Python sympy 模块，Integer() 实例源码

def test_quadratic_perfect_square():

    # B**2 - 4*A*C > 0
    # B**2 - 4*A*C is a perfect square
    assert diop_solve(48*x*y) == set([(Integer(0), t), (t, Integer(0))])
    assert diop_solve(4*x**2 - 5*x*y + y**2 + 2) == \
        set([(-Integer(1), -Integer(3)),(-Integer(1), -Integer(2)),(Integer(1), Integer(2)),(Integer(1), Integer(3))])
    assert diop_solve(-2*x**2 - 3*x*y + 2*y**2 -2*x - 17*y + 25) == set([(Integer(4), Integer(15))])
    assert diop_solve(12*x**2 + 13*x*y + 3*y**2 - 2*x + 3*y - 12) == \
        set([(-Integer(6), Integer(9)), (-Integer(2), Integer(5)), (Integer(4), -Integer(4)), (-Integer(6), Integer(16))])
    assert diop_solve(8*x**2 + 10*x*y + 2*y**2 - 32*x - 13*y - 23) == \
        set([(-Integer(44), Integer(47)), (Integer(22), -Integer(85))])
    assert diop_solve(4*x**2 - 4*x*y - 3*y- 8*x - 3) == \
        set([(-Integer(1), -Integer(9)), (-Integer(6), -Integer(9)), (Integer(0), -Integer(1)), (Integer(1), -Integer(1))])
    assert diop_solve(- 4*x*y - 4*y**2 - 3*y- 5*x - 10) == \
        set([(-Integer(2), Integer(0)), (-Integer(11), -Integer(1)), (-Integer(5), Integer(5))])
    assert diop_solve(x**2 - y**2 - 2*x - 2*y) == set([(t, -t), (-t, -t - 2)])
    assert diop_solve(x**2 - 9*y**2 - 2*x - 6*y) == set([(-3*t + 2, -t), (3*t, -t)])
    assert diop_solve(4*x**2 - 9*y**2 - 4*x - 12*y - 3) == set([(-3*t - 3, -2*t - 3), (3*t + 1, -2*t - 1)])

def test_commutator():
    A = Operator('A')
    B = Operator('B')
    c = Commutator(A, B)
    c_tall = Commutator(A**2, B)
    assert str(c) == '[A,B]'
    assert pretty(c) == '[A,B]'
    assert upretty(c) == u('[A,B]')
    assert latex(c) == r'\left[A,B\right]'
    sT(c, "Commutator(Operator(Symbol('A')),Operator(Symbol('B')))")
    assert str(c_tall) == '[A**2,B]'
    ascii_str = \
"""\
[ 2  ]\n\
[A ,B]\
"""
    ucode_str = \
u("""\
? 2  ?\n\
?A ,B?\
""")
    assert pretty(c_tall) == ascii_str
    assert upretty(c_tall) == ucode_str
    assert latex(c_tall) == r'\left[\left(A\right)^{2},B\right]'
    sT(c_tall, "Commutator(Pow(Operator(Symbol('A')), Integer(2)),Operator(Symbol('B')))")

def test_scalars():
    x = symbols('x', complex=True)
    assert Dagger(x) == conjugate(x)
    assert Dagger(I*x) == -I*conjugate(x)

    i = symbols('i', real=True)
    assert Dagger(i) == i

    p = symbols('p')
    assert isinstance(Dagger(p), adjoint)

    i = Integer(3)
    assert Dagger(i) == i

    A = symbols('A', commutative=False)
    assert Dagger(A).is_commutative is False

def _represent_NumberOp(self, basis, **options):
        ndim_info = options.get('ndim', 4)
        format = options.get('format', 'sympy')
        options['spmatrix'] = 'lil'
        vector = matrix_zeros(ndim_info, 1, **options)
        if isinstance(self.n, Integer):
            if self.n >= ndim_info:
                return ValueError("N-Dimension too small")
            value = Integer(1)
            if format == 'scipy.sparse':
                vector[int(self.n), 0] = 1.0
                vector = vector.tocsr()
            elif format == 'numpy':
                vector[int(self.n), 0] = 1.0
            else:
                vector[self.n, 0] = Integer(1)
            return vector
        else:
            return ValueError("Not Numerical State")

def _represent_NumberOp(self, basis, **options):
        ndim_info = options.get('ndim', 4)
        format = options.get('format', 'sympy')
        options['spmatrix'] = 'lil'
        vector = matrix_zeros(1, ndim_info, **options)
        if isinstance(self.n, Integer):
            if self.n >= ndim_info:
                return ValueError("N-Dimension too small")
            if format == 'scipy.sparse':
                vector[0, int(self.n)] = 1.0
                vector = vector.tocsr()
            elif format == 'numpy':
                vector[0, int(self.n)] = 1.0
            else:
                vector[0, self.n] = Integer(1)
            return vector
        else:
            return ValueError("Not Numerical State")

def eval(cls, a, b):
        if not (a and b):
            return S.Zero
        if a == b:
            return Integer(2)*a**2
        if a.is_commutative or b.is_commutative:
            return Integer(2)*a*b

