Python sympy.log方法代碼示例

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def Simple_manifold_with_scalar_function_derivative():
    coords = (x,y,z) = symbols('x y z')
    basis = (e1, e2, e3, grad) = MV.setup('e_1 e_2 e_3',metric='[1,1,1]',coords=coords)
    # Define surface
    mfvar = (u,v) = symbols('u v')
    X = u*e1+v*e2+(u**2+v**2)*e3
    print X
    MF = Manifold(X,mfvar)

    # Define field on the surface.
    g = (v+1)*log(u)

    # Method 1: Using old Manifold routines.
    VectorDerivative = (MF.rbasis[0]/MF.E_sq)*diff(g,u) + (MF.rbasis[1]/MF.E_sq)*diff(g,v)
    print 'Vector derivative =', VectorDerivative.subs({u:1,v:0})

    # Method 2: Using new Manifold routines.
    dg = MF.Grad(g)
    print 'Vector derivative =', dg.subs({u:1,v:0})
    return 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def Simple_manifold_with_scalar_function_derivative():
    Print_Function()
    coords = (x,y,z) = symbols('x y z')
    basis = (e1, e2, e3, grad) = MV.setup('e_1 e_2 e_3',metric='[1,1,1]',coords=coords)
    # Define surface
    mfvar = (u,v) = symbols('u v')
    X = u*e1+v*e2+(u**2+v**2)*e3
    print '\\f{X}{u,v} =',X
    MF = Manifold(X,mfvar)
    (eu,ev) = MF.Basis()
    # Define field on the surface.
    g = (v+1)*log(u)

    print '\\f{g}{u,v} =',g

    # Method 1: Using old Manifold routines.
    VectorDerivative = (MF.rbasis[0]/MF.E_sq)*diff(g,u) + (MF.rbasis[1]/MF.E_sq)*diff(g,v)
    print '\\eval{\\nabla g}{u=1,v=0} =', VectorDerivative.subs({u:1,v:0})

    # Method 2: Using new Manifold routines.
    dg = MF.Grad(g)
    print '\\eval{\\f{Grad}{g}}{u=1,v=0} =', dg.subs({u:1,v:0})
    dg = MF.grad*g
    print '\\eval{\\nabla g}{u=1,v=0} =', dg.subs({u:1,v:0})
    return 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def grad_log_norm_symbolic(self):
        import sympy
        D = self.dimension
        X = self.eigenvalues_symbol
        B = [1] * D
        for d in range(D):
            for dd in range(D):
                if d != dd:
                    B[d] = B[d] * (X[d] - X[dd])
        B = [1 / b for b in B]

        p_D = sympy.pi ** D

        tmp = [b * sympy.exp(x_) for x_, b in zip(X, B)]
        tmp = sum(tmp)
        symbolic_norm_for_bingham = 2 * p_D * tmp

        return [
            sympy.simplify(sympy.diff(
                sympy.log(symbolic_norm_for_bingham),
                x_
            ))
            for x_ in X
        ] 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def _compute_local_sens_gnmax(logq, sigma, num_classes, order):
  """Implements Algorithm 3 (computes an upper bound on local sensitivity).

  (See Proposition 13 for proof of correctness.)
  """
  logq0 = _compute_logq0(sigma, order)
  logq1 = _compute_logq1(sigma, order, num_classes)
  if logq1 <= logq <= logq0:
    logq = logq1

  beta = _compute_rdp_gnmax(sigma, logq, order)
  beta_bu_q = _compute_rdp_gnmax(
      sigma, math.log(_compute_bu_gnmax(math.exp(logq), sigma, num_classes)),
      order)
  beta_bl_q = _compute_rdp_gnmax(
      sigma, math.log(_compute_bl_gnmax(math.exp(logq), sigma, num_classes)),
      order)
  return max(beta_bu_q - beta, beta - beta_bl_q) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def test_expression_builder_complex(self, test_time_series):
        from sympy import symbols, cos, sin, pi, log, sqrt

