void
arb_rising_fmpq_ui(arb_t y, const fmpq_t x, ulong n, long prec)
{
if (n == 0)
{
arb_one(y);
}
else if (n == 1)
{
arb_set_fmpq(y, x, prec);
}
else
{
long wp;
wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
bsplit(y, fmpq_numref(x), fmpq_denref(x), 0, n, wp);
if (fmpz_is_one(fmpq_denref(x)))
{
arb_set_round(y, y, prec);
}
else
{
arb_t t;
arb_init(t);
arb_set_fmpz(t, fmpq_denref(x));
arb_pow_ui(t, t, n, wp);
arb_div(y, y, t, prec);
arb_clear(t);
}
}
}
void
arb_mat_set_fmpz_mat(arb_mat_t dest, const fmpz_mat_t src)
{
slong i, j;
if (arb_mat_ncols(dest) != 0)
{
for (i = 0; i < arb_mat_nrows(dest); i++)
for (j = 0; j < arb_mat_ncols(dest); j++)
arb_set_fmpz(arb_mat_entry(dest, i, j),
fmpz_mat_entry(src, i, j));
}
}
void arb_from_interval(arb_t x, const fmpz_t c, const slong k, const slong prec)
{
/* we build the ball that gives exactly (c 2^k, (c+1) 2^k) */
/* center: (2c+1) 2^(k-1) */
/* radius: 2^(k-1) */
if (prec <= 0 || prec < fmpz_bits(c) + 2)
{
fprintf(stderr, "not enough precision");
abort();
}
arb_set_fmpz(x, c);
arb_mul_2exp_si(x, x, 1);
arb_add_si(x, x, 1, prec);
arb_mul_2exp_si(x, x, k-1);
arb_add_error_2exp_si(x, k-1);
}
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;
}
void
acb_randtest_maybe_half_int(acb_t x, flint_rand_t state, long prec, long size)
{
if (n_randint(state, 8) == 0)
{
fmpz_t t;
fmpz_init(t);
fmpz_randtest(t, state, 1 + n_randint(state, prec));
arb_set_fmpz(acb_realref(x), t);
arb_zero(acb_imagref(x));
acb_mul_2exp_si(x, x, -1);
fmpz_clear(t);
}
else
{
acb_randtest(x, state, prec, size);
}
}
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;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("rising2_ui....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 1000; iter++)
{
arb_t a, u, v, u2, v2;
fmpz *f;
arb_ptr g;
ulong n;
slong i, prec;
arb_init(a);
arb_init(u);
arb_init(v);
arb_init(u2);
arb_init(v2);
arb_randtest(a, state, 1 + n_randint(state, 4000), 10);
arb_randtest(u, state, 1 + n_randint(state, 4000), 10);
arb_randtest(v, state, 1 + n_randint(state, 4000), 10);
n = n_randint(state, 120);
f = _fmpz_vec_init(n + 1);
g = _arb_vec_init(n + 1);
prec = 2 + n_randint(state, 4000);
arb_rising2_ui(u, v, a, n, prec);
arith_stirling_number_1u_vec(f, n, n + 1);
for (i = 0; i <= n; i++)
arb_set_fmpz(g + i, f + i);
_arb_poly_evaluate(u2, g, n + 1, a, prec);
_arb_poly_derivative(g, g, n + 1, prec);
_arb_poly_evaluate(v2, g, n, a, prec);
if (!arb_overlaps(u, u2) || !arb_overlaps(v, v2))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("n = %wu\n", n);
flint_printf("a = "); arb_printd(a, 15); flint_printf("\n\n");
flint_printf("u = "); arb_printd(u, 15); flint_printf("\n\n");
flint_printf("u2 = "); arb_printd(u2, 15); flint_printf("\n\n");
flint_printf("v = "); arb_printd(v, 15); flint_printf("\n\n");
flint_printf("v2 = "); arb_printd(v2, 15); flint_printf("\n\n");
abort();
}
arb_set(u2, a);
arb_rising2_ui(u2, v, u2, n, prec);
if (!arb_equal(u2, u))
{
flint_printf("FAIL: aliasing 1\n\n");
