我们从Python开源项目中，提取了以下50个代码示例，用于说明如何使用sympy.Rational()。
def __init__(self, n, index): self.dim = n if index == '1-n': self.degree = 3 data = [ (fr(1, 2*n), fsd(n, (sqrt(fr(n, 3)), 1))), ] else: assert index == '2-n' self.degree = 5 r = sqrt(fr(3, 5)) data = [ (fr(25*n**2 - 115*n + 162, 162), z(n)), (fr(70 - 25*n, 162), fsd(n, (r, 1))), (fr(25, 324), fsd(n, (r, 2))), ] self.points, self.weights = untangle(data) reference_volume = 2**n self.weights *= reference_volume return
def __init__(self, n): self.degree = 5 r = sqrt(fr(7, 15)) s, t = [sqrt((7 + i*sqrt(24)) / 15) for i in [+1, -1]] data = [ (fr(5*n**2 - 15*n+14, 14), z(n)), (fr(25, 168), _s2(n, +r)), (fr(25, 168), _s2(n, -r)), (fr(-25*(n-2), 168), fsd(n, (r, 1))), (fr(5, 48), _s11(n, +s, -t)), (fr(5, 48), _s11(n, -s, +t)), (fr(-5*(n-2), 48), fsd(n, (s, 1))), (fr(-5*(n-2), 48), fsd(n, (t, 1))), ] self.points, self.weights = untangle(data) reference_volume = 2**n self.weights *= reference_volume return
def __init__(self): self.name = 'Phillips' c = 3*sqrt(385) r, s = [sqrt((105 + i*c) / 140) for i in [+1, -1]] t = sqrt(fr(3, 5)) B1, B2 = [(77 - i*c) / 891 for i in [+1, -1]] B3 = fr(25, 324) self.degree = 7 data = [ (B1, _symm_r_0(r)), (B2, _symm_r_0(s)), (B3, _pm2(t, t)) ] self.points, self.weights = untangle(data) self.weights *= 4 return
def __init__(self): self.name = 'Meister' self.degree = 7 r = fr(2, 3) s = fr(1, 3) data = [ (fr(1024, 6720), _z()), (fr(576, 6720), _symm_s(r)), (fr(576, 6720), _symm_r_0(r)), (-fr(9, 6720), _symm_s(s)), (fr(117, 6720), _symm_s_t(1, s)), (fr(47, 6720), _symm_s(1)), ] self.points, self.weights = untangle(data) self.weights *= 4 return
def __init__(self, index): self.name = 'Irwin({})'.format(index) if index == 1: self.degree = 3 data = [ (fr(14, 48), _symm_s(1)), (-fr(1, 48), _symm_s_t(3, 1)) ] else: assert index == 2 self.degree = 5 data = [ (fr(889, 2880), _symm_s(1)), (-fr(98, 2880), _symm_s_t(3, 1)), (fr(5, 2880), _symm_s(3)), (fr(11, 2880), _symm_s_t(5, 1)), ] self.points, self.weights = untangle(data) self.weights *= 4 return
def vii(): # article: # nu, xi = numpy.sqrt((15 + plus_minus * 3*numpy.sqrt(5))) # A = 3/5 # B = 1/30 # book: nu, xi = [sqrt((5 - p_m * sqrt(5)) / 4) for p_m in [+1, -1]] A = fr(2, 5) B = fr(1, 20) data = [ (A, numpy.array([[0, 0, 0]])), (B, pm_roll(3, [nu, xi])), ] return 5, data
def __init__(self, degree): self.degree = degree if degree == 2: data = [ (fr(1, 4), _s31((5 - sqrt(5))/20)), ] else: assert degree == 3 data = [ (-fr(4, 5), _s4()), (+fr(9, 20), _s31(fr(1, 6))), ] self.bary, self.weights = untangle(data) self.points = self.bary[:, 1:] return
def __init__(self, index): if index == 4: self.degree = 2 data = [ (fr(1, 4), _s31(0.1381966011250105)) ] else: assert index == 5 self.degree = 3 data = [ (-fr(4, 5), _s4()), (fr(9, 20), _s31(fr(1, 6))), ] self.bary, self.weights = untangle(data) self.points = self.bary[:, 1:] return
def __init__(self, index): if index == 1: self.degree = 2 data = [ (fr(1, 4), _r(1/sqrt(5))), ] elif index == 2: self.degree = 2 data = [ (fr(1, 4), _r(-1/sqrt(5))), ] else: assert index == 3 self.degree = 3 data = [ (-fr(4, 5), _s4()), (fr(9, 20), _r(fr(1, 3))), ] self.bary, self.weights = untangle(data) self.points = self.bary[:, 1:] return
