def _vcjh_fn(self, sym): k = self._order eta = self._cfg.get('solver-elements-' + self.name, 'vcjh-eta') # Expand shorthand forms of eta for common schemes etacommon = dict(dg='0', sd='k/(k+1)', hu='(k+1)/k') eta_k = sy.S(etacommon.get(eta, eta), locals=dict(k=k)) lkm1, lk, lkp1 = [sy.legendre_poly(m, sym) for m in [k - 1, k, k + 1]] return (sy.S(1)/2 * (lk + (eta_k*lkm1 + lkp1)/(1 + eta_k)))
def wigner_d(l, m, m_prime, beta, wdsympy=False): """Computation of Wigner-d-functions for the rotation of a T-matrix Args: l (int): Degree :math:`l` (1, ..., lmax) m (int): Order :math:`m` (-min(l,mmax),...,min(l,mmax)) m_prime (int): Order :math:`m_prime` (-min(l,mmax),...,min(l,mmax)) beta (float): Second Euler angle in rad wdsympy (bool): If True, Wigner-d-functions come from the sympy toolbox Returns: real value of Wigner-d-function """ wig_d = np.zeros(l + 1, dtype=complex) if wdsympy == False: if beta < 0: aa = m bb = m_prime m = bb m_prime = aa if m == 0 and m_prime == 0: for nn in range(1, l + 1): wig_d[nn] = sympy.legendre_poly(nn, np.cos(beta)) else: # recursion formulation (Mishchenko, Scattering, Absorption and Emission of Light by small Particles, p.365 (B.22 - B.24)) l_min = max(abs(m), abs(m_prime)) wig_d[l_min - 1] = 0 if m_prime >= m: zeta = 1 else: zeta = (-1)**(m - m_prime) wig_d[l_min] = ( zeta * 2.0**(-l_min) * (factorial(2 * l_min) / (factorial(abs(m - m_prime)) * factorial(abs(m + m_prime))))** 0.5 * (1 - np.cos(beta))**(abs(m - m_prime) / 2) * (1 + np.cos(beta))**(abs(m + m_prime) / 2)) for ll in range(l_min, l): wig_d[ll + 1] = ( ((2 * ll + 1) * (ll * (ll + 1) * np.cos(beta) - m * m_prime) * wig_d[ll] - (ll + 1) * (ll**2 - m**2)**0.5 * (ll**2 - m_prime**2)**0.5 * wig_d[ll - 1]) / (ll * ((ll + 1)**2 - m**2)**0.5 * ((ll + 1)**2 - m_prime**2)**0.5)) else: wig_d[l] = complex(Rotation.d(l, m, m_prime, beta).doit()) return wig_d[l].real
def __init__(self, npts): # Form a suitable Legendre poly Pn = sy.legendre_poly(npts, x) dPn = Pn.diff() # Roots self.points = mp.polyroots(map(mp.mpf, sy.Poly(Pn).all_coeffs())) # Weights self.weights = [2/((1 - p**2)*dPn.evalf(mp.dps, subs={x: p})**2) for p in self.points]
def test_CRootOf_eval_rational(): p = legendre_poly(4, x, polys=True) roots = [r.eval_rational(n=18) for r in p.real_roots()] for r in roots: assert isinstance(r, Rational) roots = [str(r.n(17)) for r in roots] assert roots == [ "-0.86113631159405258", "-0.33998104358485626", "0.33998104358485626", "0.86113631159405258", ]
def __init__(self, npts): # Form a suitable Legendre poly Pn = sy.legendre_poly(npts - 1, x) dPn = Pn.diff() # Roots roots = mp.polyroots(map(mp.mpf, sy.Poly(dPn).all_coeffs())) self.points = [mp.mpf(-1)] + roots + [mp.mpf(1)] # Weights wts0 = mp.mpf(2)/(npts*(npts - 1)) wtsi = [2/(npts*(npts - 1)*Pn.evalf(mp.dps, subs={x: p})**2) for p in self.points[1:-1]] self.weights = [wts0] + wtsi + [wts0]
def test_RootOf_eval_rational(): p = legendre_poly(4, x, polys=True) roots = [r.eval_rational(S(1) / 10**20) for r in p.real_roots()] for r in roots: assert isinstance(r, Rational) # All we know is that the Rational instance will be at most 1/10^20 from # the exact root. So if we evaluate to 17 digits, it must be exactly equal # to: roots = [str(r.n(17)) for r in roots] assert roots == [ "-0.86113631159405258", "-0.33998104358485626", "0.33998104358485626", "0.86113631159405258", ]