        abc = list("abcdefghij")
        syms = symbols(abc)
        expression = syms[0]
        for s in syms:
            expression = expression + s
        expression = (
            (expression * expression)
            + sqrt(sin(pi * 0.1 ** syms[1]))
            + log(23 + syms[5] ** 1.234)
            + cos(syms[3] ** (1 + 0.1 ** syms[4]))
            + sqrt(log(abs(syms[8]) + 1))
        )
        dps1 = COGNITE_CLIENT.datapoints.synthetic.query(
            expressions=[expression],
            start=datetime(2017, 1, 1),
            end="now",
            limit=100,
            variables={v: test_time_series[tsi] for v, tsi in zip(abc, range(10))},
        )[0]
        assert 100 == len(dps1) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def test_singularities_non_rational():
    x = Symbol('x', real=True)

    assert singularities(exp(1/x), x) == (0)
    assert singularities(log((x - 2)**2), x) == (2) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def Distorted_manifold_with_scalar_function():
    coords = symbols('x y z')
    (ex,ey,ez,grad) = MV.setup('ex ey ez',metric='[1,1,1]',coords=coords)
    mfvar = (u,v) = symbols('u v')
    X = 2*u*ex+2*v*ey+(u**3+v**3/2)*ez
    MF = Manifold(X,mfvar,I=MV.I)

    (eu,ev) = MF.Basis()

    g = (v+1)*log(u)
    dg = MF.Grad(g)
    print 'g =',g
    print 'dg =',dg
    print 'dg(1,0) =',dg.subs({u:1,v:0})
    G = u*eu+v*ev
    dG = MF.Grad(G)
    print 'G =',G
    print 'P(G) =',MF.Proj(G)
    print 'zcoef =',simplify(2*(u**2 + v**2)*(-4*u**2 - 4*v**2 - 1))
    print 'dG =',dG
    print 'P(dG) =',MF.Proj(dG)
    PS = u*v*eu^ev
    print 'PS =',PS
    print 'dPS =',MF.Grad(PS)
    print 'P(dPS) =',MF.Proj(MF.Grad(PS))
    return 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def Distorted_manifold_with_scalar_function():
    Print_Function()
    coords = symbols('x y z')
    (ex,ey,ez,grad) = MV.setup('e_x e_y e_z',metric='[1,1,1]',coords=coords)
    mfvar = (u,v) = symbols('u v')
    X = 2*u*ex+2*v*ey+(u**3+v**3/2)*ez
    MF = Manifold(X,mfvar,I=MV.I)

    (eu,ev) = MF.Basis()

    g = (v+1)*log(u)
    dg = MF.Grad(g)
    print 'g =',g
    print 'dg =',dg
    print '\\eval{dg}{u=1,v=0} =',dg.subs({u:1,v:0})
    G = u*eu+v*ev
    dG = MF.Grad(G)
    print 'G =',G
    print 'P(G) =',MF.Proj(G)
    print 'dG =',dG
    print 'P(dG) =',MF.Proj(dG)
    PS = u*v*eu^ev
    print 'P(S) =',PS
    print 'dP(S) =',MF.Grad(PS)
    print 'P(dP(S)) =',MF.Proj(MF.Grad(PS))
    return 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def Simple_manifold_with_vector_function_derivative():
    Print_Function()
    coords = (x,y,z) = symbols('x y z')
    basis = (ex, ey, ez, grad) = \
            MV.setup('e_x e_y e_z',metric='[1,1,1]',coords=coords)
    # Define surface
    mfvar = (u,v) = symbols('u v')
    X = u*ex+v*ey+(u**2+v**2)*ez
    print '\\f{X}{u,v} =',X
    MF = Manifold(X,mfvar)
    (eu,ev) = MF.Basis()

    # Define field on the surface.
    g = (v+1)*log(u)

    print '\\mbox{Scalar Function: } g =',g
    dg = MF.grad*g
    dg.Fmt(3,'\\mbox{Scalar Function Derivative: } \\nabla g')
    print '\\eval{\\nabla g}{(1,0)} =',dg.subs({u:1,v:0})