def _gen5_3(n): '''Spherical product Lobatto formula. ''' data = [] s = sqrt(n+3) for k in range(1, n+1): rk = sqrt((k+2) * (n+3)) Bk = fr(2**(k-n) * (n+1), (k+1) * (k+2) * (n+3)) arr = [rk] + (n-k) * [s] data += [ (Bk, pm_array0(n, arr, range(k-1, n))) ] B0 = 1 - sum([item[0]*len(item[1]) for item in data]) data += [ (B0, numpy.full((1, n), 0)) ] return 5, data
def _xi11(d, a): assert d > 1 b = fr(1 - (d-1) * a, 2) if d == 2: out = numpy.array([ [b, b, a], [b, a, b], [a, b, b], ]) else: assert d == 3 out = numpy.array([ [b, b, a, a], [b, a, b, a], [b, a, a, b], [a, b, a, b], [a, a, b, b], [a, b, b, a], ]) return out
def __init__(self, n): self.dim = n self.degree = 3 r = fr(1, n+1) s = fr(1, n) prod = (n+1) * (n+2) * (n+3) A = fr((3-n) * (n+1)**3, prod) B = fr(3, prod) C = fr(n**3, prod) data = [ (A, [(n+1) * [r]]), (B, rd(n+1, [(1, 1)])), (C, rd(n+1, [(s, n)])), ] self.bary, self.weights = untangle(data) self.points = self.bary[:, 1:] return
def __init__(self, n): self.dim = n self.degree = 7 r = 1 s = sqrt(fr(1, n)) t = sqrt(fr(1, 2)) B = fr(8-n, n * (n+2) * (n+4)) C = fr(n**3, 2**n * n * (n+2) * (n+4)) D = fr(4, n * (n+2) * (n+4)) data = [ (B, fsd(n, (r, 1))), (C, pm(n, s)), # ERR Stroud's book wrongly states (t, t,..., t)_FS instead of # (t, t, 0, ..., 0)_FS. (D, fsd(n, (t, 2))), ] self.points, self.weights = untangle(data) self.weights *= integrate_monomial_over_unit_nsphere(n * [0]) return
def integrate_monomial_over_unit_nsphere(alpha): ''' Gerald B. Folland, How to Integrate a Polynomial over a Sphere, The American Mathematical Monthly, Vol. 108, No. 5 (May, 2001), pp. 446-448, <https://doi.org/10.2307/2695802>. ''' if any(a % 2 == 1 for a in alpha): return 0 # Use lgamma since other with ordinary gamma, numerator and denominator # might overflow. # return 2 * math.exp( # math.fsum([math.lgamma(0.5*(a+1)) for a in alpha]) # - math.lgamma(math.fsum([0.5*(a+1) for a in alpha])) # ) # Explicitly cast a to int (from numpy.int64) to work around bug # <https://github.com/sympy/sympy/issues/13618>. return 2 * ( prod([gamma(Rational(int(a)+1, 2)) for a in alpha]) / gamma(sum([Rational(int(a)+1, 2) for a in alpha])) )
def __init__(self, degree): self.degree = degree if degree == 2: data = { 's2': [ [fr(1, 3), fr(1, 6)], ] } else: assert degree == 3 data = { 's3': [ [-fr(27, 48)], ], 's2': [ [+fr(25, 48), fr(1, 5)], ] } self.bary, self.weights = untangle2(data) self.points = self.bary[:, 1:] return
def vi(): sqrt74255 = sqrt(74255) nu = sqrt(42) xi, eta = [sqrt((6615 - p_m * 21 * sqrt74255) / 454) for p_m in [+1, -1]] A = fr(5, 588) B, C = [ (5272105 + p_m * 18733 * sqrt74255) / 43661940 for p_m in [+1, -1] ] data = [ (A, fsd(2, (nu, 1))), (B, pm(2, xi)), (C, pm(2, eta)), ] return 7, data
def dotest(s): x = Symbol("x") y = Symbol("y") l = [ Rational(2), Float("1.3"), x, y, pow(x, y)*y, 5, 5.5 ] for x in l: for y in l: s(x, y) return True