def max_rE_gains_3d(order, numeric=True): """max rE for a given order is the largest root of the order+1 Legendre polynomial""" x = sp.symbols('x') lp = sp.legendre_poly(order + 1, x) # there are more efficient methods to find the roots of the Legendre # polynomials, but this is good enough for our purposes # See discussion at: # https://math.stackexchange.com/questions/12160/roots-of-legendre-polynomial if order < 5 and not numeric: roots = sp.roots(lp) else: roots = sp.nroots(lp) # the roots can be in the keys of a dictionary or in a list, # this works for either one max_rE = np.max([*roots]) return [sp.legendre(n, max_rE) for n in range(order + 1)]
def test_CRootOf_evalf(): real = rootof(x**3 + x + 3, 0).evalf(n=20) assert real.epsilon_eq(Float("-1.2134116627622296341")) re, im = rootof(x**3 + x + 3, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(-Float("1.45061224918844152650")) re, im = rootof(x**3 + x + 3, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(Float("1.45061224918844152650")) p = legendre_poly(4, x, polys=True) roots = [str(r.n(17)) for r in p.real_roots()] # magnitudes are given by # sqrt(3/S(7) - 2*sqrt(6/S(5))/7) # and # sqrt(3/S(7) + 2*sqrt(6/S(5))/7) assert roots == [ "-0.86113631159405258", "-0.33998104358485626", "0.33998104358485626", "0.86113631159405258", ] re = rootof(x**5 - 5 * x + 12, 0).evalf(n=20) assert re.epsilon_eq(Float("-1.84208596619025438271")) re, im = rootof(x**5 - 5 * x + 12, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("-1.709561043370328882010")) re, im = rootof(x**5 - 5 * x + 12, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("+1.709561043370328882010")) re, im = rootof(x**5 - 5 * x + 12, 3).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("-0.719798681483861386681")) re, im = rootof(x**5 - 5 * x + 12, 4).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("+0.719798681483861386681")) # issue 6393 assert str(rootof(x**5 + 2 * x**4 + x**3 - 68719476736, 0).n(3)) == '147.' eq = (531441 * x**11 + 3857868 * x**10 + 13730229 * x**9 + 32597882 * x**8 + 55077472 * x**7 + 60452000 * x**6 + 32172064 * x**5 - 4383808 * x**4 - 11942912 * x**3 - 1506304 * x**2 + 1453312 * x + 512) a, b = rootof(eq, 1).n(2).as_real_imag() c, d = rootof(eq, 2).n(2).as_real_imag() assert a == c assert b < d assert b == -d # issue 6451 r = rootof(legendre_poly(64, x), 7) assert r.n(2) == r.n(100).n(2) # issue 8617 ans = [w.n(2) for w in solve(x**3 - x - 4)] assert rootof(exp(x)**3 - exp(x) - 4, 0).n(2) in ans # issue 9019 r0 = rootof(x**2 + 1, 0, radicals=False) r1 = rootof(x**2 + 1, 1, radicals=False) assert r0.n(4) == -1.0 * I assert r1.n(4) == 1.0 * I # make sure verification is used in case a max/min traps the "root" assert str(rootof(4 * x**5 + 16 * x**3 + 12 * x**2 + 7, 0).n(3)) == '-0.976' # watch out for UnboundLocalError c = CRootOf(90720 * x**6 - 4032 * x**4 + 84 * x**2 - 1, 0) assert c._eval_evalf(2) # doesn't fail # watch out for imaginary parts that don't want to evaluate assert str( RootOf( x**16 + 32 * x**14 + 508 * x**12 + 5440 * x**10 + 39510 * x**8 + 204320 * x**6 + 755548 * x**4 + 1434496 * x**2 + 877969, 10).n(2)) == '-3.4*I' assert abs(RootOf(x**4 + 10 * x**2 + 1, 0).n(2)) < 0.4 # check reset and args r = [RootOf(x**3 + x + 3, i) for i in range(3)] r[0]._reset() for ri in r: i = ri._get_interval() n = ri.n(2) assert i != ri._get_interval() ri._reset() assert i == ri._get_interval() assert i == i.func(*i.args)
