Python sympy.simplify方法代码示例

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def generate_algebra_simplify_sample(vlist, ops, min_depth, max_depth):
  """Randomly generate an algebra simplify dataset sample.

  Given an input expression, produce the simplified expression.

  See go/symbolic-math-dataset.

  Args:
    vlist: Variable list. List of chars that can be used in the expression.
    ops: List of ExprOp instances. The allowed operators for the expression.
    min_depth: Expression trees will not have a smaller depth than this. 0 means
        there is just a variable. 1 means there is one operation.
    max_depth: Expression trees will not have a larger depth than this. To make
        all trees have the same depth, set this equal to `min_depth`.

  Returns:
    sample: String representation of the input.
    target: String representation of the solution.
  """
  depth = random.randrange(min_depth, max_depth + 1)
  expr = random_expr(depth, vlist, ops)

  sample = str(expr)
  target = format_sympy_expr(sympy.simplify(sample))
  return sample, target 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def testAlgebraInverse(self):
    dataset_objects = algorithmic_math.math_dataset_init(26)
    counter = 0
    for d in algorithmic_math.algebra_inverse(26, 0, 3, 10):
      counter += 1
      decoded_input = dataset_objects.int_decoder(d["inputs"])
      solve_var, expression = decoded_input.split(":")
      lhs, rhs = expression.split("=")

      # Solve for the solve-var.
      result = sympy.solve("%s-(%s)" % (lhs, rhs), solve_var)
      target_expression = dataset_objects.int_decoder(d["targets"])

      # Check that the target and sympy's solutions are equivalent.
      self.assertEqual(
          0, sympy.simplify(str(result[0]) + "-(%s)" % target_expression))
    self.assertEqual(counter, 10) 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def testCalculusIntegrate(self):
    dataset_objects = algorithmic_math.math_dataset_init(
        8, digits=5, functions={"log": "L"})
    counter = 0
    for d in algorithmic_math.calculus_integrate(8, 0, 3, 10):
      counter += 1
      decoded_input = dataset_objects.int_decoder(d["inputs"])
      var, expression = decoded_input.split(":")
      target = dataset_objects.int_decoder(d["targets"])

      for fn_name, fn_char in six.iteritems(dataset_objects.functions):
        target = target.replace(fn_char, fn_name)

      # Take the derivative of the target.
      derivative = str(sympy.diff(target, var))

      # Check that the derivative of the integral equals the input.
      self.assertEqual(0, sympy.simplify("%s-(%s)" % (expression, derivative)))
    self.assertEqual(counter, 10) 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def simplify_surd(value, sample_args, context=None):
  """E.g., "Simplify (2 + 5*sqrt(3))**2."."""
  del value  # unused
  if context is None:
    context = composition.Context()

  entropy, sample_args = sample_args.peel()

  while True:
    base = random.randint(2, 20)
    if sympy.Integer(base).is_prime:
      break
  num_primes_less_than_20 = 8
  entropy -= math.log10(num_primes_less_than_20)
  exp = _sample_surd(base, entropy, max_power=2, multiples_only=False)
  simplified = sympy.expand(sympy.simplify(exp))

  template = random.choice([
      'Simplify {exp}.',
  ])
  return example.Problem(
      question=example.question(context, template, exp=exp),
      answer=simplified) 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def bessel_basis(n, k):
    zeros = Jn_zeros(n, k)
    normalizer = []
    for order in range(n):
        normalizer_tmp = []
        for i in range(k):
            normalizer_tmp += [0.5 * Jn(zeros[order, i], order + 1)**2]
        normalizer_tmp = 1 / np.array(normalizer_tmp)**0.5
        normalizer += [normalizer_tmp]

    f = spherical_bessel_formulas(n)
    x = sym.symbols('x')
    bess_basis = []
    for order in range(n):
        bess_basis_tmp = []
        for i in range(k):
            bess_basis_tmp += [
                sym.simplify(normalizer[order][i] *
                             f[order].subs(x, zeros[order, i] * x))
            ]
        bess_basis += [bess_basis_tmp]
    return bess_basis 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def associated_legendre_polynomials(k, zero_m_only=True):
    z = sym.symbols('z')
    P_l_m = [[0] * (j + 1) for j in range(k)]