def test_measure_partial(): #Basic test of collapse of entangled two qubits (Bell States) state = Qubit('01') + Qubit('10') assert measure_partial(state, (0,)) == \ [(Qubit('10'), Rational(1, 2)), (Qubit('01'), Rational(1, 2))] assert measure_partial(state, (0,)) == \ measure_partial(state, (1,))[::-1] #Test of more complex collapse and probability calculation state1 = sqrt(2)/sqrt(3)*Qubit('00001') + 1/sqrt(3)*Qubit('11111') assert measure_partial(state1, (0,)) == \ [(sqrt(2)/sqrt(3)*Qubit('00001') + 1/sqrt(3)*Qubit('11111'), 1)] assert measure_partial(state1, (1, 2)) == measure_partial(state1, (3, 4)) assert measure_partial(state1, (1, 2, 3)) == \ [(Qubit('00001'), Rational(2, 3)), (Qubit('11111'), Rational(1, 3))] #test of measuring multiple bits at once state2 = Qubit('1111') + Qubit('1101') + Qubit('1011') + Qubit('1000') assert measure_partial(state2, (0, 1, 3)) == \ [(Qubit('1000'), Rational(1, 4)), (Qubit('1101'), Rational(1, 4)), (Qubit('1011')/sqrt(2) + Qubit('1111')/sqrt(2), Rational(1, 2))] assert measure_partial(state2, (0,)) == \ [(Qubit('1000'), Rational(1, 4)), (Qubit('1111')/sqrt(3) + Qubit('1101')/sqrt(3) + Qubit('1011')/sqrt(3), Rational(3, 4))]
def test_units(): assert (5*m/s * day) / km == 432 assert foot / meter == Rational('0.3048') # amu is a pure mass so mass/mass gives a number, not an amount (mol) assert str(grams/(amu).n(2)) == '6.0e+23' # Light from the sun needs about 8.3 minutes to reach earth t = (1*au / speed_of_light).evalf() / minute assert abs(t - 8.31) < 0.1 assert sqrt(m**2) == m assert (sqrt(m))**2 == m t = Symbol('t') assert integrate(t*m/s, (t, 1*s, 5*s)) == 12*m*s assert (t * m/s).integrate((t, 1*s, 5*s)) == 12*m*s
def __pow__(self, other): if self.data is None: raise ValueError("No power without ndarray data.") numpy = import_module('numpy') free = self.free marray = self.data for metric in free: marray = numpy.tensordot( marray, numpy.tensordot( metric[0]._tensortype.data, marray, (1, 0) ), (0, 0) ) pow2 = marray[()] return pow2 ** (Rational(1, 2) * other)
def test_legendre_poly(): raises(ValueError, lambda: legendre_poly(-1, x)) assert legendre_poly(1, x, polys=True) == Poly(x) assert legendre_poly(0, x) == 1 assert legendre_poly(1, x) == x assert legendre_poly(2, x) == Q(3, 2)*x**2 - Q(1, 2) assert legendre_poly(3, x) == Q(5, 2)*x**3 - Q(3, 2)*x assert legendre_poly(4, x) == Q(35, 8)*x**4 - Q(30, 8)*x**2 + Q(3, 8) assert legendre_poly(5, x) == Q(63, 8)*x**5 - Q(70, 8)*x**3 + Q(15, 8)*x assert legendre_poly(6, x) == Q( 231, 16)*x**6 - Q(315, 16)*x**4 + Q(105, 16)*x**2 - Q(5, 16) assert legendre_poly(1).dummy_eq(x) assert legendre_poly(1, polys=True) == Poly(x)
def test_laguerre_poly(): raises(ValueError, lambda: laguerre_poly(-1, x)) assert laguerre_poly(1, x, polys=True) == Poly(-x + 1) assert laguerre_poly(0, x) == 1 assert laguerre_poly(1, x) == -x + 1 assert laguerre_poly(2, x) == Q(1, 2)*x**2 - Q(4, 2)*x + 1 assert laguerre_poly(3, x) == -Q(1, 6)*x**3 + Q(9, 6)*x**2 - Q(18, 6)*x + 1 assert laguerre_poly(4, x) == Q( 1, 24)*x**4 - Q(16, 24)*x**3 + Q(72, 24)*x**2 - Q(96, 24)*x + 1 assert laguerre_poly(5, x) == -Q(1, 120)*x**5 + Q(25, 120)*x**4 - Q( 200, 120)*x**3 + Q(600, 120)*x**2 - Q(600, 120)*x + 1 assert laguerre_poly(6, x) == Q(1, 720)*x**6 - Q(36, 720)*x**5 + Q(450, 720)*x**4 - Q(2400, 720)*x**3 + Q(5400, 720)*x**2 - Q(4320, 720)*x + 1 assert laguerre_poly(0, x, a) == 1 assert laguerre_poly(1, x, a) == -x + a + 1 assert laguerre_poly(2, x, a) == x**2/2 + (-a - 2)*x + a**2/2 + 3*a/2 + 1 assert laguerre_poly(3, x, a) == -x**3/6 + (a/2 + Q( 3)/2)*x**2 + (-a**2/2 - 5*a/2 - 3)*x + a**3/6 + a**2 + 11*a/6 + 1 assert laguerre_poly(1).dummy_eq(-x + 1) assert laguerre_poly(1, polys=True) == Poly(-x + 1)
