/* bound (1 + 1/m)^n, m > 0, n >= 0 */
void
mag_binpow_uiui(mag_t b, ulong m, ulong n)
{
mag_t t;
if (m == 0)
{
mag_inf(b);
return;
}
mag_init(t);
/* bound by exp(n/m) <= 1 + (n/m) + (n/m)^2 */
if (m > n)
{
mag_set_ui(t, n); /* x = n/m */
mag_div_ui(t, t, m);
mag_mul(b, t, t); /* x^2 */
mag_add(b, b, t); /* x */
mag_one(t);
mag_add(b, b, t); /* 1 */
}
else
{
mag_one(b);
mag_div_ui(b, b, m);
mag_one(t);
mag_add(t, t, b);
mag_pow_ui(b, t, n);
}
mag_clear(t);
}
void
mag_geom_series(mag_t res, const mag_t x, ulong n)
{
if (mag_is_zero(x))
{
if (n == 0)
mag_one(res);
else
mag_zero(res);
}
else if (mag_is_inf(x))
{
mag_inf(res);
}
else
{
mag_t t;
mag_init(t);
mag_one(t);
mag_sub_lower(t, t, x);
if (mag_is_zero(t))
{
mag_inf(res);
}
else
{
mag_pow_ui(res, x, n);
mag_div(res, res, t);
}
mag_clear(t);
}
}
void
acb_hypgeom_mag_chi(mag_t chi, ulong n)
{
mag_t p, q;
ulong k;
mag_init(p);
mag_init(q);
if (n % 2 == 0)
{
mag_one(p);
mag_one(q);
}
else
{
/* upper bound for pi/2 */
mag_set_ui_2exp_si(p, 843314857, -28);
mag_one(q);
}
for (k = n; k >= 2; k -= 2)
{
mag_mul_ui(p, p, k);
mag_mul_ui_lower(q, q, k - 1);
}
mag_div(chi, p, q);
mag_clear(p);
mag_clear(q);
}
void
arb_sin_cos_pi(arb_t s, arb_t c, const arb_t x, long prec)
{
arb_t t;
arb_t u;
fmpz_t v;
if (arf_cmpabs_2exp_si(arb_midref(x), FLINT_MAX(65536, (4*prec))) > 0)
{
arf_zero(arb_midref(s));
mag_one(arb_radref(s));
arf_zero(arb_midref(c));
mag_one(arb_radref(c));
return;
}
arb_init(t);
arb_init(u);
fmpz_init(v);
arb_mul_2exp_si(t, x, 1);
arf_get_fmpz(v, arb_midref(t), ARF_RND_NEAR);
arb_sub_fmpz(t, t, v, prec);
arb_const_pi(u, prec);
arb_mul(t, t, u, prec);
arb_mul_2exp_si(t, t, -1);
switch (fmpz_fdiv_ui(v, 4))
{
case 0:
arb_sin_cos(s, c, t, prec);
break;
case 1:
arb_sin_cos(c, s, t, prec);
arb_neg(c, c);
break;
case 2:
arb_sin_cos(s, c, t, prec);
arb_neg(s, s);
arb_neg(c, c);
break;
default:
arb_sin_cos(c, s, t, prec);
arb_neg(s, s);
break;
}
fmpz_clear(v);
arb_clear(t);
arb_clear(u);
}
void
mag_expinv(mag_t res, const mag_t x)
{
if (mag_is_zero(x))
{
mag_one(res);
}
else if (mag_is_inf(x))
{
mag_zero(res);
}
else if (fmpz_sgn(MAG_EXPREF(x)) <= 0)
{
mag_one(res);
}
else if (fmpz_cmp_ui(MAG_EXPREF(x), 2 * MAG_BITS) > 0)
{
fmpz_t t;
fmpz_init(t);
/* If x > 2^60, exp(-x) < 2^(-2^60 / log(2)) */
/* -1/log(2) < -369/256 */
fmpz_set_si(t, -369);
fmpz_mul_2exp(t, t, 2 * MAG_BITS - 8);
mag_one(res);
mag_mul_2exp_fmpz(res, res, t);
fmpz_clear(t);
}
else
{
fmpz_t t;
slong e = MAG_EXP(x);
fmpz_init(t);
fmpz_set_ui(t, MAG_MAN(x));
if (e >= MAG_BITS)
fmpz_mul_2exp(t, t, e - MAG_BITS);
else
fmpz_tdiv_q_2exp(t, t, MAG_BITS - e);
/* upper bound for 1/e */
mag_set_ui_2exp_si(res, 395007543, -30);
mag_pow_fmpz(res, res, t);
fmpz_clear(t);
}
}
static void
arb_sqrt1pm1_tiny(arb_t r, const arb_t z, slong prec)
{
mag_t b, c;
arb_t t;
mag_init(b);
mag_init(c);
arb_init(t);
/* if |z| < 1, then |(sqrt(1+z)-1) - (z/2-z^2/8)| <= |z|^3/(1-|z|)/16 */
arb_get_mag(b, z);
mag_one(c);
mag_sub_lower(c, c, b);
mag_pow_ui(b, b, 3);
mag_div(b, b, c);
mag_mul_2exp_si(b, b, -4);