    P_l_m[0][0] = 1
    if k > 0:
        P_l_m[1][0] = z

        for j in range(2, k):
            P_l_m[j][0] = sym.simplify(((2 * j - 1) * z * P_l_m[j - 1][0] -
                                        (j - 1) * P_l_m[j - 2][0]) / j)
        if not zero_m_only:
            for i in range(1, k):
                P_l_m[i][i] = sym.simplify((1 - 2 * i) * P_l_m[i - 1][i - 1])
                if i + 1 < k:
                    P_l_m[i + 1][i] = sym.simplify(
                        (2 * i + 1) * z * P_l_m[i][i])
                for j in range(i + 2, k):
                    P_l_m[j][i] = sym.simplify(
                        ((2 * j - 1) * z * P_l_m[j - 1][i] -
                         (i + j - 1) * P_l_m[j - 2][i]) / (j - i))

    return P_l_m 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def generate_algebra_simplify_sample(vlist, ops, min_depth, max_depth):
  """Randomly generate an algebra simplify dataset sample.

  Given an input expression, produce the simplified expression.

  Args:
    vlist: Variable list. List of chars that can be used in the expression.
    ops: List of ExprOp instances. The allowed operators for the expression.
    min_depth: Expression trees will not have a smaller depth than this. 0 means
        there is just a variable. 1 means there is one operation.
    max_depth: Expression trees will not have a larger depth than this. To make
        all trees have the same depth, set this equal to `min_depth`.

  Returns:
    sample: String representation of the input.
    target: String representation of the solution.
  """
  depth = random.randrange(min_depth, max_depth + 1)
  expr = random_expr(depth, vlist, ops)

  sample = str(expr)
  target = format_sympy_expr(sympy.simplify(sample))
  return sample, target 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def get_equations(self):
        """
        :return: Functions to calculate A, B and f given state x and input u
        """
        f = sp.zeros(3, 1)

        x = sp.Matrix(sp.symbols('x y theta', real=True))
        u = sp.Matrix(sp.symbols('v w', real=True))

        f[0, 0] = u[0, 0] * sp.cos(x[2, 0])
        f[1, 0] = u[0, 0] * sp.sin(x[2, 0])
        f[2, 0] = u[1, 0]

        f = sp.simplify(f)
        A = sp.simplify(f.jacobian(x))
        B = sp.simplify(f.jacobian(u))

        f_func = sp.lambdify((x, u), f, 'numpy')
        A_func = sp.lambdify((x, u), A, 'numpy')
        B_func = sp.lambdify((x, u), B, 'numpy')

        return f_func, A_func, B_func 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def get_equations(self):
        """
        :return: Functions to calculate A, B and f given state x and input u
        """
        f = sp.zeros(6, 1)

        x = sp.Matrix(sp.symbols('rx ry vx vy t w', real=True))
        u = sp.Matrix(sp.symbols('gimbal T', real=True))

        f[0, 0] = x[2, 0]
        f[1, 0] = x[3, 0]
        f[2, 0] = 1 / self.m * sp.sin(x[4, 0] + u[0, 0]) * u[1, 0]
        f[3, 0] = 1 / self.m * (sp.cos(x[4, 0] + u[0, 0]) * u[1, 0] - self.m * self.g)
        f[4, 0] = x[5, 0]
        f[5, 0] = 1 / self.I * (-sp.sin(u[0, 0]) * u[1, 0] * self.r_T)

        f = sp.simplify(f)
        A = sp.simplify(f.jacobian(x))
        B = sp.simplify(f.jacobian(u))

        f_func = sp.lambdify((x, u), f, 'numpy')
        A_func = sp.lambdify((x, u), A, 'numpy')
        B_func = sp.lambdify((x, u), B, 'numpy')

        return f_func, A_func, B_func 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def derivatives_in_spherical_coordinates():
    Print_Function()
    X = (r,th,phi) = symbols('r theta phi')
    curv = [[r*cos(phi)*sin(th),r*sin(phi)*sin(th),r*cos(th)],[1,r,r*sin(th)]]
    (er,eth,ephi,grad) = MV.setup('e_r e_theta e_phi',metric='[1,1,1]',coords=X,curv=curv)

    f = MV('f','scalar',fct=True)
    A = MV('A','vector',fct=True)
    B = MV('B','grade2',fct=True)

    print('f =',f)
    print('A =',A)
    print('B =',B)

    print('grad*f =',grad*f)
    print('grad|A =',grad|A)
    print('-I*(grad^A) =',(-MV.I*(grad^A)).simplify())
    print('grad^B =',grad^B) 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def derivatives_in_spherical_coordinates():
    Print_Function()
    X = (r,th,phi) = symbols('r theta phi')
    s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True)
    (er,eth,ephi) = s3d.mv()
    grad = s3d.grad

    f = s3d.mv('f','scalar',f=True)
    A = s3d.mv('A','vector',f=True)
    B = s3d.mv('B','bivector',f=True)