def main(): x = Symbol("x") a = Symbol("a") h = Symbol("h") show( limit(sqrt(x**2 - 5*x + 6) - x, x, oo), -Rational(5)/2 ) show( limit(x*(sqrt(x**2 + 1) - x), x, oo), Rational(1)/2 ) show( limit(x - sqrt3(x**3 - 1), x, oo), Rational(0) ) show( limit(log(1 + exp(x))/x, x, -oo), Rational(0) ) show( limit(log(1 + exp(x))/x, x, oo), Rational(1) ) show( limit(sin(3*x)/x, x, 0), Rational(3) ) show( limit(sin(5*x)/sin(2*x), x, 0), Rational(5)/2 ) show( limit(((x - 1)/(x + 1))**x, x, oo), exp(-2))
def _print_Rational(self, e): """Print a Rational object. :param e: The expression. :rtype : bce.dom.mathml.all.Base :return: The printed MathML object. """ assert isinstance(e, _sympy.Rational) if e.q == 1: # Don't do division if the denominator is 1. return _mathml.NumberComponent(str(e.p)) return _mathml.FractionComponent( _mathml.NumberComponent(str(e.p)), _mathml.NumberComponent(str(e.q)) )
def convert_float_string_to_rational(symbol): """Convert a string that contains a float number to SymPy's internal Rational type. :type symbol: str :param symbol: The string. :return: Converted value. """ # Find the position of decimal dot. dot_pos = symbol.find(".") if dot_pos < 0: raise ValueError("Missing decimal dot.") # Get the value of its numerator and denominator. numerator = int(symbol[:dot_pos] + symbol[dot_pos + 1:]) denominator = 10 ** (len(symbol) - 1 - dot_pos) return _sympy.Rational(numerator, denominator)
def as_coefficients_dict(self): """Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0. If an expression is not an Add it is considered to have a single term. Examples ======== >>> from sympy.abc import a, x >>> (3*x + a*x + 4).as_coefficients_dict() {1: 4, x: 3, a*x: 1} >>> _[a] 0 >>> (3*a*x).as_coefficients_dict() {a*x: 3} """ c, m = self.as_coeff_Mul() if not c.is_Rational: c = S.One m = self d = defaultdict(int) d.update({m: c}) return d
def primitive(self): """Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float). Examples ======== >>> from sympy.abc import x >>> (3*(x + 1)**2).primitive() (3, (x + 1)**2) >>> a = (6*x + 2); a.primitive() (2, 3*x + 1) >>> b = (x/2 + 3); b.primitive() (1/2, x + 6) >>> (a*b).primitive() == (1, a*b) True """ if not self: return S.One, S.Zero c, r = self.as_coeff_Mul(rational=True) if c.is_negative: c, r = -c, -r return c, r
def test_definition(): # want to test if the system can have several units of the same dimension dm = Quantity("dm", length, Rational(1, 10)) base = (m, s) base_dim = (m.dimension, s.dimension) ms = UnitSystem(base, (c, dm), "MS", "MS system") assert set(ms._base_units) == set(base) assert set(ms._units) == set((m, s, c, dm)) #assert ms._units == DimensionSystem._sort_dims(base + (velocity,)) assert ms.name == "MS" assert ms.descr == "MS system" assert ms._system._base_dims == DimensionSystem.sort_dims(base_dim) assert set(ms._system._dims) == set(base_dim + (velocity,))