        # [xA,yB]  ->  xy*[A,B]
        # from sympy.physics.qmul import QMul
        ca, nca = a.args_cnc()
        cb, ncb = b.args_cnc()
        c_part = ca + cb
        if c_part:
            return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb)))

        # Canonical ordering of arguments
        #The Commutator [A,B] is on canonical form if A < B.
        if a.compare(b) == 1:
            return cls(b, a)

def do_arg(a):
    if isinstance(a, str):
        arg = Symbol(a, integer=True)
    elif isinstance(a, (Integer, Float)):
        arg = a
    elif isinstance(a, ArithmeticExpression):
        arg = a.expr
        if isinstance(arg, (Symbol, Variable)):
            arg = Symbol(arg.name, integer=True)
        else:
            arg = convert_to_integer_expression(arg)
    else:
        raise Exception('Wrong instance in do_arg')

    return arg
# ...

# ... TODO improve. this version is not working with function calls

def extract_arg(self, name):
        """
        returns an argument as a variable, given its name

        name: str
            variable name
        """
        if name is None:
            return None

        var = None
        if isinstance(name, (Integer, Float)):
            var = Integer(name)
        elif isinstance(name, str):
            if name in namespace:
                var = namespace[name]
            else:
                raise Exception("could not find {} in namespace ".format(name))
        elif isinstance(name, ArithmeticExpression):
            var = do_arg(name)
        else:
            raise Exception("Unexpected type {0} for {1}".format(type(name), name))

        return var

def monomial_basis(*degrees):
    """
    Returns the product basis of (kx, ky, kz), with monomials of the given degrees.

    :param degrees: Degree of the monomials. Multiple degrees can be given, in which case the basis consists of the monomials of all given degrees.
    :type degrees: int

    Example:

        >>> import kdotp_symmetry as kp
        >>> kp.monomial_basis(*range(3))
        [1, kx, ky, kz, kx**2, kx*ky, kx*kz, ky**2, ky*kz, kz**2]
    """
    if any(deg < 0 for deg in degrees):
        raise ValueError('Degrees must be non-negative integers')
    basis = []
    for deg in sorted(degrees):
        monomial_tuples = combinations_with_replacement(K_VEC, deg)
        basis.extend(
            reduce(operator.mul, m, sp.Integer(1)) for m in monomial_tuples
        )
    return basis

def has_integer_powers(self):
        """
        Check if the dimension object has only integer powers.

        All the dimension powers should be integers, but rational powers may
        appear in intermediate steps. This method may be used to check that the
        final result is well-defined.
        """

        for dpow in self.get_dimensional_dependencies().values():
            if not isinstance(dpow, (int, Integer)):
                return False
        else:
            return True


# base dimensions (MKS)

def test_commutator():
    A = Operator('A')
    B = Operator('B')
    c = Commutator(A, B)
    c_tall = Commutator(A**2, B)
    assert str(c) == '[A,B]'
    assert pretty(c) == '[A,B]'
    assert upretty(c) == u'[A,B]'
    assert latex(c) == r'\left[A,B\right]'
    sT(c, "Commutator(Operator(Symbol('A')),Operator(Symbol('B')))")
    assert str(c_tall) == '[A**2,B]'
    ascii_str = \
"""\
[ 2  ]\n\
[A ,B]\
"""
    ucode_str = \
u("""\
? 2  ?\n\
?A ,B?\
""")
    assert pretty(c_tall) == ascii_str
    assert upretty(c_tall) == ucode_str
    assert latex(c_tall) == r'\left[A^{2},B\right]'
    sT(c_tall, "Commutator(Pow(Operator(Symbol('A')), Integer(2)),Operator(Symbol('B')))")

def test_scalars():
    x = symbols('x', complex=True)
    assert Dagger(x) == conjugate(x)
    assert Dagger(I*x) == -I*conjugate(x)

    i = symbols('i', real=True)
    assert Dagger(i) == i

    p = symbols('p')
    assert isinstance(Dagger(p), adjoint)

    i = Integer(3)
    assert Dagger(i) == i

    A = symbols('A', commutative=False)
    assert Dagger(A).is_commutative is False

def test_IntQubit():
    iqb = IntQubit(8)
    assert iqb.as_int() == 8
    assert iqb.qubit_values == (1, 0, 0, 0)

    iqb = IntQubit(7, 4)
    assert iqb.qubit_values == (0, 1, 1, 1)
    assert IntQubit(3) == IntQubit(3, 2)

