void
_arb_poly_revert_series_lagrange_fast(arb_ptr Qinv, arb_srcptr Q, long Qlen, long n, long prec)
{
long i, j, k, m;
arb_ptr R, S, T, tmp;
arb_t t;
if (n <= 2)
{
if (n >= 1)
arb_zero(Qinv);
if (n == 2)
arb_inv(Qinv + 1, Q + 1, prec);
return;
}
m = n_sqrt(n);
arb_init(t);
R = _arb_vec_init((n - 1) * m);
S = _arb_vec_init(n - 1);
T = _arb_vec_init(n - 1);
arb_zero(Qinv);
arb_inv(Qinv + 1, Q + 1, prec);
_arb_poly_inv_series(Ri(1), Q + 1, FLINT_MIN(Qlen, n) - 1, n - 1, prec);
for (i = 2; i <= m; i++)
_arb_poly_mullow(Ri(i), Ri((i + 1) / 2), n - 1, Ri(i / 2), n - 1, n - 1, prec);
for (i = 2; i < m; i++)
arb_div_ui(Qinv + i, Ri(i) + i - 1, i, prec);
_arb_vec_set(S, Ri(m), n - 1);
for (i = m; i < n; i += m)
{
arb_div_ui(Qinv + i, S + i - 1, i, prec);
for (j = 1; j < m && i + j < n; j++)
{
arb_mul(t, S + 0, Ri(j) + i + j - 1, prec);
for (k = 1; k <= i + j - 1; k++)
arb_addmul(t, S + k, Ri(j) + i + j - 1 - k, prec);
arb_div_ui(Qinv + i + j, t, i + j, prec);
}
if (i + 1 < n)
{
_arb_poly_mullow(T, S, n - 1, Ri(m), n - 1, n - 1, prec);
tmp = S; S = T; T = tmp;
}
}
arb_clear(t);
_arb_vec_clear(R, (n - 1) * m);
_arb_vec_clear(S, n - 1);
_arb_vec_clear(T, n - 1);
}
void
arb_root_ui_exp(arb_t res, const arb_t x, ulong k, slong prec)
{
arb_log(res, x, prec + 4);
arb_div_ui(res, res, k, prec + 4);
arb_exp(res, res, prec);
}
void
_arb_poly_integral(arb_ptr res, arb_srcptr poly, slong len, slong prec)
{
slong k = len - 1;
for (k = len - 1; k > 0; k--)
arb_div_ui(res + k, poly + k - 1, k, prec);
arb_zero(res);
}
void
_arb_poly_sinh_cosh_series_basecase(arb_ptr s, arb_ptr c, arb_srcptr h, slong hlen,
slong n, slong prec)
{
slong j, k, alen = FLINT_MIN(n, hlen);
arb_ptr a;
arb_t t, u;
arb_sinh_cosh(s, c, h, prec);
if (hlen == 1)
{
_arb_vec_zero(s + 1, n - 1);
_arb_vec_zero(c + 1, n - 1);
return;
}
arb_init(t);
arb_init(u);
a = _arb_vec_init(alen);
for (k = 1; k < alen; k++)
arb_mul_ui(a + k, h + k, k, prec);
for (k = 1; k < n; k++)
{
arb_zero(t);
arb_zero(u);
for (j = 1; j < FLINT_MIN(k + 1, hlen); j++)
{
arb_addmul(t, a + j, s + k - j, prec);
arb_addmul(u, a + j, c + k - j, prec);
}
arb_div_ui(c + k, t, k, prec);
arb_div_ui(s + k, u, k, prec);
}
arb_clear(t);
arb_clear(u);
_arb_vec_clear(a, alen);
}
void
arb_div_2expm1_ui(arb_t y, const arb_t x, ulong n, long prec)
{
if (n < FLINT_BITS)
{
arb_div_ui(y, x, (1UL << n) - 1, prec);
}
else if (n < 1024 + prec / 32 || n > LONG_MAX / 4)
{
arb_t t;
fmpz_t e;
arb_init(t);
fmpz_init_set_ui(e, n);
arb_one(t);
arb_mul_2exp_fmpz(t, t, e);
arb_sub_ui(t, t, 1, prec);
arb_div(y, x, t, prec);
arb_clear(t);
fmpz_clear(e);
}
else
{
arb_t s, t;
long i, b;
arb_init(s);
arb_init(t);