def test_units(): assert convert_to((5*m/s * day) / km, 1) == 432 assert convert_to(foot / meter, meter) == Rational('0.3048') # amu is a pure mass so mass/mass gives a number, not an amount (mol) # TODO: need better simplification routine: assert str(convert_to(grams/amu, grams).n(2)) == '6.0e+23' # Light from the sun needs about 8.3 minutes to reach earth t = (1*au / speed_of_light) / minute # TODO: need a better way to simplify expressions containing units: t = convert_to(convert_to(t, meter / minute), meter) assert t == 49865956897/5995849160 # TODO: fix this, it should give `m` without `Abs` assert sqrt(m**2) == Abs(m) assert (sqrt(m))**2 == m t = Symbol('t') assert integrate(t*m/s, (t, 1*s, 5*s)) == 12*m*s assert (t * m/s).integrate((t, 1*s, 5*s)) == 12*m*s
def test_triangle_orth_exact(): x = numpy.array([Rational(1, 3), Rational(1, 7)]) L = 2 exacts = [ [sympy.sqrt(2)], [-Rational(8, 7), 8*sympy.sqrt(3)/21], [-197*sympy.sqrt(6)/441, -136*sympy.sqrt(2)/147, -26*sympy.sqrt(30)/441], ] bary = numpy.array([ x[0], x[1], 1-x[0]-x[1] ]) vals = orthopy.triangle.tree(L, bary, 'normal', symbolic=True) for val, ex in zip(vals, exacts): for v, e in zip(val, ex): assert v == e return
def test_triangle_orth_1_exact(): x = numpy.array([ [Rational(1, 5), Rational(2, 5), Rational(3, 5)], [Rational(1, 7), Rational(2, 7), Rational(3, 7)], ]) L = 2 exacts = [ [[1, 1, 1]], [ [-Rational(34, 35), Rational(2, 35), Rational(38, 35)], [Rational(2, 35), Rational(4, 35), Rational(6, 35)] ], ] bary = numpy.array([ x[0], x[1], 1-x[0]-x[1] ]) vals = orthopy.triangle.tree(L, bary, '1', symbolic=True) for val, ex in zip(vals, exacts): for v, e in zip(val, ex): assert numpy.all(v == e) return
def __init__(self, n): self.degree = 3 data = [ (fr(3-n, 3), z(n)), (fr(1, 6), fsd(n, (1, 1))), ] self.points, self.weights = untangle(data) reference_volume = 2**n self.weights *= reference_volume return
def __init__(self, n): self.degree = 3 data = [ (fr(2, 3), z(n)), (fr(1, 3 * 2**n), pm(n, 1)), ] self.points, self.weights = untangle(data) reference_volume = 2**n self.weights *= reference_volume return
def __init__(self, n, index): self.dim = n if index == 2: self.degree = 2 r = sqrt(3) / 6 data = [ (1.0, numpy.array([numpy.full(n, 2*r)])), (+r, _s(n, -1, r)), (-r, _s(n, +1, r)), ] else: assert index == 3 self.degree = 3 n2 = n // 2 if n % 2 == 0 else (n-1)//2 i_range = range(1, 2*n+1) pts = [[ [sqrt(fr(2, 3)) * cos((2*k-1)*i*pi / n) for i in i_range], [sqrt(fr(2, 3)) * sin((2*k-1)*i*pi / n) for i in i_range], ] for k in range(1, n2+1)] if n % 2 == 1: sqrt3pm = numpy.full(2*n, 1/sqrt(3)) sqrt3pm[1::2] *= -1 pts.append(sqrt3pm) pts = numpy.vstack(pts).T data = [(fr(1, 2*n), pts)] self.points, self.weights = untangle(data) reference_volume = 2**n self.weights *= reference_volume return
def __init__(self, n): self.degree = 5 r = sqrt(fr(2, 5)) data = [ (fr(8 - 5*n, 9), z(n)), (fr(5, 18), fsd(n, (r, 1))), (fr(1, 9 * 2**n), pm(n, 1)), ] self.points, self.weights = untangle(data) reference_volume = 2**n self.weights *= reference_volume return