arb_mul(t, z, z, prec);
arb_mul_2exp_si(t, t, -2);
arb_sub(r, z, t, prec);
arb_mul_2exp_si(r, r, -1);
if (mag_is_finite(b))
arb_add_error_mag(r, b);
else
arb_indeterminate(r);
mag_clear(b);
mag_clear(c);
arb_clear(t);
}
static void
acb_log1p_tiny(acb_t r, const acb_t z, slong prec)
{
mag_t b, c;
acb_t t;
int real;
mag_init(b);
mag_init(c);
acb_init(t);
real = acb_is_real(z);
/* if |z| < 1, then |log(1+z) - [z - z^2/2]| <= |z|^3/(1-|z|) */
acb_get_mag(b, z);
mag_one(c);
mag_sub_lower(c, c, b);
mag_pow_ui(b, b, 3);
mag_div(b, b, c);
acb_mul(t, z, z, prec);
acb_mul_2exp_si(t, t, -1);
acb_sub(r, z, t, prec);
if (real && mag_is_finite(b))
arb_add_error_mag(acb_realref(r), b);
else
acb_add_error_mag(r, b);
mag_clear(b);
mag_clear(c);
acb_clear(t);
}
void
_hypgeom_precompute(hypgeom_t hyp, const fmpz_poly_t P, const fmpz_poly_t Q)
{
slong k;
fmpz_t t;
fmpz_init(t);
hyp->r = fmpz_poly_degree(Q) - fmpz_poly_degree(P);
hyp->boundC = hypgeom_root_norm(P);
hyp->boundD = hypgeom_root_norm(Q);
hyp->boundK = 1 + FLINT_MAX(hyp->boundC, 2 * hyp->boundD);
mag_one(hyp->MK);
for (k = 1; k <= hyp->boundK; k++)
{
fmpz_poly_evaluate_si(t, P, k);
mag_mul_fmpz(hyp->MK, hyp->MK, t);
fmpz_poly_evaluate_si(t, Q, k);
mag_div_fmpz(hyp->MK, hyp->MK, t);
}
fmpz_clear(t);
}
/* Differential equation for F(a,b,c,y+z):
(y+z)(y-1+z) F''(z) + ((y+z)(a+b+1) - c) F'(z) + a b F(z) = 0
Coefficients in the Taylor series are bounded by
A * binomial(N+k, k) * nu^k
using the Cauchy-Kovalevskaya majorant method.
See J. van der Hoeven, "Fast evaluation of holonomic functions near
and in regular singularities"
*/
static void
bound(mag_t A, mag_t nu, mag_t N,
const acb_t a, const acb_t b, const acb_t c, const acb_t y,
const acb_t f0, const acb_t f1)
{
mag_t M0, M1, t, u;
acb_t d;
acb_init(d);
mag_init(M0);
mag_init(M1);
mag_init(t);
mag_init(u);
/* nu = max(1/|y-1|, 1/|y|) = 1/min(|y-1|, |y|) */
acb_get_mag_lower(t, y);
acb_sub_ui(d, y, 1, MAG_BITS);
acb_get_mag_lower(u, d);
mag_min(t, t, u);
mag_one(u);
mag_div(nu, u, t);
/* M0 = 2 nu |ab| */
acb_get_mag(t, a);
acb_get_mag(u, b);
mag_mul(M0, t, u);
mag_mul(M0, M0, nu);
mag_mul_2exp_si(M0, M0, 1);
/* M1 = 2 nu |(a+b+1)y-c| + 2|a+b+1| */
acb_add(d, a, b, MAG_BITS);
acb_add_ui(d, d, 1, MAG_BITS);
acb_get_mag(t, d);
acb_mul(d, d, y, MAG_BITS);
acb_sub(d, d, c, MAG_BITS);
acb_get_mag(u, d);
mag_mul(u, u, nu);
mag_add(M1, t, u);
mag_mul_2exp_si(M1, M1, 1);
/* N = max(sqrt(2 M0), 2 M1) / nu */
mag_mul_2exp_si(M0, M0, 1);
mag_sqrt(M0, M0);
mag_mul_2exp_si(M1, M1, 1);
mag_max(N, M0, M1);
mag_div(N, N, nu);
/* A = max(|f0|, |f1| / (nu (N+1)) */
acb_get_mag(t, f0);
acb_get_mag(u, f1);
mag_div(u, u, nu);
mag_div(u, u, N); /* upper bound for dividing by N+1 */
mag_max(A, t, u);
acb_clear(d);
mag_clear(M0);
mag_clear(M1);
mag_clear(t);
mag_clear(u);
}
static void
acb_hypgeom_mag_Cn(mag_t Cn, int R, const mag_t nu, const mag_t sigma, ulong n)
{
if (R == 1)
{
mag_one(Cn);
}
else
{
acb_hypgeom_mag_chi(Cn, n);
if (R == 3)
{
mag_t tmp;
mag_init(tmp);
mag_mul(tmp, nu, nu);
mag_mul(tmp, tmp, sigma);