/* x / (2^n - 1) = sum_{k>=1} x * 2^(-k*n)*/
arb_mul_2exp_si(s, x, -n);
arb_set(t, s);
b = 1;
for (i = 2; i <= prec / n + 1; i++)
{
arb_mul_2exp_si(t, t, -n);
arb_add(s, s, t, prec);
b = i;
}
/* error bound: sum_{k>b} x * 2^(-k*n) <= x * 2^(-b*n - (n-1)) */
arb_mul_2exp_si(t, x, -b*n - (n-1));
arb_abs(t, t);
arb_add_error(s, t);
arb_set(y, s);
arb_clear(s);
arb_clear(t);
}
}
/*
Bound for scaled Bessel function: 2/(2 pi x)^(1/2)
Bound for tail of integral: 2 N (k / (pi N))^(k / 2) / (k - 2).
*/
void
scaled_bessel_tail_bound(arb_t b, ulong k, const arb_t N, slong prec)
{
arb_const_pi(b, prec);
arb_mul(b, b, N, prec);
arb_ui_div(b, k, b, prec);
arb_sqrt(b, b, prec);
arb_pow_ui(b, b, k, prec);
arb_mul(b, b, N, prec);
arb_mul_ui(b, b, 2, prec);
arb_div_ui(b, b, k - 2, prec);
}
void
acb_zeta_si(acb_t z, slong s, slong prec)
{
if (s >= 0)
{
arb_zeta_ui(acb_realref(z), s, prec);
}
else
{
arb_bernoulli_ui(acb_realref(z), 1-s, prec);
arb_div_ui(acb_realref(z), acb_realref(z), 1-s, prec);
arb_neg(acb_realref(z), acb_realref(z));
}
arb_zero(acb_imagref(z));
return;
}
/* series of c^(d+x) */
static __inline__ void
_arb_poly_pow_cpx(arb_ptr res, const arb_t c, const arb_t d, long trunc, long prec)
{
long i;
arb_t logc;
arb_init(logc);
arb_log(logc, c, prec);
arb_mul(res + 0, logc, d, prec);
arb_exp(res + 0, res + 0, prec);
for (i = 1; i < trunc; i++)
{
arb_mul(res + i, res + i - 1, logc, prec);
arb_div_ui(res + i, res + i, i, prec);
}
arb_clear(logc);
}
int main()
{
long iter;
flint_rand_t state;
printf("div_ui....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000; iter++)
{
arb_t a, b, c, d;
ulong x;
long prec;
arb_init(a);
arb_init(b);
arb_init(c);
arb_init(d);
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);
x = n_randtest(state);
prec = 2 + n_randint(state, 2000);
arb_set_ui(b, x);
arb_div_ui(c, a, x, prec);
arb_div(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);
}
/* aliasing */
for (iter = 0; iter < 10000; iter++)
{
arb_t a, b, c;
ulong x;
long prec;
arb_init(a);
arb_init(b);
arb_init(c);
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);
x = n_randtest(state);
prec = 2 + n_randint(state, 2000);
arb_set_ui(b, x);
arb_div_ui(c, a, x, prec);
arb_div_ui(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);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("cos_pi_fmpq_algebraic....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
arb_t c1, c2;
ulong p, q, g;
slong prec;
prec = 2 + n_randint(state, 5000);
q = 1 + n_randint(state, 500);
p = n_randint(state, q / 2 + 1);
g = n_gcd(q, p);
q /= g;
p /= g;
arb_init(c1);
arb_init(c2);
_arb_cos_pi_fmpq_algebraic(c1, p, q, prec);
arb_const_pi(c2, prec);