    #test Dual Classes
    iqb = IntQubit(3)
    iqb_bra = IntQubitBra(3)
    assert iqb.dual_class() == IntQubitBra
    assert iqb_bra.dual_class() == IntQubit

    iqb = IntQubit(5)
    iqb_bra = IntQubitBra(5)
    assert iqb._eval_innerproduct_IntQubitBra(iqb_bra) == Integer(1)

    iqb = IntQubit(4)
    iqb_bra = IntQubitBra(5)
    assert iqb._eval_innerproduct_IntQubitBra(iqb_bra) == Integer(0)
    raises(ValueError, lambda: IntQubit(4, 1))

def _expand_powers(factors):
    """
    Helper function for normal_ordered_form and normal_order: Expand a
    power expression to a multiplication expression so that that the
    expression can be handled by the normal ordering functions.
    """

    new_factors = []
    for factor in factors.args:
        if (isinstance(factor, Pow)
                and isinstance(factor.args[1], Integer)
                and factor.args[1] > 0):
            for n in range(factor.args[1]):
                new_factors.append(factor.args[0])
        else:
            new_factors.append(factor)

    return new_factors

def intify(base):
    def inner(string):
        left, right = re.split('[^0-9]', string, 1)
        return sympy.Integer(int(right, base) * (base ** int(left)))
    return inner

def test_acceleration():
    e = (1 + 1/n)**n
    assert round(richardson(e, n, 10, 20).evalf(), 10) == round(E.evalf(), 10)

    A = Sum(Integer(-1)**(k + 1) / k, (k, 1, n))
    assert round(shanks(A, n, 25).evalf(), 4) == round(log(2).evalf(), 4)
    assert round(shanks(A, n, 25, 5).evalf(), 10) == round(log(2).evalf(), 10)

def shanks(A, k, n, m=1):
    """
    Calculate an approximation for lim k->oo A(k) using the n-term Shanks
    transformation S(A)(n). With m > 1, calculate the m-fold recursive
    Shanks transformation S(S(...S(A)...))(n).

    The Shanks transformation is useful for summing Taylor series that
    converge slowly near a pole or singularity, e.g. for log(2):

        >>> from sympy.abc import k, n
        >>> from sympy import Sum, Integer
        >>> from sympy.series.acceleration import shanks
        >>> A = Sum(Integer(-1)**(k+1) / k, (k, 1, n))
        >>> print(round(A.subs(n, 100).doit().evalf(), 10))
        0.6881721793
        >>> print(round(shanks(A, n, 25).evalf(), 10))
        0.6931396564
        >>> print(round(shanks(A, n, 25, 5).evalf(), 10))
        0.6931471806

    The correct value is 0.6931471805599453094172321215.
    """
    table = [A.subs(k, Integer(j)).doit() for j in range(n + m + 2)]
    table2 = table[:]

    for i in range(1, m + 1):
        for j in range(i, n + m + 1):
            x, y, z = table[j - 1], table[j], table[j + 1]
            table2[j] = (z*x - y**2) / (z + x - 2*y)
        table = table2[:]
    return table[n]

def test_Tuple_mul():
    assert Tuple(1, 2, 3)*2 == Tuple(1, 2, 3, 1, 2, 3)
    assert 2*Tuple(1, 2, 3) == Tuple(1, 2, 3, 1, 2, 3)
    assert Tuple(1, 2, 3)*Integer(2) == Tuple(1, 2, 3, 1, 2, 3)
    assert Integer(2)*Tuple(1, 2, 3) == Tuple(1, 2, 3, 1, 2, 3)

    raises(TypeError, lambda: Tuple(1, 2, 3)*S.Half)
    raises(TypeError, lambda: S.Half*Tuple(1, 2, 3))

def test_sympy__core__numbers__Integer():
    from sympy.core.numbers import Integer
    assert _test_args(Integer(7))

def set_v_steps(self, v_steps):
        if v_steps is None:
            self._v_steps = None
            return
        if isinstance(v_steps, int):
            v_steps = Integer(v_steps)
        elif not isinstance(v_steps, Integer):
            raise ValueError("v_steps must be an int or sympy Integer.")
        if v_steps <= Integer(0):
            raise ValueError("v_steps must be positive.")
        self._v_steps = v_steps