if (n != 1)
mag_mul_ui(tmp, tmp, n);
mag_add(Cn, Cn, tmp);
mag_pow_ui(tmp, nu, n);
mag_mul(Cn, Cn, tmp);
mag_clear(tmp);
}
}
}
void
acb_hypgeom_erf_asymp(acb_t res, const acb_t z, slong prec, slong prec2)
{
acb_t a, t, u;
acb_init(a);
acb_init(t);
acb_init(u);
acb_one(a);
acb_mul_2exp_si(a, a, -1);
acb_mul(t, z, z, prec2);
acb_hypgeom_u_asymp(u, a, a, t, -1, prec2);
acb_neg(t, t);
acb_exp(t, t, prec2);
acb_mul(u, u, t, prec2);
acb_const_pi(t, prec2);
acb_sqrt(t, t, prec2);
acb_mul(t, t, z, prec2);
acb_div(u, u, t, prec2);
/* branch cut term: -1 or 1 */
if (arb_contains_zero(acb_realref(z)))
{
arb_zero(acb_imagref(t));
arf_zero(arb_midref(acb_realref(t)));
mag_one(arb_radref(acb_realref(t)));
}
else
{
acb_set_si(t, arf_sgn(arb_midref(acb_realref(z))));
}
acb_sub(t, t, u, prec);
if (arb_is_zero(acb_imagref(z)))
arb_zero(acb_imagref(t));
else if (arb_is_zero(acb_realref(z)))
arb_zero(acb_realref(t));
acb_set(res, t);
acb_clear(a);
acb_clear(t);
acb_clear(u);
}
void
arb_sinc(arb_t z, const arb_t x, slong prec)
{
mag_t c, r;
mag_init(c);
mag_init(r);
mag_set_ui_2exp_si(c, 5, -1);
arb_get_mag_lower(r, x);
if (mag_cmp(c, r) < 0)
{
/* x is not near the origin */
_arb_sinc_direct(z, x, prec);
}
else if (mag_cmp_2exp_si(arb_radref(x), 1) < 0)
{
/* determine error magnitude using the derivative bound */
if (arb_is_exact(x))
{
mag_zero(c);
}
else
{
_arb_sinc_derivative_bound(r, x);
mag_mul(c, arb_radref(x), r);
}
/* evaluate sinc at the midpoint of x */
if (arf_is_zero(arb_midref(x)))
{
arb_one(z);
}
else
{
arb_get_mid_arb(z, x);
_arb_sinc_direct(z, z, prec);
}
/* add the error */
mag_add(arb_radref(z), arb_radref(z), c);
}
else
{
/* x has a large radius and includes points near the origin */
arf_zero(arb_midref(z));
mag_one(arb_radref(z));
}
mag_clear(c);
mag_clear(r);
}
void
_arb_sinc_derivative_bound(mag_t d, const arb_t x)
{
/* |f'(x)| < min(arb_get_mag(x), 1) / 2 */
mag_t r, one;
mag_init(r);
mag_init(one);
arb_get_mag(r, x);
mag_one(one);
mag_min(d, r, one);
mag_mul_2exp_si(d, d, -1);
mag_clear(r);
mag_clear(one);
}
void
arb_root_ui_algebraic(arb_t res, const arb_t x, ulong k, slong prec)
{
mag_t r, msubr, m1k, t;
if (arb_is_exact(x))
{
arb_root_arf(res, arb_midref(x), k, prec);
return;
}
if (!arb_is_nonnegative(x))
{
arb_indeterminate(res);
return;
}
mag_init(r);
mag_init(msubr);
mag_init(m1k);
mag_init(t);
/* x = [m-r, m+r] */
mag_set(r, arb_radref(x));
/* m - r */
arb_get_mag_lower(msubr, x);
/* m^(1/k) */
arb_root_arf(res, arb_midref(x), k, prec);
/* bound for m^(1/k) */
arb_get_mag(m1k, res);
/* C = min(1, log(1+r/(m-r))/k) */
mag_div(t, r, msubr);
mag_log1p(t, t);
mag_div_ui(t, t, k);
if (mag_cmp_2exp_si(t, 0) > 0)
mag_one(t);
/* C m^(1/k) */
mag_mul(t, m1k, t);
mag_add(arb_radref(res), arb_radref(res), t);
mag_clear(r);
mag_clear(msubr);
mag_clear(m1k);
mag_clear(t);
}
void
arb_bernoulli_fmpz(arb_t res, const fmpz_t n, slong prec)
{
if (fmpz_cmp_ui(n, UWORD_MAX) <= 0)
{
if (fmpz_sgn(n) >= 0)
arb_bernoulli_ui(res, fmpz_get_ui(n), prec);
else
arb_zero(res);
}
else if (fmpz_is_odd(n))