flint_printf("a = "); arb_printd(a, 15); flint_printf("\n\n");
flint_printf("u = "); arb_printd(u, 15); flint_printf("\n\n");
flint_printf("u2 = "); arb_printd(u2, 15); flint_printf("\n\n");
flint_printf("n = %wu\n", n);
abort();
}
arb_set(v2, a);
arb_rising2_ui(u, v2, v2, n, prec);
if (!arb_equal(v2, v))
{
flint_printf("FAIL: aliasing 2\n\n");
flint_printf("a = "); arb_printd(a, 15); flint_printf("\n\n");
flint_printf("v = "); arb_printd(v, 15); flint_printf("\n\n");
flint_printf("v2 = "); arb_printd(v2, 15); flint_printf("\n\n");
flint_printf("n = %wu\n", n);
abort();
}
arb_clear(a);
arb_clear(u);
arb_clear(v);
arb_clear(u2);
arb_clear(v2);
_fmpz_vec_clear(f, n + 1);
_arb_vec_clear(g, n + 1);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("legendre_p_ui_root....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100 * arb_test_multiplier(); iter++)
{
ulong n, k;
slong prec;
arb_ptr roots, weights;
arb_poly_t pol;
arb_t s;
fmpq_poly_t pol2;
n = 1 + n_randint(state, 100);
prec = 20 + n_randint(state, 500);
roots = _arb_vec_init(n);
weights = _arb_vec_init(n);
arb_poly_init(pol);
fmpq_poly_init(pol2);
arb_init(s);
for (k = 0; k < n; k++)
{
if (k > n / 2 && n_randint(state, 2))
{
arb_neg(roots + k, roots + n - k - 1);
arb_set(weights + k, weights + n - k - 1);
}
else
{
arb_hypgeom_legendre_p_ui_root(roots + k, weights + k, n, k, prec);
}
}
arb_poly_product_roots(pol, roots, n, prec);
/* fmpq_poly_legendre_p(pol2, n); */
arith_legendre_polynomial(pol2, n);
arb_set_fmpz(s, pol2->coeffs + n);
arb_div_fmpz(s, s, pol2->den, prec);
arb_poly_scalar_mul(pol, pol, s, prec);
if (!arb_poly_contains_fmpq_poly(pol, pol2))
{
flint_printf("FAIL: polynomial containment\n\n");
flint_printf("n = %wu, prec = %wd\n\n", n, prec);
flint_printf("pol = "); arb_poly_printd(pol, 30); flint_printf("\n\n");
flint_printf("pol2 = "); fmpq_poly_print(pol2); flint_printf("\n\n");
flint_abort();
}
arb_zero(s);
for (k = 0; k < n; k++)
{
arb_add(s, s, weights + k, prec);
}
if (!arb_contains_si(s, 2))
{
flint_printf("FAIL: sum of weights\n\n");
flint_printf("n = %wu, prec = %wd\n\n", n, prec);
flint_printf("s = "); arb_printn(s, 30, 0); flint_printf("\n\n");
flint_abort();
}
_arb_vec_clear(roots, n);
_arb_vec_clear(weights, n);
arb_poly_clear(pol);
fmpq_poly_clear(pol2);
arb_clear(s);
}
for (iter = 0; iter < 500 * arb_test_multiplier(); iter++)
{
arb_t x1, x2, w1, w2;
ulong n, k;
slong prec1, prec2;
arb_init(x1);
arb_init(x2);
arb_init(w1);
arb_init(w2);
n = 1 + n_randtest(state) % 100000;
if (n_randint(state, 2) || n == 1)
k = n_randtest(state) % n;
else
k = n / 2 - (n_randtest(state) % (n / 2));
prec1 = 2 + n_randtest(state) % 2000;
prec2 = 2 + n_randtest(state) % 2000;
arb_hypgeom_legendre_p_ui_root(x1, w1, n, k, prec1);
if (n_randint(state, 10) == 0)
arb_hypgeom_legendre_p_ui_root(x1, NULL, n, k, prec1);
arb_hypgeom_legendre_p_ui_root(x2, w2, n, k, prec2);
if (!arb_overlaps(x1, x2) || !arb_overlaps(w1, w2))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("n = %wu, k = %wu, prec1 = %wd, prec2 = %wd\n\n", n, k, prec1, prec2);