def __init__(self, index): if index == '1-2': self.degree = 3 data = [ (1, fsd(2, (sqrt(fr(2, 3)), 1))) ] elif index == '2-2': self.degree = 5 alpha = sqrt(fr(3, 5)) data = [ (fr(64, 81), z(2)), (fr(40, 81), fsd(2, (alpha, 1))), (fr(25, 81), pm(2, alpha)), ] else: assert index == '3-2' self.degree = 7 alpha = sqrt(fr(3, 5)) xi1, xi2 = [ sqrt(fr(3, 287) * (38 - i*sqrt(583))) for i in [+1, -1] ] data = [ (fr(98, 405), fsd(2, (sqrt(fr(6, 7)), 1))), (0.5205929166673945, pm(2, xi1)), (0.2374317746906302, pm(2, xi2)), ] self.points, self.weights = untangle(data) return
def __init__(self): self.name = 'Burnside' self.degree = 5 r = sqrt(fr(7, 15)) s = sqrt(fr(7, 9)) data = [ (fr(10, 49), _symm_r_0(r)), (fr(9, 196), _symm_s(s)), ] self.points, self.weights = untangle(data) self.weights *= 4 return
def __init__(self): self.degree = 7 sqrt17770 = sqrt(17770) r, s = [ sqrt((1715 - plus_minus * 7 * sqrt17770) / 2817) for plus_minus in [+1, -1] ] t = sqrt(fr(7, 18)) u = sqrt(fr(7, 27)) B1, B2 = [ (2965 * sqrt17770 + plus_minus * 227816) / 72030 / sqrt17770 for plus_minus in [+1, -1] ] B3 = fr(324, 12005) B4 = fr(2187, 96040) data = [ (B1, fsd(3, (r, 1))), (B2, fsd(3, (s, 1))), (B3, fsd(3, (t, 2))), (B4, pm(3, u)), ] self.points, self.weights = untangle(data) self.weights *= fr(4, 3) * pi return
def __init__(self): self.degree = 3 data = [ (fr(1, 4), z()), (fr(1, 12), fs_r00(1)), (fr(1, 48), fs_rr0(1)), ] self.points, self.weights = untangle(data) self.weights *= 8 return
def __init__(self): self.degree = 5 sqrt19 = sqrt(19) t = sqrt(71440 + 6802 * sqrt19) lmbd, gmma = [ sqrt((1919 - 148*sqrt19 + i * 4*t) / 3285) for i in [+1, -1] ] xi, mu = [ -sqrt((1121 + 74*sqrt19 - i * 2*t) / 3285) for i in [+1, -1] ] mu *= -1 B, C = [ 133225 / (260072 - 1520*sqrt19 + i*(133 - 37*sqrt19)*t) for i in [+1, -1] ] data = [ (fr(32, 19), z()), (B, rss_pm(lmbd, xi)), (C, rss_pm(gmma, mu)), ] self.points, self.weights = untangle(data) return
def __init__(self): self.degree = 5 data = [ (fr(91, 450), fs_r00(1)), (fr(-20, 225), fs_rr0(1)), (fr(8, 225), fs_rrs(sqrt(fr(5, 8)), 1)), ] self.points, self.weights = untangle(data) self.weights *= 8 return
def viii(): nu = sqrt(30) eta = sqrt(10) A = fr(3, 5) B = fr(2, 75) C = fr(3, 100) data = [ (A, numpy.array([[0, 0, 0]])), (B, fsd(3, (nu, 1))), (C, pm(3, eta)), ] return 5, data
def ix(): eta = sqrt(10) xi, nu = [sqrt(15 - p_m * 5 * sqrt(5)) for p_m in [+1, -1]] A = fr(3, 5) B = fr(1, 50) data = [ (A, numpy.array([[0, 0, 0]])), (B, pm(3, eta)), (B, pm_roll(3, [xi, nu])), ] return 5, data
def viiia(): r = sqrt(fr(5, 4)) s = sqrt(fr(5, 2)) data = [ (fr(4, 25), fsd(3, (r, 1))), (fr(1, 200), pm(3, s)), ] return 5, data
def viiib(): r = sqrt(fr(5, 2)) s = sqrt(fr(5, 6)) data = [ (fr(2, 5), numpy.array([[0, 0, 0]])), (fr(1, 25), fsd(3, (r, 1))), (fr(9, 200), pm(3, s)), ] return 5, data
def __init__(self, n, index): self.dim = n if index == 'I': self.degree = 2 data = [ (fr(1, n+1), sqrt(fr(1, 2)) * _nsimplex(n)) ] elif index == 'II': self.degree = 3 nu = sqrt(fr(n, 2)) data = [ (fr(1, 2*n), fsd(n, (nu, 1))) ] elif index == 'III': self.degree = 3 nu = sqrt(fr(1, 2)) data = [ (fr(1, 2**n), pm(n, nu)) ] else: assert index == 'IV' self.degree = 5 nu = sqrt(fr(n+2, 2)) xi = sqrt(fr(n+2, 4)) A = fr(2, n+2) B = fr(4-n, 2 * (n+2)**2) C = fr(1, (n+2)**2) data = [ (A, numpy.full((1, n), 0)), (B, fsd(n, (nu, 1))), (C, fsd(n, (xi, 2))), ] self.points, self.weights = untangle(data) self.weights *= sqrt(pi)**n return