{
arb_zero(res);
}
else
{
arb_t t;
slong wp;
arb_init(t);
wp = prec + 2 * fmpz_bits(n);
/* zeta(n) ~= 1 */
arf_one(arb_midref(res));
mag_one(arb_radref(res));
mag_mul_2exp_si(arb_radref(res), arb_radref(res), WORD_MIN);
/* |B_n| = 2 * n! / (2*pi)^n * zeta(n) */
arb_gamma_fmpz(t, n, wp);
arb_mul_fmpz(t, t, n, wp);
arb_mul(res, res, t, wp);
arb_const_pi(t, wp);
arb_mul_2exp_si(t, t, 1);
arb_pow_fmpz(t, t, n, wp);
arb_div(res, res, t, prec);
arb_mul_2exp_si(res, res, 1);
if (fmpz_fdiv_ui(n, 4) == 0)
arb_neg(res, res);
arb_clear(t);
}
}
slong renf_set_embeddings_fmpz_poly(renf * nf, fmpz_poly_t pol, slong lim, slong prec)
{
slong i, n, n_exact, n_interval;
fmpq_poly_t p2;
arb_t a;
fmpz * c;
slong * k;
n = fmpz_poly_num_real_roots_upper_bound(pol);
c = _fmpz_vec_init(n);
k = (slong *) flint_malloc(n * sizeof(slong));
fmpz_poly_isolate_real_roots(NULL, &n_exact, c, k, &n_interval, pol);
if (n_exact)
{
fprintf(stderr, "ERROR (fmpz_poly_real_embeddings): rational roots\n");
abort();
}
arb_init(a);
fmpq_poly_init(p2);
fmpz_one(fmpq_poly_denref(p2));
fmpq_poly_fit_length(p2, pol->length);
_fmpz_vec_set(p2->coeffs, pol->coeffs, pol->length);
p2->length = pol->length;
for (i = 0; i < FLINT_MIN(lim, n_interval); i++)
{
arb_set_fmpz(a, c + i);
arb_mul_2exp_si(a, a, 1);
arb_add_si(a, a, 1, prec);
mag_one(arb_radref(a));
arb_mul_2exp_si(a, a, k[i] - 1);
renf_init(nf + i, p2, a, prec);
}
arb_clear(a);
fmpq_poly_clear(p2);
_fmpz_vec_clear(c, n);
flint_free(k);
return n_interval;
}
static void
phase(acb_t res, const arb_t re, const arb_t im)
{
if (arb_is_nonnegative(re) || arb_is_negative(im))
{
acb_one(res);
}
else if (arb_is_negative(re) && arb_is_nonnegative(im))
{
acb_set_si(res, -3);
}
else
{
arb_zero(acb_imagref(res));
/* -1 +/- 2 */
arf_set_si(arb_midref(acb_realref(res)), -1);
mag_one(arb_radref(acb_realref(res)));
mag_mul_2exp_si(arb_radref(acb_realref(res)), arb_radref(acb_realref(res)), 1);
}
}
/* f(z) = sin(1/z), assume on real interval */
int
f_essing(acb_ptr res, const acb_t z, void * param, slong order, slong prec)
{
if (order > 1)
flint_abort(); /* Would be needed for Taylor method. */
if ((order == 0) && acb_is_real(z) && arb_contains_zero(acb_realref(z)))
{
/* todo: arb_zero_pm_one, arb_unit_interval? */
acb_zero(res);
mag_one(arb_radref(acb_realref(res)));
}
else
{
acb_inv(res, z, prec);
acb_sin(res, res, prec);
}
return 0;
}
void
arb_atan(arb_t z, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_atan_arf(z, arb_midref(x), prec);
}
else
{
mag_t t, u;
mag_init(t);
mag_init(u);
arb_get_mag_lower(t, x);
if (mag_is_zero(t))
{
mag_set(t, arb_radref(x));
}
else
{
mag_mul_lower(t, t, t);
mag_one(u);
mag_add_lower(t, t, u);
mag_div(t, arb_radref(x), t);
}
if (mag_cmp_2exp_si(t, 0) > 0)
{
mag_const_pi(u);
mag_min(t, t, u);
}
arb_atan_arf(z, arb_midref(x), prec);
mag_add(arb_radref(z), arb_radref(z), t);
mag_clear(t);
mag_clear(u);
}
}
void
mag_pow_ui_lower(mag_t z, const mag_t x, ulong e)
{
if (e <= 2)
{
if (e == 0)
mag_one(z);
else if (e == 1)
mag_set(z, x);
else
mag_mul_lower(z, x, x);