flint_printf("x1 = "); arb_printn(x1, 100, 0); flint_printf("\n\n");
flint_printf("x2 = "); arb_printn(x2, 100, 0); flint_printf("\n\n");
flint_printf("w1 = "); arb_printn(w1, 100, 0); flint_printf("\n\n");
flint_printf("w2 = "); arb_printn(w2, 100, 0); flint_printf("\n\n");
flint_abort();
}
if (arb_rel_accuracy_bits(x1) < prec1 - 3 || arb_rel_accuracy_bits(w1) < prec1 - 3)
{
flint_printf("FAIL: accuracy\n\n");
flint_printf("n = %wu, k = %wu, prec1 = %wd\n\n", n, k, prec1);
flint_printf("acc(x1) = %wd, acc(w1) = %wd\n\n", arb_rel_accuracy_bits(x1), arb_rel_accuracy_bits(w1));
flint_printf("x1 = "); arb_printn(x1, prec1, ARB_STR_CONDENSE * 30); flint_printf("\n\n");
flint_printf("w1 = "); arb_printn(w1, prec1, ARB_STR_CONDENSE * 30); flint_printf("\n\n");
flint_abort();
}
if (arb_rel_accuracy_bits(x2) < prec2 - 3 || arb_rel_accuracy_bits(w2) < prec2 - 3)
{
flint_printf("FAIL: accuracy 2\n\n");
flint_printf("n = %wu, k = %wu, prec2 = %wd\n\n", n, k, prec2);
flint_printf("acc(x2) = %wd, acc(w2) = %wd\n\n", arb_rel_accuracy_bits(x2), arb_rel_accuracy_bits(w2));
flint_printf("x2 = "); arb_printn(x2, prec2, ARB_STR_CONDENSE * 30); flint_printf("\n\n");
flint_printf("w2 = "); arb_printn(w2, prec2, ARB_STR_CONDENSE * 30); flint_printf("\n\n");
flint_abort();
}
arb_clear(x1);
arb_clear(x2);
arb_clear(w1);
arb_clear(w2);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
long iter;
flint_rand_t state;
printf("submul_fmpz....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000; iter++)
{
arb_t a, b, c, d;
fmpz_t x;
long prec;
arb_init(a);
arb_init(b);
arb_init(c);
arb_init(d);
fmpz_init(x);
arb_randtest_special(a, state, 1 + n_randint(state, 2000), 100);
arb_randtest_special(b, state, 1 + n_randint(state, 2000), 100);
arb_randtest_special(c, state, 1 + n_randint(state, 2000), 100);
fmpz_randtest(x, state, 1 + n_randint(state, 2000));
prec = 2 + n_randint(state, 2000);
arb_set_fmpz(b, x);
arb_set(d, c);
arb_submul_fmpz(c, a, x, prec);
arb_submul(d, a, b, prec);
if (!arb_equal(c, d))
{
printf("FAIL\n\n");
printf("a = "); arb_print(a); printf("\n\n");
printf("b = "); arb_print(b); printf("\n\n");
printf("c = "); arb_print(c); printf("\n\n");
printf("d = "); arb_print(d); printf("\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
arb_clear(d);
fmpz_clear(x);
}
/* aliasing */
for (iter = 0; iter < 10000; iter++)
{
arb_t a, b, c;
fmpz_t x;
long prec;
arb_init(a);
arb_init(b);
arb_init(c);
fmpz_init(x);
arb_randtest_special(a, state, 1 + n_randint(state, 2000), 100);
arb_randtest_special(b, state, 1 + n_randint(state, 2000), 100);
arb_randtest_special(c, state, 1 + n_randint(state, 2000), 100);
fmpz_randtest(x, state, 1 + n_randint(state, 2000));
prec = 2 + n_randint(state, 2000);
arb_set_fmpz(b, x);
arb_set(c, a);
arb_submul_fmpz(c, a, x, prec);
arb_submul_fmpz(a, a, x, prec);
if (!arb_equal(a, c))
{
printf("FAIL (aliasing)\n\n");
printf("a = "); arb_print(a); printf("\n\n");
printf("b = "); arb_print(b); printf("\n\n");
printf("c = "); arb_print(c); printf("\n\n");
