void
acb_cot(acb_t r, const acb_t z, slong prec)
{
if (arb_is_zero(acb_imagref(z)))
{
arb_cot(acb_realref(r), acb_realref(z), prec);
arb_zero(acb_imagref(r));
}
else if (arb_is_zero(acb_realref(z)))
{
arb_coth(acb_imagref(r), acb_imagref(z), prec);
arb_neg(acb_imagref(r), acb_imagref(r));
arb_zero(acb_realref(r));
}
else
{
acb_t t;
acb_init(t);
if (arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 0) < 0)
{
acb_sin_cos(r, t, z, prec + 4);
acb_div(r, t, r, prec);
}
else
{
acb_mul_2exp_si(t, z, 1);
if (arf_sgn(arb_midref(acb_imagref(z))) > 0)
{
acb_mul_onei(t, t);
acb_exp(t, t, prec + 4);
acb_sub_ui(r, t, 1, prec + 4);
acb_div(r, t, r, prec + 4);
acb_mul_2exp_si(r, r, 1);
acb_sub_ui(r, r, 1, prec);
acb_mul_onei(r, r);
}
else
{
acb_div_onei(t, t);
acb_exp(t, t, prec + 4);
acb_sub_ui(r, t, 1, prec + 4);
acb_div(r, t, r, prec + 4);
acb_mul_2exp_si(r, r, 1);
acb_sub_ui(r, r, 1, prec);
acb_div_onei(r, r);
}
}
acb_clear(t);
}
}
void
acb_atanh(acb_t res, const acb_t z, slong prec)
{
acb_mul_onei(res, z);
acb_atan(res, res, prec);
acb_div_onei(res, res);
}
void
integrals_edge_factors_gc(acb_ptr res, const acb_t cab, const acb_t ba2, sec_t c, slong prec)
{
slong i;
acb_t cj, ci;
acb_init(cj);
acb_init(ci);
/* polynomial shift */
acb_vec_polynomial_shift(res, cab, c.g, prec);
/* constants cj, j = 1 */
/* c_1 = (1-zeta^-1) ba2^(-d/2) (-I)^i
* = 2 / ba2^(d/2) */
acb_pow_ui(cj, ba2, c.d / 2, prec);
if (c.d % 2)
{
acb_t t;
acb_init(t);
acb_sqrt(t, ba2, prec);
acb_mul(cj, cj, t, prec);
acb_clear(t);
}
acb_inv(cj, cj, prec);
acb_mul_2exp_si(cj, cj, 1);
_acb_vec_scalar_mul(res, res, c.g, cj, prec);
/* constant ci = -I * ba2*/
acb_one(ci);
for (i = 1; i < c.g; i++)
{
acb_mul(ci, ci, ba2, prec);
acb_div_onei(ci, ci);
acb_mul(res + i, res + i, ci, prec);
}
acb_clear(ci);
acb_clear(cj);
}
int main(int argc, char *argv[])
{
acb_t s, t, a, b;
mag_t tol;
slong prec, goal;
slong N;
ulong k;
int integral, ifrom, ito;
int i, twice, havegoal, havetol;
acb_calc_integrate_opt_t options;
ifrom = ito = -1;
for (i = 1; i < argc; i++)
{
if (!strcmp(argv[i], "-i"))
{
if (!strcmp(argv[i+1], "all"))
{
ifrom = 0;
ito = NUM_INTEGRALS - 1;
}
else
{
ifrom = ito = atol(argv[i+1]);
if (ito < 0 || ito >= NUM_INTEGRALS)
flint_abort();
}
}
}
if (ifrom == -1)
{
flint_printf("Compute integrals using acb_calc_integrate.\n");
flint_printf("Usage: integrals -i n [-prec p] [-tol eps] [-twice] [...]\n\n");
flint_printf("-i n - compute integral n (0 <= n <= %d), or \"-i all\"\n", NUM_INTEGRALS - 1);
flint_printf("-prec p - precision in bits (default p = 64)\n");
flint_printf("-goal p - approximate relative accuracy goal (default p)\n");
flint_printf("-tol eps - approximate absolute error goal (default 2^-p)\n");
flint_printf("-twice - run twice (to see overhead of computing nodes)\n");
flint_printf("-heap - use heap for subinterval queue\n");
flint_printf("-verbose - show information\n");
flint_printf("-verbose2 - show more information\n");
flint_printf("-deg n - use quadrature degree up to n\n");
flint_printf("-eval n - limit number of function evaluations to n\n");
flint_printf("-depth n - limit subinterval queue size to n\n\n");
flint_printf("Implemented integrals:\n");
for (integral = 0; integral < NUM_INTEGRALS; integral++)
flint_printf("I%d = %s\n", integral, descr[integral]);
flint_printf("\n");
return 1;
}
acb_calc_integrate_opt_init(options);
prec = 64;
twice = 0;
goal = 0;