def test_RootOf_evalf(): real = RootOf(x**3 + x + 3, 0).evalf(n=20) assert real.epsilon_eq(Float("-1.2134116627622296341")) re, im = RootOf(x**3 + x + 3, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(-Float("1.45061224918844152650")) re, im = RootOf(x**3 + x + 3, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(Float("1.45061224918844152650")) p = legendre_poly(4, x, polys=True) roots = [str(r.n(17)) for r in p.real_roots()] assert roots == [ "-0.86113631159405258", "-0.33998104358485626", "0.33998104358485626", "0.86113631159405258", ] re = RootOf(x**5 - 5 * x + 12, 0).evalf(n=20) assert re.epsilon_eq(Float("-1.84208596619025438271")) re, im = RootOf(x**5 - 5 * x + 12, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("-1.709561043370328882010")) re, im = RootOf(x**5 - 5 * x + 12, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("+1.709561043370328882010")) re, im = RootOf(x**5 - 5 * x + 12, 3).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("-0.719798681483861386681")) re, im = RootOf(x**5 - 5 * x + 12, 4).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("+0.719798681483861386681")) # issue 6393 assert str(RootOf(x**5 + 2 * x**4 + x**3 - 68719476736, 0).n(3)) == '147.' eq = (531441 * x**11 + 3857868 * x**10 + 13730229 * x**9 + 32597882 * x**8 + 55077472 * x**7 + 60452000 * x**6 + 32172064 * x**5 - 4383808 * x**4 - 11942912 * x**3 - 1506304 * x**2 + 1453312 * x + 512) a, b = RootOf(eq, 1).n(2).as_real_imag() c, d = RootOf(eq, 2).n(2).as_real_imag() assert a == c assert b < d assert b == -d # issue 6451 r = RootOf(legendre_poly(64, x), 7) assert r.n(2) == r.n(100).n(2) # issue 8617 ans = [w.n(2) for w in solve(x**3 - x - 4)] assert RootOf(exp(x)**3 - exp(x) - 4, 0).n(2) in ans # issue 9019 r0 = RootOf(x**2 + 1, 0, radicals=False) r1 = RootOf(x**2 + 1, 1, radicals=False) assert r0.n(4) == -1.0 * I assert r1.n(4) == 1.0 * I # make sure verification is used in case a max/min traps the "root" assert str(RootOf(4 * x**5 + 16 * x**3 + 12 * x**2 + 7, 0).n(3)) == '-0.976'
def test_CRootOf_evalf(): real = rootof(x**3 + x + 3, 0).evalf(n=20) assert real.epsilon_eq(Float("-1.2134116627622296341")) re, im = rootof(x**3 + x + 3, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq( Float("0.60670583138111481707")) assert im.epsilon_eq(-Float("1.45061224918844152650")) re, im = rootof(x**3 + x + 3, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(Float("1.45061224918844152650")) p = legendre_poly(4, x, polys=True) roots = [str(r.n(17)) for r in p.real_roots()] # magnitudes are given by # sqrt(3/S(7) - 2*sqrt(6/S(5))/7) # and # sqrt(3/S(7) + 2*sqrt(6/S(5))/7) assert roots == [ "-0.86113631159405258", "-0.33998104358485626", "0.33998104358485626", "0.86113631159405258", ] re = rootof(x**5 - 5*x + 12, 0).evalf(n=20) assert re.epsilon_eq(Float("-1.84208596619025438271")) re, im = rootof(x**5 - 5*x + 12, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("-1.709561043370328882010")) re, im = rootof(x**5 - 5*x + 12, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("+1.709561043370328882010")) re, im = rootof(x**5 - 5*x + 12, 3).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("-0.719798681483861386681")) re, im = rootof(x**5 - 5*x + 12, 4).