abort();
}
arb_clear(a);
arb_clear(b);
arb_clear(c);
fmpz_clear(x);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
static void
bsplit_recursive_arb(arb_t P, arb_t Q, arb_t B, arb_t T,
const hypgeom_t hyp, long a, long b, int cont, long prec)
{
if (b - a < 4)
{
fmpz_t PP, QQ, BB, TT;
fmpz_init(PP);
fmpz_init(QQ);
fmpz_init(BB);
fmpz_init(TT);
bsplit_recursive_fmpz(PP, QQ, BB, TT, hyp, a, b, cont);
arb_set_fmpz(P, PP);
arb_set_fmpz(Q, QQ);
arb_set_fmpz(B, BB);
arb_set_fmpz(T, TT);
fmpz_clear(PP);
fmpz_clear(QQ);
fmpz_clear(BB);
fmpz_clear(TT);
}
else
{
long m;
arb_t P2, Q2, B2, T2;
m = (a + b) / 2;
arb_init(P2);
arb_init(Q2);
arb_init(B2);
arb_init(T2);
bsplit_recursive_arb(P, Q, B, T, hyp, a, m, 1, prec);
bsplit_recursive_arb(P2, Q2, B2, T2, hyp, m, b, 1, prec);
if (arb_is_one(B) && arb_is_one(B2))
{
arb_mul(T, T, Q2, prec);
arb_addmul(T, P, T2, prec);
}
else
{
arb_mul(T, T, B2, prec);
arb_mul(T, T, Q2, prec);
arb_mul(T2, T2, B, prec);
arb_addmul(T, P, T2, prec);
}
arb_mul(B, B, B2, prec);
arb_mul(Q, Q, Q2, prec);
if (cont)
arb_mul(P, P, P2, prec);
arb_clear(P2);
arb_clear(Q2);
arb_clear(B2);
arb_clear(T2);
}
}
void
arb_rising2_ui_rs(arb_t u, arb_t v,
const arb_t x, ulong n, ulong m, slong prec)
{
if (n == 0)
{
arb_zero(v);
arb_one(u);
}
else if (n == 1)
{
arb_set(u, x);
arb_one(v);
}
else
{
slong wp;
ulong i, j, a, b;
arb_ptr xs;
arb_t S, T, U, V;
fmpz *A, *B;
wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
if (m == 0)
{
ulong m1, m2;
m1 = 0.6 * pow(wp, 0.4);
m2 = n_sqrt(n);
m = FLINT_MIN(m1, m2);
}
m = FLINT_MAX(m, 1);
xs = _arb_vec_init(m + 1);
A = _fmpz_vec_init(2 * m + 1);
B = A + (m + 1);
arb_init(S);
arb_init(T);
arb_init(U);
arb_init(V);
_arb_vec_set_powers(xs, x, m + 1, wp);
for (i = 0; i < n; i += m)
{
a = i;
b = FLINT_MIN(n, a + m);
if (a == 0 || b != a + m)
{
_gamma_rf_bsplit(A, a, b);
}
else
{
fmpz tt = m;
_fmpz_poly_taylor_shift(A, &tt, m + 1);
}
_fmpz_poly_derivative(B, A, b - a + 1);
arb_set_fmpz(S, A);
for (j = 1; j <= b - a; j++)
arb_addmul_fmpz(S, xs + j, A + j, wp);
arb_set_fmpz(T, B);
for (j = 1; j < b - a; j++)
arb_addmul_fmpz(T, xs + j, B + j, wp);
if (i == 0)
{
arb_set(U, S);
arb_set(V, T);
}
else
{
arb_mul(V, V, S, wp);
arb_addmul(V, U, T, wp);
arb_mul(U, U, S, wp);
}
}
arb_set(u, U);
arb_set(v, V);
_arb_vec_clear(xs, m + 1);
_fmpz_vec_clear(A, 2 * m + 1);
arb_clear(S);
arb_clear(T);
arb_clear(U);
arb_clear(V);
}
}
static int
arb_set_float_str(arb_t res, const char * inp, slong prec)
{
char * emarker;
char * buf;
int error;
slong i;
fmpz_t exp;
fmpz_t man;
slong num_int, num_frac;
int after_radix;
if (inp[0] == '+')
{
return arb_set_float_str(res, inp + 1, prec);
}
if (inp[0] == '-')
{
error = arb_set_float_str(res, inp + 1, prec);
arb_neg(res, res);
return error;
}
if (strcmp(inp, "inf") == 0)
{
arb_pos_inf(res);
return 0;
}
if (strcmp(inp, "nan") == 0)
{
arb_indeterminate(res);
return 0;
}
error = 0;
fmpz_init(exp);
fmpz_init(man);
buf = flint_malloc(strlen(inp) + 1);
emarker = strchr(inp, 'e');
/* parse exponent (0 by default) */
