void arb_div_arf(arb_t z, const arb_t x, const arf_t y, long prec) { mag_t zr, ym; int inexact; if (arf_is_zero(y)) { arb_zero_pm_inf(z); } else if (arb_is_exact(x)) { inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND); if (inexact) arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec); else mag_zero(arb_radref(z)); } else if (mag_is_inf(arb_radref(x))) { arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND); mag_inf(arb_radref(z)); } else { mag_init(ym); mag_init(zr); arf_get_mag_lower(ym, y); mag_div(zr, arb_radref(x), ym); inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND); if (inexact) arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec); else mag_swap(arb_radref(z), zr); mag_clear(ym); mag_clear(zr); } }
void arb_sqrtpos(arb_t z, const arb_t x, long prec) { if (!arb_is_finite(x)) { if (mag_is_zero(arb_radref(x)) && arf_is_pos_inf(arb_midref(x))) arb_pos_inf(z); else arb_zero_pm_inf(z); } else if (arb_contains_nonpositive(x)) { arf_t t; arf_init(t); arf_set_mag(t, arb_radref(x)); arf_add(t, arb_midref(x), t, MAG_BITS, ARF_RND_CEIL); if (arf_sgn(t) <= 0) { arb_zero(z); } else { arf_sqrt(t, t, MAG_BITS, ARF_RND_CEIL); arf_mul_2exp_si(t, t, -1); arf_set(arb_midref(z), t); arf_get_mag(arb_radref(z), t); } arf_clear(t); } else { arb_sqrt(z, x, prec); } arb_nonnegative_part(z, z, prec); }
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); }
slong _acb_poly_find_roots(acb_ptr roots, acb_srcptr poly, acb_srcptr initial, slong len, slong maxiter, slong prec) { slong iter, i, deg; slong rootmag, max_rootmag, correction, max_correction; deg = len - 1; if (deg == 0) { return 0; } else if (acb_contains_zero(poly + len - 1)) { /* if the leading coefficient contains zero, roots can be anywhere */ for (i = 0; i < deg; i++) { arb_zero_pm_inf(acb_realref(roots + i)); arb_zero_pm_inf(acb_imagref(roots + i)); } return 0; } else if (deg == 1) { acb_inv(roots + 0, poly + 1, prec); acb_mul(roots + 0, roots + 0, poly + 0, prec); acb_neg(roots + 0, roots + 0); return 1; } if (initial == NULL) _acb_poly_roots_initial_values(roots, deg, prec); else _acb_vec_set(roots, initial, deg); if (maxiter == 0) maxiter = 2 * deg + n_sqrt(prec); for (iter = 0; iter < maxiter; iter++) { max_rootmag = -ARF_PREC_EXACT; for (i = 0; i < deg; i++) { rootmag = _acb_get_mid_mag(roots + i); max_rootmag = FLINT_MAX(rootmag, max_rootmag); } _acb_poly_refine_roots_durand_kerner(roots, poly, len, prec); max_correction = -ARF_PREC_EXACT; for (i = 0; i < deg; i++) { correction = _acb_get_rad_mag(roots + i); max_correction = FLINT_MAX(correction, max_correction); } /* estimate the correction relative to the whole set of roots */ max_correction -= max_rootmag; /* flint_printf("ITER %wd MAX CORRECTION: %wd\n", iter, max_correction); */ if (max_correction < -prec / 2) maxiter = FLINT_MIN(maxiter, iter + 2); else if (max_correction < -prec / 3) maxiter = FLINT_MIN(maxiter, iter + 3); else if (max_correction < -prec / 4) maxiter = FLINT_MIN(maxiter, iter + 4); } return _acb_poly_validate_roots(roots, poly, len, prec); }