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("+0.719798681483861386681")) # issue 6393 assert str(rootof(x**5 + 2*x**4 + x**3 - 68719476736, 0).n(3)) == '147.' eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 + 55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 - 11942912*x**3 - 1506304*x**2 + 1453312*x + 512) a, b = rootof(eq, 1).n(2).as_real_imag() c, d = rootof(eq, 2).n(2).as_real_imag() assert a == c assert b < d assert b == -d # issue 6451 r = rootof(legendre_poly(64, x), 7) assert r.n(2) == r.n(100).n(2) # issue 9019 r0 = rootof(x**2 + 1, 0, radicals=False) r1 = rootof(x**2 + 1, 1, radicals=False) assert r0.n(4) == -1.0*I assert r1.n(4) == 1.0*I # make sure verification is used in case a max/min traps the "root" assert str(rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).n(3)) == '-0.976' # watch out for UnboundLocalError c = CRootOf(90720*x**6 - 4032*x**4 + 84*x**2 - 1, 0) assert c._eval_evalf(2) # doesn't fail # watch out for imaginary parts that don't want to evaluate assert str(RootOf(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 + 39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 + 877969, 10).n(2)) == '-3.4*I' assert abs(RootOf(x**4 + 10*x**2 + 1, 0).n(2)) < 0.4 # check reset and args r = [RootOf(x**3 + x + 3, i) for i in range(3)] r[0]._reset() for ri in r: i = ri._get_interval() n = ri.n(2) assert i != ri._get_interval() ri._reset() assert i == ri._get_interval() assert i == i.func(*i.args)
def test_RootOf_evalf(): real = RootOf(x**3 + x + 3, 0).evalf(n=20) assert real.epsilon_eq(Float("-1.2134116627622296341")) re, im = RootOf(x**3 + x + 3, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(-Float("1.45061224918844152650")) re, im = RootOf(x**3 + x + 3, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("0.60670583138111481707")) assert im.epsilon_eq(Float("1.45061224918844152650")) p = legendre_poly(4, x, polys=True) roots = [str(r.n(17)) for r in p.real_roots()] assert roots == [ "-0.86113631159405258", "-0.33998104358485626", "0.33998104358485626", "0.86113631159405258", ] re = RootOf(x**5 - 5 * x + 12, 0).evalf(n=20) assert re.epsilon_eq(Float("-1.84208596619025438271")) re, im = RootOf(x**5 - 5 * x + 12, 1).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("-1.709561043370328882010")) re, im = RootOf(x**5 - 5 * x + 12, 2).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("-0.351854240827371999559")) assert im.epsilon_eq(Float("+1.709561043370328882010")) re, im = RootOf(x**5 - 5 * x + 12, 3).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("-0.719798681483861386681")) re, im = RootOf(x**5 - 5 * x + 12, 4).evalf(n=20).as_real_imag() assert re.epsilon_eq(Float("+1.272897223922499190910")) assert im.epsilon_eq(Float("+0.719798681483861386681")) # issue 6393 assert str(RootOf(x**5 + 2 * x**4 + x**3 - 68719476736, 0).n(3)) == '147.' eq = ( 531441 * x**11 + 3857868 * x**10 + 13730229 * x**9 + 32597882 * x**8 + 55077472 * x**7 + 60452000 * x**6 + 32172064 * x**5 - 4383808 * x**4 - 11942912 * x**3 - 1506304 * x**2 + 1453312 * x + 512) a, b = RootOf(eq, 1).n(2).as_real_imag() c, d = RootOf(eq, 2).n(2).as_real_imag() assert a == c assert b < d assert b == -d # issue 6451 r = RootOf(legendre_poly(64, x), 7) assert r.n(2) == r.n(100).n(2) # issue 8617 ans = [w.n(2) for w in solve(x**3 - x - 4)] assert RootOf(exp(x)**3 - exp(x) - 4, 0).n(2) in ans # make sure verification is used in case a max/min traps the "root" assert str(RootOf(4 * x**5 + 16 * x**3 + 12 * x**2 + 7, 0).n(3)) == '-0.976'