def mpc_gamma(x, prec, rounding=round_fast, p1=1): re, im = x if im == fzero: return mpf_gamma(re, prec, rounding, p1), fzero # More precision is needed for enormous x. sign, man, exp, bc = re isign, iman, iexp, ibc = im if re == fzero: size = iexp + ibc else: size = max(exp + bc, iexp + ibc) if size > 5: size = int(size * math.log(size, 2)) reflect = sign or (exp + bc < -1) wp = prec + max(0, size) + 25 # Near x = 0 pole (TODO: other poles) if p1: if size < -prec - 5: return mpc_add_mpf(mpc_div(mpc_one, x, 2*prec+10), \ mpf_neg(mpf_euler(2*prec+10)), prec, rounding) elif size < -5: wp += (-2 * size) if p1: # Should be done exactly! re_orig = re re = mpf_sub(re, fone, bc + abs(exp) + 2) x = re, im if reflect: # Reflection formula wp += 15 pi = mpf_pi(wp), fzero pix = mpc_mul(x, pi, wp) t = mpc_sin_pi(x, wp) u = mpc_sub(mpc_one, x, wp) g = mpc_gamma(u, wp) w = mpc_mul(t, g, wp) return mpc_div(pix, w, wp) # Extremely close to the real line? # XXX: reflection formula if iexp + ibc < -wp: a = mpf_gamma(re_orig, wp) b = mpf_psi0(re_orig, wp) gamma_diff = mpf_div(a, b, wp) return mpf_pos(a, prec, rounding), mpf_mul(gamma_diff, im, prec, rounding) sprec, a, c = get_spouge_coefficients(wp) s = spouge_sum_complex(re, im, sprec, a, c) # gamma = exp(log(x+a)*(x+0.5) - xpa) * s repa = mpf_add(re, from_int(a), wp) logxpa = mpc_log((repa, im), wp) reph = mpf_add(re, fhalf, wp) t = mpc_sub(mpc_mul(logxpa, (reph, im), wp), (repa, im), wp) t = mpc_mul(mpc_exp(t, wp), s, prec, rounding) return t
def mpc_gamma(x, prec, rounding=round_fast, p1=1): re, im = x if im == fzero: return mpf_gamma(re, prec, rounding, p1), fzero # More precision is needed for enormous x. sign, man, exp, bc = re isign, iman, iexp, ibc = im if re == fzero: size = iexp+ibc else: size = max(exp+bc, iexp+ibc) if size > 5: size = int(size * math.log(size,2)) reflect = sign or (exp+bc < -1) wp = prec + max(0, size) + 25 # Near x = 0 pole (TODO: other poles) if p1: if size < -prec-5: return mpc_add_mpf(mpc_div(mpc_one, x, 2*prec+10), \ mpf_neg(mpf_euler(2*prec+10)), prec, rounding) elif size < -5: wp += (-2*size) if p1: # Should be done exactly! re_orig = re re = mpf_sub(re, fone, bc+abs(exp)+2) x = re, im if reflect: # Reflection formula wp += 15 pi = mpf_pi(wp), fzero pix = mpc_mul(x, pi, wp) t = mpc_sin_pi(x, wp) u = mpc_sub(mpc_one, x, wp) g = mpc_gamma(u, wp) w = mpc_mul(t, g, wp) return mpc_div(pix, w, wp) # Extremely close to the real line? # XXX: reflection formula if iexp+ibc < -wp: a = mpf_gamma(re_orig, wp) b = mpf_psi0(re_orig, wp) gamma_diff = mpf_div(a, b, wp) return mpf_pos(a, prec, rounding), mpf_mul(gamma_diff, im, prec, rounding) sprec, a, c = get_spouge_coefficients(wp) s = spouge_sum_complex(re, im, sprec, a, c) # gamma = exp(log(x+a)*(x+0.5) - xpa) * s repa = mpf_add(re, from_int(a), wp) logxpa = mpc_log((repa, im), wp) reph = mpf_add(re, fhalf, wp) t = mpc_sub(mpc_mul(logxpa, (reph, im), wp), (repa, im), wp) t = mpc_mul(mpc_exp(t, wp), s, prec, rounding) return t