    print('f =',f)
    print('A =',A)
    print('B =',B)

    print('grad*f =',grad*f)
    print('grad|A =',grad|A)
    print('-I*(grad^A) =',(-s3d.E()*(grad^A)).simplify())
    print('grad^B =',grad^B) 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def test_check_generalized_BAC_CAB_formulas(self):

        a, b, c, d, e = Ga('a b c d e').mv()

        assert str(a|(b*c)) == '-(a.c)*b + (a.b)*c'
        assert str(a|(b^c)) == '-(a.c)*b + (a.b)*c'
        assert str(a|(b^c^d)) == '(a.d)*b^c - (a.c)*b^d + (a.b)*c^d'

        expr = (a|(b^c))+(c|(a^b))+(b|(c^a)) # = (a.b)*c - (b.c)*a - ((a.b)*c - (b.c)*a)
        assert str(expr.simplify()) == '0'

        assert str(a*(b^c)-b*(a^c)+c*(a^b)) == '3*a^b^c'
        assert str(a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)) == '4*a^b^c^d'
        assert str((a^b)|(c^d)) == '-(a.c)*(b.d) + (a.d)*(b.c)'
        assert str(((a^b)|c)|d) == '-(a.c)*(b.d) + (a.d)*(b.c)'
        assert str(Ga.com(a^b, c^d)) == '-(b.d)*a^c + (b.c)*a^d + (a.d)*b^c - (a.c)*b^d'
        assert str((a|(b^c))|(d^e)) == '(-(a.b)*(c.e) + (a.c)*(b.e))*d + ((a.b)*(c.d) - (a.c)*(b.d))*e' 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def test_derivatives_in_spherical_coordinates(self):

        X = r, th, phi = symbols('r theta phi')
        s3d = Ga('e_r e_theta e_phi', g=[1, r ** 2, r ** 2 * sin(th) ** 2], coords=X, norm=True)
        er, eth, ephi = s3d.mv()
        grad = s3d.grad

        f = s3d.mv('f', 'scalar', f=True)
        A = s3d.mv('A', 'vector', f=True)
        B = s3d.mv('B', 'bivector', f=True)

        assert str(f) == 'f'
        assert str(A) == 'A__r*e_r + A__theta*e_theta + A__phi*e_phi'
        assert str(B) == 'B__rtheta*e_r^e_theta + B__rphi*e_r^e_phi + B__thetaphi*e_theta^e_phi'

        assert str(grad*f) == 'D{r}f*e_r + D{theta}f*e_theta/r + D{phi}f*e_phi/(r*sin(theta))'
        assert str((grad|A).simplify()) == '(r*D{r}A__r + 2*A__r + A__theta/tan(theta) + D{theta}A__theta + D{phi}A__phi/sin(theta))/r'
        assert str(-s3d.I()*(grad^A)) == '(A__phi/tan(theta) + D{theta}A__phi - D{phi}A__theta/sin(theta))*e_r/r + (-r*D{r}A__phi - A__phi + D{phi}A__r/sin(theta))*e_theta/r + (r*D{r}A__theta + A__theta - D{theta}A__r)*e_phi/r'

        assert latex(grad) == r'\boldsymbol{e}_{r} \frac{\partial}{\partial r} + \boldsymbol{e}_{\theta } \frac{1}{r} \frac{\partial}{\partial \theta } + \boldsymbol{e}_{\phi } \frac{1}{r \sin{\left (\theta  \right )}} \frac{\partial}{\partial \phi }'
        assert latex(B|(eth^ephi)) == r'- B^{\theta \phi } {\left (r,\theta ,\phi  \right )}'

        assert str(grad^B) == '(r*D{r}B__thetaphi - B__rphi/tan(theta) + 2*B__thetaphi - D{theta}B__rphi + D{phi}B__rtheta/sin(theta))*e_r^e_theta^e_phi/r' 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
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
        ] 

# 需要导入模块: import sympy [as 别名]
# 或者: from sympy import simplify [as 别名]
def simplifyEquation(input_str):
    """
    Only simplify the equation if there is no obvious problem
    Equation is not simplified if it includes the following terms:
        exp
        log
    """
    s_list = list()
    # these are considered the "dangerous" operation that will
    # overflow/underflow in np
    s_list.append(len(input_str.atoms(exp)))
    s_list.append(len(input_str.atoms(log)))

    if np.sum(s_list) != 0:
        # it is dangerous to simplify!
        return input_str, True
    else:
        #TODO: Removed actual simplyify (do we need it?)
        return input_str, False