}
else if (mag_is_inf(x))
{
mag_inf(z);
}
else
{
mag_t y;
int i, bits;
mag_init_set(y, x);
bits = FLINT_BIT_COUNT(e);
for (i = bits - 2; i >= 0; i--)
{
mag_mul_lower(y, y, y);
if (e & (1UL << i))
mag_mul_lower(y, y, x);
}
mag_swap(z, y);
mag_clear(y);
}
}
int main()
{
long iter;
flint_rand_t state;
printf("const_glaisher....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 250; iter++)
{
arb_t r, s, t;
fmpz_t v;
long accuracy, prec;
prec = 2 + n_randint(state, 2000);
arb_init(r);
arb_init(s);
arb_init(t);
fmpz_init(v);
arb_const_glaisher(r, prec);
arb_const_glaisher(s, prec + 100);
if (!arb_overlaps(r, s))
{
printf("FAIL: containment\n\n");
printf("prec = %ld\n", prec);
printf("r = "); arb_printd(r, prec / 3.33); printf("\n\n");
abort();
}
accuracy = arb_rel_accuracy_bits(r);
if (accuracy < prec - 4)
{
printf("FAIL: poor accuracy\n\n");
printf("prec = %ld\n", prec);
printf("r = "); arb_printd(r, prec / 3.33); printf("\n\n");
abort();
}
if (n_randint(state, 30) == 0)
{
flint_cleanup();
}
fmpz_set_str(v, "128242712910062263687534256886979172776768892732500", 10);
arb_set_fmpz(t, v);
mag_one(arb_radref(t));
fmpz_ui_pow_ui(v, 10, 50);
arb_div_fmpz(t, t, v, 170);
if (!arb_overlaps(r, t))
{
printf("FAIL: reference value\n\n");
printf("prec = %ld\n", prec);
printf("r = "); arb_printd(r, prec / 3.33); printf("\n\n");
abort();
}
arb_clear(r);
arb_clear(s);
arb_clear(t);
fmpz_clear(v);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
void
_arb_bell_sum_taylor(arb_t res, const fmpz_t n,
const fmpz_t a, const fmpz_t b, const fmpz_t mmag, long tol)
{
fmpz_t m, r, R, tmp;
mag_t B, C, D, bound;
arb_t t, u;
long wp, k, N;
if (_fmpz_sub_small(b, a) < 5)
{
arb_bell_sum_bsplit(res, n, a, b, mmag, tol);
return;
}
fmpz_init(m);
fmpz_init(r);
fmpz_init(R);
fmpz_init(tmp);
/* r = max(m - a, b - m) */
/* m = a + (b - a) / 2 */
fmpz_sub(r, b, a);
fmpz_cdiv_q_2exp(r, r, 1);
fmpz_add(m, a, r);
fmpz_mul_2exp(R, r, RADIUS_BITS);
mag_init(B);
mag_init(C);
mag_init(D);
mag_init(bound);
arb_init(t);
arb_init(u);
if (fmpz_cmp(R, m) >= 0)
{
mag_inf(C);
mag_inf(D);
}
else
{
/* C = exp(R * |F'(m)| + (1/2) R^2 * (n/(m-R)^2 + 1/(m-R))) */
/* C = exp(R * (|F'(m)| + (1/2) R * (n/(m-R) + 1)/(m-R))) */
/* D = (1/2) R * (n/(m-R) + 1)/(m-R) */
fmpz_sub(tmp, m, R);
mag_set_fmpz(D, n);
mag_div_fmpz(D, D, tmp);
mag_one(C);
mag_add(D, D, C);
mag_div_fmpz(D, D, tmp);
mag_mul_fmpz(D, D, R);
mag_mul_2exp_si(D, D, -1);
/* C = |F'(m)| */
wp = 20 + 1.05 * fmpz_bits(n);
arb_set_fmpz(t, n);
arb_div_fmpz(t, t, m, wp);
fmpz_add_ui(tmp, m, 1);
arb_set_fmpz(u, tmp);
arb_digamma(u, u, wp);
arb_sub(t, t, u, wp);
arb_get_mag(C, t);
/* C = exp(R * (C + D)) */
mag_add(C, C, D);
mag_mul_fmpz(C, C, R);
mag_exp(C, C);
}
if (mag_cmp_2exp_si(C, tol / 4 + 2) > 0)
{
_arb_bell_sum_taylor(res, n, a, m, mmag, tol);
_arb_bell_sum_taylor(t, n, m, b, mmag, tol);
arb_add(res, res, t, 2 * tol);
}
else
{
arb_ptr mx, ser1, ser2, ser3;
/* D = T(m) */
wp = 20 + 1.05 * fmpz_bits(n);
arb_set_fmpz(t, m);
arb_pow_fmpz(t, t, n, wp);
fmpz_add_ui(tmp, m, 1);