    # Define vector field on the surface

    G = v**2*eu+u**2*ev
    print '\\mbox{Vector Function: } G =',G
    dG = MF.grad*G
    dG.Fmt(3,'\\mbox{Vector Function Derivative: } \\nabla G')
    print '\\eval{\\nabla G}{(1,0)} =',dG.subs({u:1,v:0})

    return 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def log_norm(self, remove_duplicate_eigenvalues=True):
        return np.log(self.norm(remove_duplicate_eigenvalues=remove_duplicate_eigenvalues)) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def _compute_logq1(sigma, order, num_classes):
  logq0 = _compute_logq0(sigma, order)  # Most likely already cached.
  logq1 = math.log(_compute_bl_gnmax(math.exp(logq0), sigma, num_classes))
  assert logq1 <= logq0
  return logq1 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def _compute_data_dep_bound_gnmax(sigma, logq, order):
  # Applies Theorem 6 in Appendix without checking that logq satisfies necessary
  # constraints. The pre-conditions must be assured by comparing logq against
  # logq0 by the caller.
  variance = sigma**2
  mu1, mu2 = _compute_mu1_mu2_gnmax(sigma, logq)
  eps1 = mu1 / variance
  eps2 = mu2 / variance

  log1q = np.log1p(-math.exp(logq))  # log1q = log(1-q)
  log_a = (order - 1) * (
      log1q - (np.log1p(-math.exp((logq + eps2) * (1 - 1 / mu2)))))
  log_b = (order - 1) * (eps1 - logq / (mu1 - 1))

  return np.logaddexp(log1q + log_a, logq + log_b) / (order - 1) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def compute_rdp_of_smooth_sensitivity_gaussian(beta, sigma, order):
  """Computes the RDP curve for the GNSS mechanism.

  Implements Theorem 23 (https://arxiv.org/pdf/1802.08908.pdf).
  """
  if beta > 0 and not 1 < order < 1 / (2 * beta):
    raise ValueError("Order outside the (1, 1/(2*beta)) range.")

  return order * math.exp(2 * beta) / sigma**2 + (
      -.5 * math.log(1 - 2 * order * beta) + beta * order) / (
          order - 1) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def test_log():
    x = Symbol("x")
    x1 = sympy.Symbol("x")

    assert log(x) == log(x1)
    assert log(x)._sympy_() == sympy.log(x1)
    assert sympify(sympy.log(x1)) == log(x)

    y = Symbol("y")
    y1 = sympy.Symbol("y")

    assert log(x, y) == log(x, y1)
    assert log(x1, y) == log(x1, y1)
    assert log(x, y)._sympy_() == sympy.log(x1, y1)
    assert sympify(sympy.log(x1, y1)) == log(x, y) 

# 需要導入模塊: import sympy [as 別名]
# 或者: from sympy import log [as 別名]
def test_entropy():
    up = JzKet(S(1)/2, S(1)/2)
    down = JzKet(S(1)/2, -S(1)/2)
    d = Density((up, 0.5), (down, 0.5))

    # test for density object
    ent = entropy(d)
    assert entropy(d) == 0.5*log(2)
    assert d.entropy() == 0.5*log(2)

    np = import_module('numpy', min_module_version='1.4.0')
    if np:
        #do this test only if 'numpy' is available on test machine
        np_mat = represent(d, format='numpy')
        ent = entropy(np_mat)
        assert isinstance(np_mat, np.matrixlib.defmatrix.matrix)
        assert ent.real == 0.69314718055994529
        assert ent.imag == 0

    scipy = import_module('scipy', __import__kwargs={'fromlist': ['sparse']})
    if scipy and np:
        #do this test only if numpy and scipy are available
        mat = represent(d, format="scipy.sparse")
        assert isinstance(mat, scipy_sparse_matrix)
        assert ent.real == 0.69314718055994529
        assert ent.imag == 0