def mpc_psi(m, z, prec, rnd=round_fast): """ Computation of the polygamma function of arbitrary integer order m >= 0, for a complex argument z. """ if m == 0: return mpc_psi0(z, prec, rnd) re, im = z wp = prec + 20 sign, man, exp, bc = re if not man: if re == finf and im == fzero: return (fzero, fzero) if re == fnan: return fnan # Recurrence w = to_int(re) n = int(0.4*wp + 4*m) s = mpc_zero if w < n: for k in xrange(w, n): t = mpc_pow_int(z, -m-1, wp) s = mpc_add(s, t, wp) z = mpc_add_mpf(z, fone, wp) zm = mpc_pow_int(z, -m, wp) z2 = mpc_pow_int(z, -2, wp) # 1/m*(z+N)^m integral_term = mpc_div_mpf(zm, from_int(m), wp) s = mpc_add(s, integral_term, wp) # 1/2*(z+N)^(-(m+1)) s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp) a = m + 1 b = 2 k = 1 # Important: we want to sum up to the *relative* error, # not the absolute error, because psi^(m)(z) might be tiny magn = mpc_abs(s, 10) magn = magn[2]+magn[3] eps = mpf_shift(fone, magn-wp+2) while 1: zm = mpc_mul(zm, z2, wp) bern = mpf_bernoulli(2*k, wp) scal = mpf_mul_int(bern, a, wp) scal = mpf_div(scal, from_int(b), wp) term = mpc_mul_mpf(zm, scal, wp) s = mpc_add(s, term, wp) szterm = mpc_abs(term, 10) if k > 2 and mpf_le(szterm, eps): break #print k, to_str(szterm, 10), to_str(eps, 10) a *= (m+2*k)*(m+2*k+1) b *= (2*k+1)*(2*k+2) k += 1 # Scale and sign factor v = mpc_mul_mpf(s, mpf_gamma(from_int(m+1), wp), prec, rnd) if not (m & 1): v = mpf_neg(v[0]), mpf_neg(v[1]) return v
def mpc_psi(m, z, prec, rnd=round_fast): """ Computation of the polygamma function of arbitrary integer order m >= 0, for a complex argument z. """ if m == 0: return mpc_psi0(z, prec, rnd) re, im = z wp = prec + 20 sign, man, exp, bc = re if not man: if re == finf and im == fzero: return (fzero, fzero) if re == fnan: return fnan # Recurrence w = to_int(re) n = int(0.4 * wp + 4 * m) s = mpc_zero if w < n: for k in xrange(w, n): t = mpc_pow_int(z, -m - 1, wp) s = mpc_add(s, t, wp) z = mpc_add_mpf(z, fone, wp) zm = mpc_pow_int(z, -m, wp) z2 = mpc_pow_int(z, -2, wp) # 1/m*(z+N)^m integral_term = mpc_div_mpf(zm, from_int(m), wp) s = mpc_add(s, integral_term, wp) # 1/2*(z+N)^(-(m+1)) s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp) a = m + 1 b = 2 k = 1 # Important: we want to sum up to the *relative* error, # not the absolute error, because psi^(m)(z) might be tiny magn = mpc_abs(s, 10) magn = magn[2] + magn[3] eps = mpf_shift(fone, magn - wp + 2) while 1: zm = mpc_mul(zm, z2, wp) bern = mpf_bernoulli(2 * k, wp) scal = mpf_mul_int(bern, a, wp) scal = mpf_div(scal, from_int(b), wp) term = mpc_mul_mpf(zm, scal, wp) s = mpc_add(s, term, wp) szterm = mpc_abs(term, 10) if k > 2 and mpf_le(szterm, eps): break #print k, to_str(szterm, 10), to_str(eps, 10) a *= (m + 2 * k) * (m + 2 * k + 1) b *= (2 * k + 1) * (2 * k + 2) k += 1 # Scale and sign factor v = mpc_mul_mpf(s, mpf_gamma(from_int(m + 1), wp), prec, rnd) if not (m & 1): v = mpf_neg(v[0]), mpf_neg(v[1]) return v
def mpc_erf(z, prec, rnd=round_fast): re, im = z if im == fzero: return (mpf_erf(re, prec, rnd), fzero) wp = prec + 20 z2 = mpc_square(z, prec + 20) v = mpc_hyp1f1_rat((1, 2), (3, 2), mpc_neg(z2), wp, rnd) sqrtpi = mpf_sqrt(mpf_pi(wp), wp) c = mpf_rdiv_int(2, sqrtpi, wp) c = mpc_mul_mpf(z, c, wp) return mpc_mul(c, v, prec, rnd)
def mpc_erf(z, prec, rnd=round_fast): re, im = z if im == fzero: return (mpf_erf(re, prec, rnd), fzero) wp = prec + 20 z2 = mpc_mul(z, z, prec+20) v = mpc_hyp1f1_rat((1,2), (3,2), mpc_neg(z2), wp, rnd) sqrtpi = mpf_sqrt(mpf_pi(wp), wp) c = mpf_rdiv_int(2, sqrtpi, wp) c = mpc_mul_mpf(z, c, wp) return mpc_mul(c, v, prec, rnd)
def __mul__(s, t): prec, rounding = prec_rounding if not isinstance(t, mpc): if isinstance(t, int_types): return make_mpc(mpc_mul_int(s._mpc_, t, prec, rounding)) t = mpc_convert_lhs(t) if t is NotImplemented: return t if isinstance(t, mpf): return make_mpc(mpc_mul_mpf(s._mpc_, t._mpf_, prec, rounding)) t = mpc(t) return make_mpc(mpc_mul(s._mpc_, t._mpc_, prec, rounding))
