def __rmul__(s, t): prec, rounding = prec_rounding 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 return t * s
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(mpc_div(c, s, wp), (mpf_pi(wp), fzero), 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_div(mpc_one, 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_mul(z, 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_div((bern, fzero), 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