arb_gamma_fmpz(u, tmp, wp);
arb_div(t, t, u, wp);
arb_get_mag(D, t);
/* error bound: (b-a) * C * D * B^N / (1 - B), B = r/R */
/* ((b-a) * C * D * 2) * 2^(-N*RADIUS_BITS) */
/* ((b-a) * C * D * 2) */
mag_mul(bound, C, D);
mag_mul_2exp_si(bound, bound, 1);
fmpz_sub(tmp, b, a);
mag_mul_fmpz(bound, bound, tmp);
/* N = (tol + log2((b-a)*C*D*2) - mmag) / RADIUS_BITS */
if (mmag == NULL)
{
/* estimate D ~= 2^mmag */
fmpz_add_ui(tmp, MAG_EXPREF(C), tol);
fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS);
}
else
{
fmpz_sub(tmp, MAG_EXPREF(bound), mmag);
fmpz_add_ui(tmp, tmp, tol);
fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS);
}
if (fmpz_cmp_ui(tmp, 5 * tol / 4) > 0)
N = 5 * tol / 4;
else if (fmpz_cmp_ui(tmp, 2) < 0)
N = 2;
else
N = fmpz_get_ui(tmp);
/* multiply by 2^(-N*RADIUS_BITS) */
mag_mul_2exp_si(bound, bound, -N * RADIUS_BITS);
mx = _arb_vec_init(2);
ser1 = _arb_vec_init(N);
ser2 = _arb_vec_init(N);
ser3 = _arb_vec_init(N);
/* estimate (this should work for moderate n and tol) */
wp = 1.1 * tol + 1.05 * fmpz_bits(n) + 5;
/* increase precision until convergence */
while (1)
{
/* (m+x)^n / gamma(m+1+x) */
arb_set_fmpz(mx, m);
arb_one(mx + 1);
_arb_poly_log_series(ser1, mx, 2, N, wp);
for (k = 0; k < N; k++)
arb_mul_fmpz(ser1 + k, ser1 + k, n, wp);
arb_add_ui(mx, mx, 1, wp);
_arb_poly_lgamma_series(ser2, mx, 2, N, wp);
_arb_vec_sub(ser1, ser1, ser2, N, wp);
_arb_poly_exp_series(ser3, ser1, N, N, wp);
/* t = a - m, u = b - m */
arb_set_fmpz(t, a);
arb_sub_fmpz(t, t, m, wp);
arb_set_fmpz(u, b);
arb_sub_fmpz(u, u, m, wp);
arb_power_sum_vec(ser1, t, u, N, wp);
arb_zero(res);
for (k = 0; k < N; k++)
arb_addmul(res, ser3 + k, ser1 + k, wp);
if (mmag != NULL)
{
if (_fmpz_sub_small(MAG_EXPREF(arb_radref(res)), mmag) <= -tol)
break;
}
else
{
if (arb_rel_accuracy_bits(res) >= tol)
break;
}
wp = 2 * wp;
}
/* add the series truncation bound */
arb_add_error_mag(res, bound);
_arb_vec_clear(mx, 2);
_arb_vec_clear(ser1, N);
_arb_vec_clear(ser2, N);
_arb_vec_clear(ser3, N);
}
mag_clear(B);
mag_clear(C);
mag_clear(D);
mag_clear(bound);
arb_clear(t);
arb_clear(u);
fmpz_clear(m);
fmpz_clear(r);
fmpz_clear(R);
fmpz_clear(tmp);
}
void
mag_polylog_tail(mag_t u, const mag_t z, long sigma, ulong d, ulong N)
{
mag_t TN, UN, t;
if (N < 2)
{
mag_inf(u);
return;
}
mag_init(TN);
mag_init(UN);
mag_init(t);
if (mag_cmp_2exp_si(z, 0) >= 0)
{
mag_inf(u);
}
else
{
/* Bound T(N) */
mag_pow_ui(TN, z, N);
/* multiply by log(N)^d */
if (d > 0)
{
mag_log_ui(t, N);
mag_pow_ui(t, t, d);
mag_mul(TN, TN, t);
}
/* multiply by 1/k^s */
if (sigma > 0)
{
mag_set_ui_lower(t, N);
mag_pow_ui_lower(t, t, sigma);
mag_div(TN, TN, t);
}
else if (sigma < 0)
{
mag_set_ui(t, N);
mag_pow_ui(t, t, -sigma);
mag_mul(TN, TN, t);
}
/* Bound U(N) */
mag_set(UN, z);
/* multiply by (1 + 1/N)**S */
if (sigma < 0)
{
mag_binpow_uiui(t, N, -sigma);
mag_mul(UN, UN, t);
}
/* multiply by (1 + 1/(N log(N)))^d */
if (d > 0)
{
ulong nl;
/* rounds down */
nl = mag_d_log_lower_bound(N) * N * (1 - 1e-13);
mag_binpow_uiui(t, nl, d);
mag_mul(UN, UN, t);