def mpc_psi0(z, prec, rnd=round_fast): """ Computation of the digamma function (psi function of order 0) of a complex argument. """ re, im = z # Fall back to the real case if im == fzero: return (mpf_psi0(re, prec, rnd), fzero) wp = prec + 20 sign, man, exp, bc = re # Reflection formula if sign and exp+bc > 3: c = mpc_cos_pi(z, wp) s = mpc_sin_pi(z, wp) q = mpc_mul_mpf(mpc_div(c, s, wp), mpf_pi(wp), wp) p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp) return mpc_sub(p, q, prec, rnd) # Just the logarithmic term if (not sign) and bc + exp > wp: return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd) # Initial recurrence to obtain a large enough z w = to_int(re) n = int(0.11*wp) + 2 s = mpc_zero if w < n: for k in xrange(w, n): s = mpc_sub(s, mpc_reciprocal(z, wp), wp) z = mpc_add_mpf(z, fone, wp) z = mpc_sub(z, mpc_one, wp) # Logarithmic and endpoint term s = mpc_add(s, mpc_log(z, wp), wp) s = mpc_add(s, mpc_div(mpc_half, z, wp), wp) # Euler-Maclaurin remainder sum z2 = mpc_square(z, wp) t = mpc_one prev = mpc_zero k = 1 eps = mpf_shift(fone, -wp+2) while 1: t = mpc_mul(t, z2, wp) bern = mpf_bernoulli(2*k, wp) term = mpc_mpf_div(bern, mpc_mul_int(t, 2*k, wp), wp) s = mpc_sub(s, term, wp) szterm = mpc_abs(term, 10) if k > 2 and mpf_le(szterm, eps): break prev = term k += 1 return s
def mpc_psi0(z, prec, rnd=round_fast): """ Computation of the digamma function (psi function of order 0) of a complex argument. """ re, im = z # Fall back to the real case if im == fzero: return (mpf_psi0(re, prec, rnd), fzero) wp = prec + 20 sign, man, exp, bc = re # Reflection formula if sign and exp + bc > 3: c = mpc_cos_pi(z, wp) s = mpc_sin_pi(z, wp) q = mpc_mul_mpf(mpc_div(c, s, wp), mpf_pi(wp), wp) p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp) return mpc_sub(p, q, prec, rnd) # Just the logarithmic term if (not sign) and bc + exp > wp: return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd) # Initial recurrence to obtain a large enough z w = to_int(re) n = int(0.11 * wp) + 2 s = mpc_zero if w < n: for k in xrange(w, n): s = mpc_sub(s, mpc_reciprocal(z, wp), wp) z = mpc_add_mpf(z, fone, wp) z = mpc_sub(z, mpc_one, wp) # Logarithmic and endpoint term s = mpc_add(s, mpc_log(z, wp), wp) s = mpc_add(s, mpc_div(mpc_half, z, wp), wp) # Euler-Maclaurin remainder sum z2 = mpc_square(z, wp) t = mpc_one prev = mpc_zero k = 1 eps = mpf_shift(fone, -wp + 2) while 1: t = mpc_mul(t, z2, wp) bern = mpf_bernoulli(2 * k, wp) term = mpc_mpf_div(bern, mpc_mul_int(t, 2 * k, wp), wp) s = mpc_sub(s, term, wp) szterm = mpc_abs(term, 10) if k > 2 and mpf_le(szterm, eps): break prev = term k += 1 return s
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False): re, im = s if im == fzero: return mpf_zeta(re, prec, rnd, alt), fzero # slow for large s if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)): raise NotImplementedError wp = prec + 20 # Near pole r = mpc_sub(mpc_one, s, wp) asign, aman, aexp, abc = mpc_abs(r, 10) pole_dist = -2*(aexp+abc) if pole_dist > wp: if alt: q = mpf_ln2(wp) y = mpf_mul(q, mpf_euler(wp), wp) g = mpf_shift(mpf_mul(q, q, wp), -1) g = mpf_sub(y, g) z = mpc_mul_mpf(r, mpf_neg(g), wp) z = mpc_add_mpf(z, q, wp) return mpc_pos(z, prec, rnd) else: q = mpc_neg(mpc_div(mpc_one, r, wp)) q = mpc_add_mpf(q, mpf_euler(wp), wp) return mpc_pos(q, prec, rnd) else: wp += max(0, pole_dist) # Reflection formula. To be rigorous, we should reflect to the left of # re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary # slowdown for interesting values of s if mpf_lt(re, fzero): # XXX: could use the separate refl. formula for Dirichlet eta if alt: q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp) return mpc_mul(mpc_zeta(s, wp), q, prec, rnd) # XXX: -1 should be done exactly y = mpc_sub(mpc_one, s, 10*wp) a = mpc_gamma(y, wp) b = mpc_zeta(y, wp) c = mpc_sin_pi(mpc_shift(s, -1), wp) rsign, rman, rexp, rbc = re isign, iman, iexp, ibc = im mag = max(rexp+rbc, iexp+ibc) wp2 = wp + mag pi = mpf_pi(wp+wp2) pi2 = (mpf_shift(pi, 1), fzero) d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2) return mpc_mul(a,mpc_mul(b,mpc_mul(c,d,wp),wp),prec,rnd) n = int(wp/2.54 + 5) n += int(0.9*abs(to_int(im))) d = borwein_coefficients(n) ref = to_fixed(re, wp) imf = to_fixed(im, wp) tre = MPZ_ZERO tim = MPZ_ZERO one = MPZ_ONE << wp one_2wp = MPZ_ONE << (2*wp) critical_line = re == fhalf for k in xrange(n): log = log_int_fixed(k+1, wp) # A square root is much cheaper than an exp if critical_line: w = one_2wp // sqrt_fixed((k+1) << wp, wp) else: w = to_fixed(mpf_exp(from_man_exp(-ref*log, -2*wp), wp), wp) if k & 1: w *= (d[n] - d[k]) else: w *= (d[k] - d[n]) wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2*wp), wp) tre += (w * to_fixed(wre, wp)) >> wp tim += (w * to_fixed(wim, wp)) >> wp tre //= (-d[n]) tim //= (-d[n]) tre = from_man_exp(tre, -wp, wp) tim = from_man_exp(tim, -wp, wp) if alt: return mpc_pos((tre, tim), prec, rnd) else: q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp) return mpc_div((tre, tim), q, prec, rnd)
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False): re, im = s if im == fzero: return mpf_zeta(re, prec, rnd, alt), fzero # slow for large s if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)): raise NotImplementedError wp = prec + 20 # Near pole r = mpc_sub(mpc_one, s, wp) asign, aman, aexp, abc = mpc_abs(r, 10) pole_dist = -2 * (aexp + abc) if pole_dist > wp: if alt: q = mpf_ln2(wp) y = mpf_mul(q, mpf_euler(wp), wp) g = mpf_shift(mpf_mul(q, q, wp), -1) g = mpf_sub(y, g) z = mpc_mul_mpf(r, mpf_neg(g), wp) z = mpc_add_mpf(z, q, wp) return mpc_pos(z, prec, rnd) else: q = mpc_neg(mpc_div(mpc_one, r, wp)) q = mpc_add_mpf(q, mpf_euler(wp), wp) return mpc_pos(q, prec, rnd) else: wp += max(0, pole_dist) # Reflection formula. To be rigorous, we should reflect to the left of # re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary # slowdown for interesting values of s if mpf_lt(re, fzero): # XXX: could use the separate refl. formula for Dirichlet eta if alt: q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp) return mpc_mul(mpc_zeta(s, wp), q, prec, rnd) # XXX: -1 should be done exactly y = mpc_sub(mpc_one, s, 10 * wp) a = mpc_gamma(y, wp) b = mpc_zeta(y, wp) c = mpc_sin_pi(mpc_shift(s, -1), wp) rsign, rman, rexp, rbc = re isign, iman, iexp, ibc = im mag = max(rexp + rbc, iexp + ibc) wp2 = wp + mag pi = mpf_pi(wp + wp2) pi2 = (mpf_shift(pi, 1), fzero) d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2) return mpc_mul(a, mpc_mul(b, mpc_mul(c, d, wp), wp), prec, rnd) n = int(wp / 2.54 + 5) n += int(0.9 * abs(to_int(im))) d = borwein_coefficients(n) ref = to_fixed(re, wp) imf = to_fixed(im, wp) tre = MPZ_ZERO tim = MPZ_ZERO one = MPZ_ONE << wp one_2wp = MPZ_ONE << (2 * wp) critical_line = re == fhalf for k in xrange(n): log = log_int_fixed(k + 1, wp) # A square root is much cheaper than an exp if critical_line: w = one_2wp // sqrt_fixed((k + 1) << wp, wp) else: w = to_fixed(mpf_exp(from_man_exp(-ref * log, -2 * wp), wp), wp) if k & 1: w *= (d[n] - d[k]) else: w *= (d[k] - d[n]) wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2 * wp), wp) tre += (w * to_fixed(wre, wp)) >> wp tim += (w * to_fixed(wim, wp)) >> wp tre //= (-d[n]) tim //= (-d[n]) tre = from_man_exp(tre, -wp, wp) tim = from_man_exp(tim, -wp, wp) if alt: return mpc_pos((tre, tim), prec, rnd) else: q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp) return mpc_div((tre, tim), q, prec, rnd)