def vrange(self):
        """
        Yields v_steps+1 sympy numbers ranging from
        v_min to v_max.
        """
        d = (self.v_max - self.v_min) / self.v_steps
        for i in xrange(self.v_steps + 1):
            a = self.v_min + (d * Integer(i))
            yield a

def vrange2(self):
        """
        Yields v_steps pairs of sympy numbers ranging from
        (v_min, v_min + step) to (v_max - step, v_max).
        """
        d = (self.v_max - self.v_min) / self.v_steps
        a = self.v_min + (d * Integer(0))
        for i in xrange(self.v_steps):
            b = self.v_min + (d * Integer(i + 1))
            yield a, b
            a = b

def __setitem__(self, i, args):
        """
        Parses and adds a PlotMode to the function
        list.
        """
        if not (isinstance(i, (int, Integer)) and i >= 0):
            raise ValueError("Function index must "
                             "be an integer >= 0.")

        if isinstance(args, PlotObject):
            f = args
        else:
            if (not is_sequence(args)) or isinstance(args, GeometryEntity):
                args = [args]
            if len(args) == 0:
                return  # no arguments given
            kwargs = dict(bounds_callback=self.adjust_all_bounds)
            f = PlotMode(*args, **kwargs)

        if f:
            self._render_lock.acquire()
            self._functions[i] = f
            self._render_lock.release()
        else:
            raise ValueError("Failed to parse '%s'."
                    % ', '.join(str(a) for a in args))

def test_univariate():

    assert diop_solve((x - 1)*(x - 2)**2) == set([(Integer(1),), (Integer(2),)])
    assert diop_solve((x - 1)*(x - 2)) == set([(Integer(1),), (Integer(2),)])

def test_quadratic_elliptical_case():

    # Elliptical case: B**2 - 4AC < 0
    # Two test cases highlighted require lot of memory due to quadratic_congruence() method.
    # This above method should be replaced by Pernici's square_mod() method when his PR gets merged.

    #assert diop_solve(42*x**2 + 8*x*y + 15*y**2 + 23*x + 17*y - 4915) == set([(-Integer(11), -Integer(1))])
    assert diop_solve(4*x**2 + 3*y**2 + 5*x - 11*y + 12) == set([])
    assert diop_solve(x**2 + y**2 + 2*x + 2*y + 2) == set([(-Integer(1), -Integer(1))])
    #assert diop_solve(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) == set([(-Integer(15), Integer(6))])
    assert diop_solve(10*x**2 + 12*x*y + 12*y**2 - 34) == \
        set([(Integer(1), -Integer(2)), (-Integer(1), -Integer(1)),(Integer(1), Integer(1)), (-Integer(1), Integer(2))])

def is_pell_transformation_ok(eq):
    """
    Test whether X*Y, X, or Y terms are present in the equation
    after transforming the equation using the transformation returned
    by transformation_to_pell(). If they are not present we are good.
    Moreover, coefficient of X**2 should be a divisor of coefficient of
    Y**2 and the constant term.
    """
    A, B = transformation_to_DN(eq)
    u = (A*Matrix([X, Y]) + B)[0]
    v = (A*Matrix([X, Y]) + B)[1]
    simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())

    coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])

    for term in [X*Y, X, Y]:
        if term in coeff.keys():
            return False

    for term in [X**2, Y**2, Integer(1)]:
        if term not in coeff.keys():
            coeff[term] = Integer(0)

    if coeff[X**2] != 0:
        return isinstance(S(coeff[Y**2])/coeff[X**2], Integer) and isinstance(S(coeff[Integer(1)])/coeff[X**2], Integer)

    return True

def test_solve_poly_system():
    assert solve_poly_system([x - 1], x) == [(S.One,)]

    assert solve_poly_system([y - x, y - x - 1], x, y) is None

    assert solve_poly_system([y - x**2, y + x**2], x, y) == [(S.Zero, S.Zero)]

    assert solve_poly_system([2*x - 3, 3*y/2 - 2*x, z - 5*y], x, y, z) == \
        [(Rational(3, 2), Integer(2), Integer(10))]

    assert solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) == \
        [(0, 0), (2, -sqrt(2)), (2, sqrt(2))]

    assert solve_poly_system([y - x**2, y + x**2 + 1], x, y) == \
        [(-I*sqrt(S.Half), -S.Half), (I*sqrt(S.Half), -S.Half)]