}
/* T(N) / (1 - U(N)) */
if (mag_cmp_2exp_si(UN, 0) >= 0)
{
mag_inf(u);
}
else
{
mag_one(t);
mag_sub_lower(t, t, UN);
mag_div(u, TN, t);
}
}
mag_clear(TN);
mag_clear(UN);
mag_clear(t);
}
void
acb_rising_ui_get_mag(mag_t bound, const acb_t s, ulong n)
{
if (n == 0)
{
mag_one(bound);
return;
}
if (n == 1)
{
acb_get_mag(bound, s);
return;
}
if (!acb_is_finite(s))
{
mag_inf(bound);
return;
}
if (arf_sgn(arb_midref(acb_realref(s))) >= 0)
{
acb_rising_get_mag2_right(bound, acb_realref(s), acb_imagref(s), n);
}
else
{
arb_t a;
long k;
mag_t bound2, t, u;
arb_init(a);
mag_init(bound2);
mag_init(t);
mag_init(u);
arb_get_mag(u, acb_imagref(s));
mag_mul(u, u, u);
mag_one(bound);
for (k = 0; k < n; k++)
{
arb_add_ui(a, acb_realref(s), k, MAG_BITS);
if (arf_sgn(arb_midref(a)) >= 0)
{
acb_rising_get_mag2_right(bound2, a, acb_imagref(s), n - k);
mag_mul(bound, bound, bound2);
break;
}
else
{
arb_get_mag(t, a);
mag_mul(t, t, t);
mag_add(t, t, u);
mag_mul(bound, bound, t);
}
}
arb_clear(a);
mag_clear(bound2);
mag_clear(t);
mag_clear(u);
}
mag_sqrt(bound, bound);
}
/* computes the factors that are independent of n (all are upper bounds) */
void
acb_hypgeom_u_asymp_bound_factors(int * R, mag_t alpha,
mag_t nu, mag_t sigma, mag_t rho, mag_t zinv,
const acb_t a, const acb_t b, const acb_t z)
{
mag_t r, u, zre, zim, zlo, sigma_prime;
acb_t t;
mag_init(r);
mag_init(u);
mag_init(zre);
mag_init(zim);
mag_init(zlo);
mag_init(sigma_prime);
acb_init(t);
/* lower bounds for |re(z)|, |im(z)|, |z| */
arb_get_mag_lower(zre, acb_realref(z));
arb_get_mag_lower(zim, acb_imagref(z));
acb_get_mag_lower(zlo, z); /* todo: hypot */
/* upper bound for 1/|z| */
mag_one(u);
mag_div(zinv, u, zlo);
/* upper bound for r = |b - 2a| */
acb_mul_2exp_si(t, a, 1);
acb_sub(t, b, t, MAG_BITS);
acb_get_mag(r, t);
/* determine region */
*R = 0;
if (mag_cmp(zlo, r) >= 0)
{
int znonneg = arb_is_nonnegative(acb_realref(z));
if (znonneg && mag_cmp(zre, r) >= 0)
{
*R = 1;
}
else if (mag_cmp(zim, r) >= 0 || znonneg)
{
*R = 2;
}
else
{
mag_mul_2exp_si(u, r, 1);
if (mag_cmp(zlo, u) >= 0)
*R = 3;
}
}
if (R == 0)
{
mag_inf(alpha);
mag_inf(nu);
mag_inf(sigma);
mag_inf(rho);
}
else
{
/* sigma = |(b-2a)/z| */
mag_mul(sigma, r, zinv);
/* nu = (1/2 + 1/2 sqrt(1-4 sigma^2))^(-1/2) <= 1 + 2 sigma^2 */
if (mag_cmp_2exp_si(sigma, -1) <= 0)
{
mag_mul(nu, sigma, sigma);
mag_mul_2exp_si(nu, nu, 1);
mag_one(u);
mag_add(nu, nu, u);
}
else
{
mag_inf(nu);
}
/* modified sigma for alpha, beta, rho when in R3 */
if (*R == 3)
mag_mul(sigma_prime, sigma, nu);
else
mag_set(sigma_prime, sigma);
/* alpha = 1/(1-sigma') */
mag_one(alpha);
mag_sub_lower(alpha, alpha, sigma_prime);
mag_one(u);
mag_div(alpha, u, alpha);
/* rho = |2a^2-2ab+b|/2 + sigma'*(1+sigma'/4)/(1-sigma')^2 */
mag_mul_2exp_si(rho, sigma_prime, -2);
mag_one(u);
mag_add(rho, rho, u);
mag_mul(rho, rho, sigma_prime);
mag_mul(rho, rho, alpha);
mag_mul(rho, rho, alpha);
acb_sub(t, a, b, MAG_BITS);
acb_mul(t, t, a, MAG_BITS);
acb_mul_2exp_si(t, t, 1);
acb_add(t, t, b, MAG_BITS);
acb_get_mag(u, t);
