int main() { #define ni 5 slong n, i, f; double b[ni][2] = { { 0, 1}, {-1, 1}, {0, 10}, {-20, 3}, {0, 3.14} }; const slong prec = 40; #define nf 5 arb_func_t func[nf] = { (arb_func_t)&f_1x2, (arb_func_t)&f_pol, (arb_func_t)&f_thsh, (arb_func_t)&f_thsh_shift, (arb_func_t)&f_aj }; #if VERBOSE char * fn[nf] = { "1/(1+x^2)", "1/p(x)", "th(sh(x))", "th(sh(x+.7I))" , "1/y" }; #endif params_t p[nf]; arf_t tmin, tmax; mag_t max; flint_printf("max_func..."); fflush(stdout); p[1].len = 3; p[1].z = _acb_vec_init(3); acb_set_d_d(p[1].z + 0, 2, 1); acb_set_d_d(p[1].z + 1, 2, .1); acb_set_d_d(p[1].z + 2, 1, .1); p[4] = p[1]; arf_init(tmin); arf_init(tmax); mag_init(max); for (i = 0; i < ni; i++) { arf_set_d(tmin, b[i][0]); arf_set_d(tmax, b[i][1]); for (f = 0; f < nf; f++) { for (n = 5; n < 100; n *= 2) { slong count; count = mag_func_arf(max, func[f], (void *)&p[f], tmin, tmax, n, prec); #if VERBOSE flint_printf("\nmax %s on [%lf, %lf] <= ",fn[f],b[i][0],b[i][1]); mag_printd(max,8); flint_printf(" [asked %ld, did %ld]", n, count); #endif } } } mag_clear(max); arf_clear(tmin); arf_clear(tmax); printf("PASS\n"); return 0; }
void acb_lambertw_principal_d(acb_t res, const acb_t z) { double za, zb, wa, wb, ewa, ewb, t, u, q, r; int k, maxk = 15; za = arf_get_d(arb_midref(acb_realref(z)), ARF_RND_DOWN); zb = arf_get_d(arb_midref(acb_imagref(z)), ARF_RND_DOWN); /* make sure we end up on the right branch */ if (za < -0.367 && zb > -1e-20 && zb <= 0.0 && arf_sgn(arb_midref(acb_imagref(z))) < 0) zb = -1e-20; wa = za; wb = zb; if (fabs(wa) > 2.0 || fabs(wb) > 2.0) { t = atan2(wb, wa); wa = 0.5 * log(wa * wa + wb * wb); wb = t; } else if (fabs(wa) > 0.25 || fabs(wb) > 0.25) { /* We have W(z) ~= -1 + (2(ez+1))^(1/2) near the branch point. Changing the exponent to 1/4 gives a much worse local guess which however does the job on a larger domain. */ wa *= 5.43656365691809; wb *= 5.43656365691809; wa += 2.0; t = atan2(wb, wa); r = pow(wa * wa + wb * wb, 0.125); wa = r * cos(0.25 * t); wb = r * sin(0.25 * t); wa -= 1.0; } for (k = 0; k < maxk; k++) { t = exp(wa); ewa = t * cos(wb); ewb = t * sin(wb); t = (ewa * wa - ewb * wb); q = t + ewa; t -= za; u = (ewb * wa + ewa * wb); r = u + ewb; u -= zb; ewa = q * t + r * u; ewb = q * u - r * t; r = 1.0 / (q * q + r * r); ewa *= r; ewb *= r; if ((ewa*ewa + ewb*ewb) < (wa*wa + wb*wb) * 1e-12) maxk = FLINT_MIN(maxk, k + 2); wa -= ewa; wb -= ewb; } acb_set_d_d(res, wa, wb); }