arb_mul_ui(c2, c2, p, prec);
arb_div_ui(c2, c2, q, prec);
arb_cos(c2, c2, prec);
if (!arb_overlaps(c1, c2))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("p/q = %wu/%wu", p, q); flint_printf("\n\n");
flint_printf("c1 = "); arb_printd(c1, 15); flint_printf("\n\n");
flint_printf("c2 = "); arb_printd(c2, 15); flint_printf("\n\n");
abort();
}
if (arb_rel_accuracy_bits(c1) < prec - 2)
{
flint_printf("FAIL: accuracy\n\n");
flint_printf("p/q = %wu/%wu", p, q); flint_printf("\n\n");
flint_printf("prec=%wd eff=%wd\n", prec, arb_rel_accuracy_bits(c1));
flint_printf("c1 = "); arb_printd(c1, 15); flint_printf("\n\n");
flint_printf("c2 = "); arb_printd(c2, 15); flint_printf("\n\n");
abort();
}
arb_clear(c1);
arb_clear(c2);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
gc_integrals_precomp(acb_ptr res, acb_srcptr u, slong d1, slong d, slong g, const gc_int_t gc, int flag, slong prec)
{
slong l;
arb_t w, x;
acb_t y, yxi;
void (*sqrt_pol) (acb_t y, acb_srcptr u, slong d1, slong d,
const arb_t x, slong prec);
arb_init(w);
arb_init(x);
acb_init(y);
acb_init(yxi);
#if DEBUG
flint_printf("\ngc integral : d1 = %ld, d = %ld, g = %ld, n = %ld, prec = %ld",
d1, d, g, gc->n, prec);
#endif
sqrt_pol = &sqrt_pol_turn;
if (flag & AJ_ROOT_DEF)
sqrt_pol = &sqrt_pol_def;
else if (flag & AJ_ROOT_TURN)
sqrt_pol = &sqrt_pol_turn;
/* compute integral */
_acb_vec_zero(res, g);
for (l = 0; l < gc->len; l++)
{
/* compute 1/y(x) */
sqrt_pol(y, u, d1, d, gc->x + l, prec);
acb_inv(y, y, prec);
/* differentials */
acb_vec_add_geom_arb(res, g, y, gc->x + l, prec);
/* now on -x */
arb_neg(x, gc->x + l);
sqrt_pol(y, u, d1, d, x, prec);
acb_inv(y, y, prec);
acb_vec_add_geom_arb(res, g, y, x, prec);
}
if (gc->n % 2)
{
arb_zero(x);
/* FIXME: pb with turn */
sqrt_pol_def(y, u, d1, d, x, prec);
#if DEBUG > 1
flint_printf("\nend integration sum");
_acb_vec_printd(res, g, 30, "\n");
flint_printf("\nroots (d1=%ld, d=%ld)\n",d1,d);
_acb_vec_printd(u, d, 30, "\n");
flint_printf("\n -> y = ");
acb_printd(y, 30);
#endif
acb_inv(y, y, prec);
acb_add(res + 0, res + 0, y, prec);
}
/* multiply by weight = Pi / n */
arb_const_pi(w, prec);
arb_div_ui(w, w, gc->n, prec);
_acb_vec_scalar_mul_arb(res, res, g, w, prec);
#if DEBUG > 1
flint_printf("\nend integration ");
_acb_vec_printd(res, g, 30, "\n");
#endif
arb_clear(x);
arb_clear(w);
acb_clear(y);
acb_clear(yxi);
}
void
acb_calc_cauchy_bound(arb_t bound, acb_calc_func_t func, void * param,
const acb_t x, const arb_t radius, slong maxdepth, slong prec)
{
slong i, n, depth, wp;
arb_t pi, theta, v, s1, c1, s2, c2, st, ct;
acb_t t, u;
arb_t b;
arb_init(pi);
arb_init(theta);
arb_init(v);
arb_init(s1);
arb_init(c1);