havetol = havegoal = 0;
acb_init(a);
acb_init(b);
acb_init(s);
acb_init(t);
mag_init(tol);
for (i = 1; i < argc; i++)
{
if (!strcmp(argv[i], "-prec"))
{
prec = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-twice"))
{
twice = 1;
}
else if (!strcmp(argv[i], "-goal"))
{
goal = atol(argv[i+1]);
if (goal < 0)
{
flint_printf("expected goal >= 0\n");
return 1;
}
havegoal = 1;
}
else if (!strcmp(argv[i], "-tol"))
{
arb_t x;
arb_init(x);
arb_set_str(x, argv[i+1], 10);
arb_get_mag(tol, x);
arb_clear(x);
havetol = 1;
}
else if (!strcmp(argv[i], "-deg"))
{
options->deg_limit = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-eval"))
{
options->eval_limit = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-depth"))
{
options->depth_limit = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-verbose"))
{
options->verbose = 1;
}
else if (!strcmp(argv[i], "-verbose2"))
{
options->verbose = 2;
}
else if (!strcmp(argv[i], "-heap"))
{
options->use_heap = 1;
}
}
if (!havegoal)
goal = prec;
if (!havetol)
mag_set_ui_2exp_si(tol, 1, -prec);
for (integral = ifrom; integral <= ito; integral++)
{
flint_printf("I%d = %s ...\n", integral, descr[integral]);
for (i = 0; i < 1 + twice; i++)
{
TIMEIT_ONCE_START
switch (integral)
{
case 0:
acb_set_d(a, 0);
acb_set_d(b, 100);
acb_calc_integrate(s, f_sin, NULL, a, b, goal, tol, options, prec);
break;
case 1:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_atanderiv, NULL, a, b, goal, tol, options, prec);
acb_mul_2exp_si(s, s, 2);
break;
case 2:
acb_set_d(a, 0);
acb_one(b);
acb_mul_2exp_si(b, b, goal);
acb_calc_integrate(s, f_atanderiv, NULL, a, b, goal, tol, options, prec);
arb_add_error_2exp_si(acb_realref(s), -goal);
acb_mul_2exp_si(s, s, 1);
break;
case 3:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_circle, NULL, a, b, goal, tol, options, prec);
acb_mul_2exp_si(s, s, 2);
break;
case 4:
acb_set_d(a, 0);
acb_set_d(b, 8);
acb_calc_integrate(s, f_rump, NULL, a, b, goal, tol, options, prec);
break;
case 5:
acb_set_d(a, 1);
acb_set_d(b, 101);
acb_calc_integrate(s, f_floor, NULL, a, b, goal, tol, options, prec);
break;
case 6:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_helfgott, NULL, a, b, goal, tol, options, prec);
break;
case 7:
acb_zero(s);
acb_set_d_d(a, -1.0, -1.0);
acb_set_d_d(b, 2.0, -1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 2.0, -1.0);
acb_set_d_d(b, 2.0, 1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 2.0, 1.0);
acb_set_d_d(b, -1.0, 1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, -1.0, 1.0);
acb_set_d_d(b, -1.0, -1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_const_pi(t, prec);
acb_div(s, s, t, prec);
acb_mul_2exp_si(s, s, -1);
acb_div_onei(s, s);
break;
case 8:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_essing, NULL, a, b, goal, tol, options, prec);
break;
case 9:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_essing2, NULL, a, b, goal, tol, options, prec);
break;
case 10:
acb_set_d(a, 0);
acb_set_d(b, 10000);
acb_calc_integrate(s, f_factorial1000, NULL, a, b, goal, tol, options, prec);
break;
case 11:
acb_set_d_d(a, 1.0, 0.0);
acb_set_d_d(b, 1.0, 1000.0);
acb_calc_integrate(s, f_gamma, NULL, a, b, goal, tol, options, prec);
break;
case 12:
acb_set_d(a, -10.0);
acb_set_d(b, 10.0);
acb_calc_integrate(s, f_sin_plus_small, NULL, a, b, goal, tol, options, prec);
break;
case 13:
acb_set_d(a, -1020.0);
acb_set_d(b, -1010.0);