int main() { flint_printf("spherical_y...."); fflush(stdout); { slong i, n, m; acb_t z, w, x, y; acb_init(z); acb_init(w); acb_init(x); acb_init(y); i = 0; arb_set_str(acb_realref(x), "0.2", 64); arb_set_str(acb_imagref(x), "0.3", 64); arb_set_str(acb_realref(y), "0.3", 64); arb_set_str(acb_imagref(y), "0.4", 64); for (n = -4; n <= 4; n++) { for (m = -4; m <= 4; m++) { acb_hypgeom_spherical_y(z, n, m, x, y, 64); acb_set_d_d(w, testdata[2 * i], testdata[2 * i + 1]); mag_set_d(arb_radref(acb_realref(w)), 1e-13); mag_set_d(arb_radref(acb_imagref(w)), 1e-13); if (!acb_overlaps(z, w)) { flint_printf("FAIL: value\n\n"); flint_printf("n = %wd, m = %wd\n", n, m); flint_printf("z = "); acb_printd(z, 20); flint_printf("\n\n"); flint_printf("w = "); acb_printd(w, 20); flint_printf("\n\n"); abort(); } i++; } } acb_clear(z); acb_clear(w); acb_clear(x); acb_clear(y); } flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
int main(int argc, char *argv[]) { int function, i, numtests; slong prec, goal; double t, total, logtotal; acb_t a, b, c, z, r, s; acb_init(a); acb_init(b); acb_init(c); acb_init(z); acb_init(r); acb_init(s); /* J(0,pi x) I(0,pi x) K(0,pi x) */ for (function = 0; function < 3; function++) { total = 0.0; logtotal = 0.0; if (function < 2) numtests = 40; else numtests = 30; for (i = 0; i < numtests; i++) { // printf("%2d ", i + 1); fflush(stdout); TIMEIT_START prec = 96; for (;;) { if (function == 0) { acb_set_d_d(a, input_1f1[i][0], input_1f1[i][1]); acb_set_d_d(b, input_1f1[i][2], input_1f1[i][3]); acb_set_d_d(z, input_1f1[i][4], input_1f1[i][5]); acb_hypgeom_m(r, a, b, z, 0, prec); } else if (function == 1) { acb_set_d_d(a, input_1f1[i][0], input_1f1[i][1]); acb_set_d_d(b, input_1f1[i][2], input_1f1[i][3]); acb_set_d_d(z, input_1f1[i][4], input_1f1[i][5]); acb_hypgeom_u(r, a, b, z, prec); } else if (function == 2) { acb_set_d_d(a, input_2f1[i][0], input_2f1[i][1]); acb_set_d_d(b, input_2f1[i][2], input_2f1[i][3]); acb_set_d_d(c, input_2f1[i][4], input_2f1[i][5]); acb_set_d_d(z, input_2f1[i][6], input_2f1[i][7]); acb_hypgeom_2f1(r, a, b, c, z, 0, prec); } else { acb_set_d_d(a, input_2f1[i][0], input_2f1[i][1]); acb_set_d_d(b, input_2f1[i][2], input_2f1[i][3]); acb_set_d_d(c, input_2f1[i][4], input_2f1[i][5]); acb_set_d_d(z, input_2f1[i][6], input_2f1[i][7]); acb_mul_2exp_si(z, z, 1); acb_sub_ui(z, z, 1, prec); acb_neg(z, z); acb_hypgeom_legendre_q(r, a, c, z, 0, prec); } if (function == 3) { if (acb_rel_accuracy_bits(r) >= goal) break; } else { if (arb_can_round_arf(acb_realref(r), goal, ARF_RND_NEAR) && arb_can_round_arf(acb_imagref(r), goal, ARF_RND_NEAR)) break; } prec *= 2; } TIMEIT_STOP_VAL(t) total += t; logtotal += log(t); #if 1 printf("%8g, ", t); #else printf("%8ld %8g ", prec, t); _acb_print(r, 25); printf("\n"); #endif } printf("---------------------------------------------------------------\n"); printf("Mean %g s; geometric mean %g\n", total, exp(logtotal / numtests)); printf("---------------------------------------------------------------\n"); } acb_clear(a); acb_clear(b); acb_clear(c); acb_clear(z); acb_clear(r); acb_clear(s); }
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; }