arb_init(s2);
arb_init(c2);
arb_init(st);
arb_init(ct);
acb_init(t);
acb_init(u);
arb_init(b);
wp = prec + 20;
arb_const_pi(pi, wp);
arb_zero_pm_inf(b);
for (depth = 0, n = 16; depth < maxdepth; n *= 2, depth++)
{
arb_zero(b);
/* theta = 2 pi / n */
arb_div_ui(theta, pi, n, wp);
arb_mul_2exp_si(theta, theta, 1);
/* sine and cosine of i*theta and (i+1)*theta */
arb_zero(s1);
arb_one(c1);
arb_sin_cos(st, ct, theta, wp);
arb_set(s2, st);
arb_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 */
arb_union(acb_realref(t), c1, c2, wp);
arb_union(acb_imagref(t), s1, s2, wp);
acb_mul_arb(t, t, radius, wp);
acb_add(t, t, x, prec);
/* next angle */
arb_mul(v, c2, ct, wp);
arb_mul(c1, s2, st, wp);
arb_sub(c1, v, c1, wp);
arb_mul(v, c2, st, wp);
arb_mul(s1, s2, ct, wp);
arb_add(s1, v, s1, wp);
arb_swap(c1, c2);
arb_swap(s1, s2);
func(u, t, param, 1, prec);
acb_abs(v, u, prec);
arb_add(b, b, v, prec);
}
arb_div_ui(b, b, n, prec);
if (arb_is_positive(b))
break;
}
arb_set(bound, b);
arb_clear(pi);
arb_clear(theta);
arb_clear(v);
acb_clear(t);
acb_clear(u);
arb_clear(b);
arb_clear(s1);
arb_clear(c1);
arb_clear(s2);
arb_clear(c2);
arb_clear(st);
arb_clear(ct);
}
void
_arb_poly_exp_series(arb_ptr f, arb_srcptr h, slong hlen, slong n, slong prec)
{
hlen = FLINT_MIN(hlen, n);
if (hlen == 1)
{
arb_exp(f, h, prec);
_arb_vec_zero(f + 1, n - 1);
}
else if (n == 2)
{
arb_exp(f, h, prec);
arb_mul(f + 1, f, h + 1, prec); /* safe since hlen >= 2 */
}
else if (_arb_vec_is_zero(h + 1, hlen - 2)) /* h = a + bx^d */
{
slong i, j, d = hlen - 1;
arb_t t;
arb_init(t);
arb_set(t, h + d);
arb_exp(f, h, prec);
for (i = 1, j = d; j < n; j += d, i++)
{
arb_mul(f + j, f + j - d, t, prec);
arb_div_ui(f + j, f + j, i, prec);
_arb_vec_zero(f + j - d + 1, hlen - 2);
}
_arb_vec_zero(f + j - d + 1, n - (j - d + 1));
arb_clear(t);
}
else if (hlen <= arb_poly_newton_exp_cutoff)
{
_arb_poly_exp_series_basecase(f, h, hlen, n, prec);
}
else
{
arb_ptr g, t;
arb_t u;
int fix;
g = _arb_vec_init((n + 1) / 2);
fix = (hlen < n || h == f || !arb_is_zero(h));
if (fix)
{
t = _arb_vec_init(n);
_arb_vec_set(t + 1, h + 1, hlen - 1);
}
else
t = (arb_ptr) h;
arb_init(u);
arb_exp(u, h, prec);
_arb_poly_exp_series_newton(f, g, t, n, prec, 0, arb_poly_newton_exp_cutoff);
if (!arb_is_one(u))
_arb_vec_scalar_mul(f, f, n, u, prec);
_arb_vec_clear(g, (n + 1) / 2);
if (fix)
_arb_vec_clear(t, n);
arb_clear(u);
}
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("bernoulli_poly_ui....");
fflush(stdout);
flint_randinit(state);
/* test multiplication theorem */
for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++)
{
arb_t x, t, res1, res2;
ulong n, m, k;
slong prec;