acb_calc_integrate(s, f_exp, NULL, a, b, goal, tol, options, prec);
break;
case 14:
acb_set_d(a, 0);
acb_set_d(b, ceil(sqrt(goal * 0.693147181) + 1.0));
acb_calc_integrate(s, f_gaussian, NULL, a, b, goal, tol, options, prec);
acb_mul(b, b, b, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 15:
acb_set_d(a, 0.0);
acb_set_d(b, 1.0);
acb_calc_integrate(s, f_spike, NULL, a, b, goal, tol, options, prec);
break;
case 16:
acb_set_d(a, 0.0);
acb_set_d(b, 8.0);
acb_calc_integrate(s, f_monster, NULL, a, b, goal, tol, options, prec);
break;
case 17:
acb_set_d(a, 0);
acb_set_d(b, ceil(goal * 0.693147181 + 1.0));
acb_calc_integrate(s, f_sech, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
acb_mul_2exp_si(b, b, 1);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 18:
acb_set_d(a, 0);
acb_set_d(b, ceil(goal * 0.693147181 / 3.0 + 2.0));
acb_calc_integrate(s, f_sech3, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_mul_ui(b, b, 3, prec);
acb_exp(b, b, prec);
acb_mul_2exp_si(b, b, 3);
acb_div_ui(b, b, 3, prec);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 19:
if (goal < 0)
abort();
/* error bound 2^-N (1+N) when truncated at 2^-N */
N = goal + FLINT_BIT_COUNT(goal);
acb_one(a);
acb_mul_2exp_si(a, a, -N);
acb_one(b);
acb_calc_integrate(s, f_log_div1p, NULL, a, b, goal, tol, options, prec);
acb_set_ui(b, N + 1);
acb_mul_2exp_si(b, b, -N);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 20:
if (goal < 0)
abort();
/* error bound (N+1) exp(-N) when truncated at N */
N = goal + FLINT_BIT_COUNT(goal);
acb_zero(a);
acb_set_ui(b, N);
acb_calc_integrate(s, f_log_div1p_transformed, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
acb_mul_ui(b, b, N + 1, prec);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 21:
acb_zero(s);
N = 10;
acb_set_d_d(a, 0.5, -0.5);
acb_set_d_d(b, 0.5, 0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 0.5, 0.5);
acb_set_d_d(b, -0.5, 0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, -0.5, 0.5);
acb_set_d_d(b, -0.5, -0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, -0.5, -0.5);
acb_set_d_d(b, 0.5, -0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_const_pi(t, prec);
acb_div(s, s, t, prec);
acb_mul_2exp_si(s, s, -1);
acb_div_onei(s, s);
break;
case 22:
acb_zero(s);
N = 1000;
acb_set_d_d(a, 100.0, 0.0);
acb_set_d_d(b, 100.0, N);
acb_calc_integrate(t, f_zeta_frac, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 100, N);
acb_set_d_d(b, 0.5, N);
acb_calc_integrate(t, f_zeta_frac, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_div_onei(s, s);
arb_zero(acb_imagref(s));
acb_set_ui(t, N);
acb_dirichlet_hardy_theta(t, t, NULL, NULL, 1, prec);
acb_add(s, s, t, prec);
acb_const_pi(t, prec);
acb_div(s, s, t, prec);
acb_add_ui(s, s, 1, prec);
break;
case 23:
acb_set_d(a, 0.0);
acb_set_d(b, 1000.0);
acb_calc_integrate(s, f_lambertw, NULL, a, b, goal, tol, options, prec);
break;
case 24:
acb_set_d(a, 0.0);
acb_const_pi(b, prec);
acb_calc_integrate(s, f_max_sin_cos, NULL, a, b, goal, tol, options, prec);
break;
case 25:
acb_set_si(a, -1);
acb_set_si(b, 1);
acb_calc_integrate(s, f_erf_bent, NULL, a, b, goal, tol, options, prec);
break;
case 26:
acb_set_si(a, -10);
acb_set_si(b, 10);
acb_calc_integrate(s, f_airy_ai, NULL, a, b, goal, tol, options, prec);
break;
case 27:
acb_set_si(a, 0);
acb_set_si(b, 10);
acb_calc_integrate(s, f_horror, NULL, a, b, goal, tol, options, prec);
break;
case 28:
acb_set_d_d(a, -1, -1);
acb_set_d_d(b, -1, 1);
acb_calc_integrate(s, f_sqrt, NULL, a, b, goal, tol, options, prec);
break;
case 29:
acb_set_d(a, 0);
acb_set_d(b, ceil(sqrt(goal * 0.693147181) + 1.0));
acb_calc_integrate(s, f_gaussian_twist, NULL, a, b, goal, tol, options, prec);
acb_mul(b, b, b, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
arb_add_error(acb_realref(s), acb_realref(b));
arb_add_error(acb_imagref(s), acb_realref(b));
break;
case 30:
acb_set_d(a, 0);
acb_set_d(b, ceil(goal * 0.693147181 + 1.0));
acb_calc_integrate(s, f_exp_airy, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
acb_mul_2exp_si(b, b, 1);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 31:
acb_zero(a);
acb_const_pi(b, prec);
acb_calc_integrate(s, f_sin_cos_frac, NULL, a, b, goal, tol, options, prec);
break;
case 32:
acb_zero(a);
acb_set_ui(b, 3);
acb_calc_integrate(s, f_sin_near_essing, NULL, a, b, goal, tol, options, prec);
break;
case 33:
acb_zero(a);
acb_zero(b);
k = 3;
scaled_bessel_select_N(acb_realref(b), k, prec);
acb_calc_integrate(s, f_scaled_bessel, &k, a, b, goal, tol, options, prec);
scaled_bessel_tail_bound(acb_realref(a), k, acb_realref(b), prec);
arb_add_error(acb_realref(s), acb_realref(a));
break;
case 34:
acb_zero(a);
acb_zero(b);
k = 15;
scaled_bessel_select_N(acb_realref(b), k, prec);
acb_calc_integrate(s, f_scaled_bessel, &k, a, b, goal, tol, options, prec);
scaled_bessel_tail_bound(acb_realref(a), k, acb_realref(b), prec);
arb_add_error(acb_realref(s), acb_realref(a));
break;
case 35:
acb_set_d_d(a, -1, -1);
acb_set_d_d(b, -1, 1);
acb_calc_integrate(s, f_rsqrt, NULL, a, b, goal, tol, options, prec);
break;
default:
abort();
}
TIMEIT_ONCE_STOP
}
flint_printf("I%d = ", integral);
acb_printn(s, 3.333 * prec, 0);
flint_printf("\n\n");
}
acb_clear(a);
acb_clear(b);
acb_clear(s);
acb_clear(t);
mag_clear(tol);
flint_cleanup();
return 0;
}
void
acb_hypgeom_fresnel_erf(acb_t res1, acb_t res2, const acb_t z, slong prec)
{
acb_t t, u, v, w1, w2;
acb_init(t);
acb_init(v);
acb_init(w1);
if (arb_is_zero(acb_imagref(z)))
{
acb_mul_onei(t, z);
acb_add(w1, z, t, 2 * prec);
acb_hypgeom_erf(t, w1, prec + 4);
acb_mul_2exp_si(t, t, 1);
acb_mul_onei(v, t);
acb_add(t, t, v, prec);
if (res1 != NULL) acb_set_arb(res1, acb_realref(t));
if (res2 != NULL) acb_set_arb(res2, acb_imagref(t));
}
else if (arb_is_zero(acb_realref(z)))
{
acb_mul_onei(t, z);
acb_sub(w1, t, z, 2 * prec);
acb_hypgeom_erf(t, w1, prec + 4);
acb_mul_2exp_si(t, t, 1);
acb_mul_onei(v, t);
acb_add(t, t, v, prec);
if (res1 != NULL) acb_set_arb(res1, acb_realref(t));
if (res1 != NULL) acb_mul_onei(res1, res1);
if (res2 != NULL) acb_set_arb(res2, acb_imagref(t));
if (res2 != NULL) acb_div_onei(res2, res2);
}
else
{
acb_init(u);
acb_init(w2);
/* w1 = (1+i)z, w2 = (1-i)z */
acb_mul_onei(t, z);
acb_add(w1, z, t, 2 * prec);
acb_sub(w2, z, t, 2 * prec);
acb_hypgeom_erf(t, w1, prec + 4);
acb_hypgeom_erf(u, w2, prec + 4);
/* S = (1+i) (t - ui) = (1+i) t + (1-i) u */
/* C = (1-i) (t + ui) = (1-i) t + (1+i) u */
acb_mul_onei(v, t);
if (res1 != NULL) acb_add(res1, t, v, prec);
if (res2 != NULL) acb_sub(res2, t, v, prec);
acb_mul_onei(v, u);
if (res1 != NULL) acb_add(res1, res1, u, prec);
if (res1 != NULL) acb_sub(res1, res1, v, prec);
if (res2 != NULL) acb_add(res2, res2, u, prec);
if (res2 != NULL) acb_add(res2, res2, v, prec);
acb_clear(u);
acb_clear(w2);
}
acb_clear(t);
acb_clear(v);
acb_clear(w1);
}