - 1, d + m - 1 + 0, wp);
for (k = 0; k + 2 * m <= n; k += m)
{
for (i = 0; i < m - 1; i++)
{
fmpz_set_ui(h, k);
_fmpz_poly_evaluate_horner_fmpz(c, d + i * m, m - i, h);
if (i == 0)
fmprb_add_fmpz(t, t, c, wp);
else
fmprb_addmul_fmpz(t, xs + i, c, wp);
}
fmprb_add(t, t, xs + m - 1, wp);
fmprb_mul(y, y, t, wp);
}
fmprb_set_round(y, y, prec);
fmprb_clear(t);
fmprb_clear(u);
fmprb_clear(v);
_fmprb_vec_clear(xs, m + 1);
_fmpz_vec_clear(d, m * m);
_fmpz_vec_clear(s, m + 1);
fmpz_clear(c);
fmpz_clear(h);
}
void
_fmprb_poly_rgamma_series(fmprb_ptr res, fmprb_srcptr h, long hlen, long len, long prec)
{
int reflect;
long i, rflen, r, n, wp;
fmprb_ptr t, u, v;
fmprb_struct f[2];
hlen = FLINT_MIN(hlen, len);
wp = prec + FLINT_BIT_COUNT(prec);
t = _fmprb_vec_init(len);
u = _fmprb_vec_init(len);
v = _fmprb_vec_init(len);
fmprb_init(f);
fmprb_init(f + 1);
/* use zeta values at small integers */
if (fmprb_is_int(h) && (fmpr_cmpabs_ui(fmprb_midref(h), prec / 2) < 0))
{
r = fmpr_get_si(fmprb_midref(h), FMPR_RND_DOWN);
gamma_lgamma_series_at_one(u, len, wp);
_fmprb_vec_neg(u, u, len);
_fmprb_poly_exp_series(t, u, len, len, wp);
if (r == 1)
{
_fmprb_vec_swap(v, t, len);
}
else if (r <= 0)
{
fmprb_set(f, h);
fmprb_one(f + 1);
rflen = FLINT_MIN(len, 2 - r);
_fmprb_poly_rising_ui_series(u, f, FLINT_MIN(2, len), 1 - r, rflen, wp);
_fmprb_poly_mullow(v, t, len, u, rflen, len, wp);
}
else
{
fmprb_one(f);
fmprb_one(f + 1);
rflen = FLINT_MIN(len, r);
_fmprb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r - 1, rflen, wp);
/* TODO: use div_series? */
_fmprb_poly_inv_series(u, v, rflen, len, wp);
_fmprb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* otherwise use Stirling series */
gamma_stirling_choose_param_fmprb(&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) */
fmprb_sub_ui(f, h, r + 1, wp);
fmprb_neg(f, f);
gamma_stirling_eval_fmprb_series(t, f, n, len, wp);
_fmprb_poly_exp_series(u, t, len, len, wp);
for (i = 1; i < len; i += 2)
fmprb_neg(u + i, u + i);
/* v = sin(pi x) */
fmprb_const_pi(f + 1, wp);
fmprb_mul(f, h, f + 1, wp);
_fmprb_poly_sin_series(v, f, 2, len, wp);
_fmprb_poly_mullow(t, u, len, v, len, len, wp);
/* rf(1-h,r) * pi */
if (r == 0)
{
fmprb_const_pi(u, wp);
_fmprb_vec_scalar_div(v, t, len, u, wp);
}
else
{
fmprb_sub_ui(f, h, 1, wp);
fmprb_neg(f, f);
fmprb_set_si(f + 1, -1);
rflen = FLINT_MIN(len, r + 1);
_fmprb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
fmprb_const_pi(u, wp);
_fmprb_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 */
_fmprb_poly_inv_series(u, v, rflen, len, wp);
_fmprb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* rgamma(h) = rgamma(h+r) rf(h,r) */
if (r == 0)
{
fmprb_add_ui(f, h, r, wp);
gamma_stirling_eval_fmprb_series(t, f, n, len, wp);
_fmprb_vec_neg(t, t, len);
_fmprb_poly_exp_series(v, t, len, len, wp);
}
else
{
fmprb_set(f, h);
fmprb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_fmprb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);
fmprb_add_ui(f, h, r, wp);
gamma_stirling_eval_fmprb_series(v, f, n, len, wp);
_fmprb_vec_neg(v, v, len);
_fmprb_poly_exp_series(u, v, len, len, wp);
_fmprb_poly_mullow(v, u, len, t, rflen, len, wp);
}
}
}
/* compose with nonconstant part */
fmprb_zero(t);
_fmprb_vec_set(t + 1, h + 1, hlen - 1);
_fmprb_poly_compose_series(res, v, len, t, hlen, len, prec);
fmprb_clear(f);
fmprb_clear(f + 1);
_fmprb_vec_clear(t, len);
_fmprb_vec_clear(u, len);
_fmprb_vec_clear(v, len);
}