if (emarker != NULL)
{
/* allow e+42 as well as e42 */
if (emarker[1] == '+')
{
if (!(emarker[2] >= '0' && emarker[2] <= '9'))
error = 1;
else
error = fmpz_set_str(exp, emarker + 2, 10);
}
else
error = fmpz_set_str(exp, emarker + 1, 10);
if (error)
goto cleanup;
}
/* parse floating-point part */
{
num_int = 0;
num_frac = 0;
after_radix = 0;
for (i = 0; inp + i != emarker && inp[i] != '\0'; i++)
{
if (inp[i] == '.' && !after_radix)
{
after_radix = 1;
}
else if (inp[i] >= '0' && inp[i] <= '9')
{
buf[num_int + num_frac] = inp[i];
num_frac += after_radix;
num_int += !after_radix;
}
else
{
error = 1;
goto cleanup;
}
}
buf[num_int + num_frac] = '\0';
/* put trailing zeros into the exponent */
while (num_int + num_frac > 1 && buf[num_int + num_frac - 1] == '0')
{
buf[num_int + num_frac - 1] = '\0';
num_frac--;
}
fmpz_sub_si(exp, exp, num_frac);
error = fmpz_set_str(man, buf, 10);
if (error)
goto cleanup;
}
if (fmpz_is_zero(man))
{
arb_zero(res);
}
else if (fmpz_is_zero(exp))
{
arb_set_round_fmpz(res, man, prec);
}
else
{
arb_t t;
arb_init(t);
arb_set_ui(t, 10);
arb_set_fmpz(res, man);
if (fmpz_sgn(exp) > 0)
{
arb_pow_fmpz_binexp(t, t, exp, prec + 4);
arb_mul(res, res, t, prec);
}
else
{
fmpz_neg(exp, exp);
arb_pow_fmpz_binexp(t, t, exp, prec + 4);
arb_div(res, res, t, prec);
}
arb_clear(t);
}
cleanup:
fmpz_clear(exp);
fmpz_clear(man);
flint_free(buf);
if (error)
arb_indeterminate(res);
return error;
}
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 acb_modular_transform(acb_t w, const psl2z_t g, const acb_t z, slong prec)
{
#define a (&g->a)
#define b (&g->b)
#define c (&g->c)
#define d (&g->d)
#define x acb_realref(z)
#define y acb_imagref(z)
if (fmpz_is_zero(c))
{
/* (az+b)/d, where we must have a = d = 1 */
acb_add_fmpz(w, z, b, prec);
}
else if (fmpz_is_zero(a))
{
/* b/(cz+d), where -bc = 1, c = 1 => -1/(z+d) */
acb_add_fmpz(w, z, d, prec);
acb_inv(w, w, prec);
acb_neg(w, w);
}
else if (0)
{
acb_t t, u;
acb_init(t);
acb_init(u);
acb_set_fmpz(t, b);
acb_addmul_fmpz(t, z, a, prec);
acb_set_fmpz(u, d);
acb_addmul_fmpz(u, z, c, prec);
acb_div(w, t, u, prec);
acb_clear(t);
acb_clear(u);
}
else
{
/* (az+b)/(cz+d) = (re+im*i)/den where
re = bd + (bc+ad)x + ac(x^2+y^2)
im = (ad-bc)y
den = c^2(x^2+y^2) + 2cdx + d^2
*/
fmpz_t t;
arb_t re, im, den;
arb_init(re);
arb_init(im);
arb_init(den);
fmpz_init(t);
arb_mul(im, x, x, prec);
arb_addmul(im, y, y, prec);
fmpz_mul(t, b, d);
arb_set_fmpz(re, t);
fmpz_mul(t, b, c);
fmpz_addmul(t, a, d);
arb_addmul_fmpz(re, x, t, prec);
fmpz_mul(t, a, c);
arb_addmul_fmpz(re, im, t, prec);
fmpz_mul(t, d, d);
arb_set_fmpz(den, t);
fmpz_mul(t, c, d);
fmpz_mul_2exp(t, t, 1);
arb_addmul_fmpz(den, x, t, prec);
fmpz_mul(t, c, c);
arb_addmul_fmpz(den, im, t, prec);
fmpz_mul(t, a, d);
fmpz_submul(t, b, c);
arb_mul_fmpz(im, y, t, prec);
arb_div(acb_realref(w), re, den, prec);
arb_div(acb_imagref(w), im, den, prec);
arb_clear(re);
arb_clear(im);
arb_clear(den);
fmpz_clear(t);
}
#undef a
#undef b
#undef c
#undef d
#undef x
#undef y
}