    f_1 = x**2 + y + z - 1
    f_2 = x + y**2 + z - 1
    f_3 = x + y + z**2 - 1

    a, b = sqrt(2) - 1, -sqrt(2) - 1

    assert solve_poly_system([f_1, f_2, f_3], x, y, z) == \
        [(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)]

    solution = [(1, -1), (1, 1)]

    assert solve_poly_system([Poly(x**2 - y**2), Poly(x - 1)]) == solution
    assert solve_poly_system([x**2 - y**2, x - 1], x, y) == solution
    assert solve_poly_system([x**2 - y**2, x - 1]) == solution

    assert solve_poly_system(
        [x + x*y - 3, y + x*y - 4], x, y) == [(-3, -2), (1, 2)]

    raises(NotImplementedError, lambda: solve_poly_system([x**3 - y**3], x, y))
    raises(PolynomialError, lambda: solve_poly_system([1/x], x))

def __new__(cls, *args, **hints):
        if not len(args) in [1, 2]:
            raise ValueError('1 or 2 parameters expected, got %s' % args)

        if len(args) == 1:
            args = (args[0], Integer(1))

        if len(args) == 2:
            args = (args[0], Integer(args[1]))

        return Operator.__new__(cls, *args)

def _eval_commutator_FermionOp(self, other, **hints):
        return Integer(0)

def _eval_innerproduct_BosonCoherentBra(self, bra, **hints):
        if self.alpha == bra.alpha:
            return Integer(1)
        else:
            return exp(-(abs(self.alpha)**2 + abs(bra.alpha)**2 - 2 * conjugate(bra.alpha) * self.alpha)/2)

def _eval_innerproduct_QubitBra(self, bra, **hints):
        if self.label == bra.label:
            return Integer(1)
        else:
            return Integer(0)

def get_target_matrix(self, format='sympy'):
        if format == 'sympy':
            return Matrix([[1, 0], [0, exp(Integer(2)*pi*I/(Integer(2)**self.k))]])
        raise NotImplementedError(
            'Invalid format for the R_k gate: %r' % format)

def _eval_commutator(self, other, **hints):
        if isinstance(other, OneQubitGate):
            if self.targets != other.targets or self.__class__ == other.__class__:
                return Integer(0)
        return Operator._eval_commutator(self, other, **hints)

def _eval_anticommutator(self, other, **hints):
        if isinstance(other, OneQubitGate):
            if self.targets != other.targets or self.__class__ == other.__class__:
                return Integer(2)*self*other
        return Operator._eval_anticommutator(self, other, **hints)

def _eval_commutator(self, other, **hints):
        return Integer(0)

def _eval_anticommutator(self, other, **hints):
        return Integer(2)*other

def _eval_anticommutator_YGate(self, other, **hints):
        return Integer(0)

def _eval_anticommutator_XGate(self, other, **hints):
        return Integer(2)*IdentityGate(self.targets[0])

def _eval_anticommutator_YGate(self, other, **hints):
        return Integer(0)

def _eval_anticommutator_ZGate(self, other, **hints):
        return Integer(0)

def _eval_commutator_ZGate(self, other, **hints):
        return Integer(2)*I*XGate(self.targets[0])

def _eval_anticommutator_YGate(self, other, **hints):
        return Integer(2)*IdentityGate(self.targets[0])

def _eval_commutator_XGate(self, other, **hints):
        return Integer(2)*I*YGate(self.targets[0])

def _eval_anticommutator_YGate(self, other, **hints):
        return Integer(0)

def _eval_commutator_ZGate(self, other, **hints):
        return Integer(0)

def _eval_commutator_TGate(self, other, **hints):
        return Integer(0)

def _eval_commutator_ZGate(self, other, **hints):
        return Integer(0)

def _eval_commutator_ZGate(self, other, **hints):
        """[CNOT(i, j), Z(i)] == 0."""
        if self.controls[0] == other.targets[0]:
            return Integer(0)
        else:
            raise NotImplementedError('Commutator not implemented: %r' % other)

def _eval_commutator_XGate(self, other, **hints):
        """[CNOT(i, j), X(j)] == 0."""
        if self.targets[0] == other.targets[0]:
            return Integer(0)
        else:
            raise NotImplementedError('Commutator not implemented: %r' % other)

def _eval_commutator_CNotGate(self, other, **hints):
        """[CNOT(i, j), CNOT(i,k)] == 0."""
        if self.controls[0] == other.controls[0]:
            return Integer(0)
        else:
            raise NotImplementedError('Commutator not implemented: %r' % other)