n = n_randint(state, 50);
m = 1 + n_randint(state, 5);
prec = 2 + n_randint(state, 200);
arb_init(x);
arb_init(t);
arb_init(res1);
arb_init(res2);
arb_randtest(x, state, 2 + n_randint(state, 200), 20);
arb_randtest(res1, state, 2 + n_randint(state, 200), 20);
arb_mul_ui(t, x, m, prec);
arb_bernoulli_poly_ui(res1, n, t, prec);
arb_zero(res2);
for (k = 0; k < m; k++)
{
arb_set_ui(t, k);
arb_div_ui(t, t, m, prec);
arb_add(t, t, x, prec);
arb_bernoulli_poly_ui(t, n, t, prec);
arb_add(res2, res2, t, prec);
}
if (n > 0)
{
arb_ui_pow_ui(t, m, n - 1, prec);
arb_mul(res2, res2, t, prec);
}
else
{
arb_div_ui(res2, res2, m, prec);
}
if (!arb_overlaps(res1, res2))
{
flint_printf("FAIL: overlap\n\n");
flint_printf("n = %wu, m = %wu\n\n", n, m);
flint_printf("x = "); arb_printd(x, 15); flint_printf("\n\n");
flint_printf("res1 = "); arb_printd(res1, 15); flint_printf("\n\n");
flint_printf("res2 = "); arb_printd(res2, 15); flint_printf("\n\n");
abort();
}
arb_clear(x);
arb_clear(t);
arb_clear(res1);
arb_clear(res2);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void Lib_Arb_Div_Ui(ArbPtr f, ArbPtr g, uint32_t x, int32_t prec)
{
arb_div_ui( (arb_ptr) f, (arb_ptr) g, x, prec);
}
void
_acb_poly_zeta_em_bound(arb_ptr bound, const acb_t s, const acb_t a, ulong N, ulong M, slong len, slong wp)
{
arb_t K, C, AN, S2M;
arb_ptr F, R;
slong k;
arb_srcptr alpha = acb_realref(a);
arb_srcptr beta = acb_imagref(a);
arb_srcptr sigma = acb_realref(s);
arb_srcptr tau = acb_imagref(s);
arb_init(AN);
arb_init(S2M);
/* require alpha + N > 1, sigma + 2M > 1 */
arb_add_ui(AN, alpha, N - 1, wp);
arb_add_ui(S2M, sigma, 2*M - 1, wp);
if (!arb_is_positive(AN) || !arb_is_positive(S2M) || N < 1 || M < 1)
{
arb_clear(AN);
arb_clear(S2M);
for (k = 0; k < len; k++)
arb_pos_inf(bound + k);
return;
}
/* alpha + N, sigma + 2M */
arb_add_ui(AN, AN, 1, wp);
arb_add_ui(S2M, S2M, 1, wp);
R = _arb_vec_init(len);
F = _arb_vec_init(len);
arb_init(K);
arb_init(C);
/* bound for power integral */
bound_C(C, AN, beta, wp);
bound_K(K, AN, beta, tau, wp);
bound_I(R, AN, S2M, C, len, wp);
for (k = 0; k < len; k++)
{
arb_mul(R + k, R + k, K, wp);
arb_div_ui(K, K, k + 1, wp);
}
/* bound for rising factorial */
bound_rfac(F, s, 2*M, len, wp);
/* product (TODO: only need upper bound; write a function for this) */
_arb_poly_mullow(bound, F, len, R, len, len, wp);
/* bound for bernoulli polynomials, 4 / (2pi)^(2M) */
arb_const_pi(C, wp);
arb_mul_2exp_si(C, C, 1);
arb_pow_ui(C, C, 2 * M, wp);
arb_ui_div(C, 4, C, wp);
_arb_vec_scalar_mul(bound, bound, len, C, wp);
arb_clear(K);
arb_clear(C);
arb_clear(AN);
arb_clear(S2M);
_arb_vec_clear(R, len);
_arb_vec_clear(F, len);
}