def mpc_ei(z, prec, rnd=round_fast, e1=False): if e1: z = mpc_neg(z) a, b = z asign, aman, aexp, abc = a bsign, bman, bexp, bbc = b if b == fzero: if e1: x = mpf_neg(mpf_ei(a, prec, rnd)) if not asign: y = mpf_neg(mpf_pi(prec, rnd)) else: y = fzero return x, y else: return mpf_ei(a, prec, rnd), fzero if a != fzero: if not aman or not bman: return (fnan, fnan) wp = prec + 40 amag = aexp+abc bmag = bexp+bbc zmag = max(amag, bmag) can_use_asymp = zmag > wp if not can_use_asymp: zabsint = abs(to_int(a)) + abs(to_int(b)) can_use_asymp = zabsint > int(wp*0.693) + 20 try: if can_use_asymp: if zmag > wp: v = fone, fzero else: zre = to_fixed(a, wp) zim = to_fixed(b, wp) vre, vim = complex_ei_asymptotic(zre, zim, wp) v = from_man_exp(vre, -wp), from_man_exp(vim, -wp) v = mpc_mul(v, mpc_exp(z, wp), wp) v = mpc_div(v, z, wp) if e1: v = mpc_neg(v, prec, rnd) else: x, y = v if bsign: v = mpf_pos(x, prec, rnd), mpf_sub(y, mpf_pi(wp), prec, rnd) else: v = mpf_pos(x, prec, rnd), mpf_add(y, mpf_pi(wp), prec, rnd) return v except NoConvergence: pass #wp += 2*max(0,zmag) wp += 2*int(to_int(mpc_abs(z, 5))) zre = to_fixed(a, wp) zim = to_fixed(b, wp) vre, vim = complex_ei_taylor(zre, zim, wp) vre += euler_fixed(wp) v = from_man_exp(vre,-wp), from_man_exp(vim,-wp) if e1: u = mpc_log(mpc_neg(z),wp) else: u = mpc_log(z,wp) v = mpc_add(v, u, prec, rnd) if e1: v = mpc_neg(v) return v
def mpc_ei(z, prec, rnd=round_fast, e1=False): if e1: z = mpc_neg(z) a, b = z asign, aman, aexp, abc = a bsign, bman, bexp, bbc = b if b == fzero: if e1: x = mpf_neg(mpf_ei(a, prec, rnd)) if not asign: y = mpf_neg(mpf_pi(prec, rnd)) else: y = fzero return x, y else: return mpf_ei(a, prec, rnd), fzero if a != fzero: if not aman or not bman: return (fnan, fnan) wp = prec + 40 amag = aexp + abc bmag = bexp + bbc zmag = max(amag, bmag) can_use_asymp = zmag > wp if not can_use_asymp: zabsint = abs(to_int(a)) + abs(to_int(b)) can_use_asymp = zabsint > int(wp * 0.693) + 20 try: if can_use_asymp: if zmag > wp: v = fone, fzero else: zre = to_fixed(a, wp) zim = to_fixed(b, wp) vre, vim = complex_ei_asymptotic(zre, zim, wp) v = from_man_exp(vre, -wp), from_man_exp(vim, -wp) v = mpc_mul(v, mpc_exp(z, wp), wp) v = mpc_div(v, z, wp) if e1: v = mpc_neg(v, prec, rnd) else: x, y = v if bsign: v = mpf_pos(x, prec, rnd), mpf_sub(y, mpf_pi(wp), prec, rnd) else: v = mpf_pos(x, prec, rnd), mpf_add(y, mpf_pi(wp), prec, rnd) return v except NoConvergence: pass #wp += 2*max(0,zmag) wp += 2 * int(to_int(mpc_abs(z, 5))) zre = to_fixed(a, wp) zim = to_fixed(b, wp) vre, vim = complex_ei_taylor(zre, zim, wp) vre += euler_fixed(wp) v = from_man_exp(vre, -wp), from_man_exp(vim, -wp) if e1: u = mpc_log(mpc_neg(z), wp) else: u = mpc_log(z, wp) v = mpc_add(v, u, prec, rnd) if e1: v = mpc_neg(v) return v