mag_mul_2exp_si(u, u, -1);
mag_add(rho, rho, u);
}
mag_clear(r);
mag_clear(u);
mag_clear(zre);
mag_clear(zim);
mag_clear(zlo);
mag_clear(sigma_prime);
acb_clear(t);
}
slong
hypgeom_bound(mag_t error, int r,
slong A, slong B, slong K, const mag_t TK, const mag_t z, slong tol_2exp)
{
mag_t Tn, t, u, one, tol, num, den;
slong n, m;
mag_init(Tn);
mag_init(t);
mag_init(u);
mag_init(one);
mag_init(tol);
mag_init(num);
mag_init(den);
mag_one(one);
mag_set_ui_2exp_si(tol, UWORD(1), -tol_2exp);
/* approximate number of needed terms */
n = hypgeom_estimate_terms(z, r, tol_2exp);
/* required for 1 + O(1/k) part to be decreasing */
n = FLINT_MAX(n, K + 1);
/* required for z^k / (k!)^r to be decreasing */
m = hypgeom_root_bound(z, r);
n = FLINT_MAX(n, m);
/* We now have |R(k)| <= G(k) where G(k) is monotonically decreasing,
and can bound the tail using a geometric series as soon
as soon as G(k) < 1. */
/* bound T(n-1) */
hypgeom_term_bound(Tn, TK, K, A, B, r, z, n-1);
while (1)
{
/* bound R(n) */
mag_mul_ui(num, z, n);
mag_mul_ui(num, num, n - B);
mag_set_ui_lower(den, n - A);
mag_mul_ui_lower(den, den, n - 2*B);
if (r != 0)
{
mag_set_ui_lower(u, n);
mag_pow_ui_lower(u, u, r);
mag_mul_lower(den, den, u);
}
mag_div(t, num, den);
/* multiply bound for T(n-1) by bound for R(n) to bound T(n) */
mag_mul(Tn, Tn, t);
/* geometric series termination check */
/* u = max(1-t, 0), rounding down [lower bound] */
mag_sub_lower(u, one, t);
if (!mag_is_zero(u))
{
mag_div(u, Tn, u);
if (mag_cmp(u, tol) < 0)
{
mag_set(error, u);
break;
}
}
/* move on to next term */
n++;
}
mag_clear(Tn);
mag_clear(t);
mag_clear(u);
mag_clear(one);
mag_clear(tol);
mag_clear(num);
mag_clear(den);
return n;
}
void
acb_hypgeom_pfq_sum_rs(acb_t res, acb_t term, acb_srcptr a, slong p,
acb_srcptr b, slong q, const acb_t z, slong n, slong prec)
{
acb_ptr zpow;
acb_t s, t, u;
slong i, j, k, m;
mag_t B, C;
if (n == 0)
{
acb_zero(res);
acb_one(term);
return;
}
if (n < 0)
abort();
m = n_sqrt(n);
m = FLINT_MIN(m, 150);
mag_init(B);
mag_init(C);
acb_init(s);
acb_init(t);
acb_init(u);
zpow = _acb_vec_init(m + 1);
_acb_vec_set_powers(zpow, z, m + 1, prec);
mag_one(B);
for (k = n; k >= 0; k--)
{
j = k % m;
if (k < n)
acb_add(s, s, zpow + j, prec);
if (k > 0)
{
if (p > 0)
{
acb_add_ui(u, a, k - 1, prec);
for (i = 1; i < p; i++)
{
acb_add_ui(t, a + i, k - 1, prec);
acb_mul(u, u, t, prec);
}
if (k < n)
acb_mul(s, s, u, prec);
acb_get_mag(C, u);
mag_mul(B, B, C);
}
if (q > 0)
{
acb_add_ui(u, b, k - 1, prec);
for (i = 1; i < q; i++)
{
acb_add_ui(t, b + i, k - 1, prec);
acb_mul(u, u, t, prec);
}
if (k < n)
acb_div(s, s, u, prec);
acb_get_mag_lower(C, u);
mag_div(B, B, C);
}
if (j == 0 && k < n)
{
acb_mul(s, s, zpow + m, prec);
}
}
}
acb_get_mag(C, z);
mag_pow_ui(C, C, n);
mag_mul(B, B, C);
acb_zero(term);
if (_acb_vec_is_real(a, p) && _acb_vec_is_real(b, q) && acb_is_real(z))
arb_add_error_mag(acb_realref(term), B);
else
acb_add_error_mag(term, B);
acb_set(res, s);
mag_clear(B);
mag_clear(C);
acb_clear(s);
acb_clear(t);
acb_clear(u);
_acb_vec_clear(zpow, m + 1);
}