Python utilities.numbered_symbols函数代码示例

 def genEval(self):
     text = "        bool evaluate(\n"
     args=[]
     for name,v in self.variables:
         if v.is_Matrix:
             args.append("           const Eigen::MatrixXd & %s" % name)
         else:
             args.append("           double %s" % name)
     args.append("           bool evalF=true,bool evalJ=true")
     text += ",\n".join(args) + ") {\n"
     text += "           if (evalF) {\n"
     (interm, expr) = cse(self.function,numbered_symbols("__x"));
     for dummy,exp in interm:
         text += "               double %s = %s;\n" % (str(dummy),ccode(exp))
     for i in range(self.function.rows):
         text += "               F(%d) = %s;\n" % (i,ccode(expr[0][i]))
     text += "           }\n"
     text += "           if (evalJ) {\n"
     (interm, expr) = cse(self.J,numbered_symbols("__x"));
     for dummy,exp in interm:
         text += "               double %s = %s;\n" % (str(dummy),ccode(exp))
     for i in range(self.J.rows):
         for j in range(self.J.cols):
             text += "               J(%d,%d) = %s;\n" % (i,j,ccode(expr[0][i,j]))
     text += "           }\n"
     text += "           return true;\n"
     text += "       }\n"
     return text

def viete(f, roots=None, *gens, **args):
    """
    Generate Viete's formulas for ``f``.

    Examples
    ========

    >>> from sympy.polys.polyfuncs import viete
    >>> from sympy import symbols

    >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')

    >>> viete(a*x**2 + b*x + c, [r1, r2], x)
    [(r1 + r2, -b/a), (r1*r2, c/a)]

    """
    allowed_flags(args, [])

    if isinstance(roots, Basic):
        gens, roots = (roots,) + gens, None

    try:
        f, opt = poly_from_expr(f, *gens, **args)
    except PolificationFailed as exc:
        raise ComputationFailed('viete', 1, exc)

    if f.is_multivariate:
        raise MultivariatePolynomialError(
            "multivariate polynomials are not allowed")

    n = f.degree()

    if n < 1:
        raise ValueError(
            "can't derive Viete's formulas for a constant polynomial")

    if roots is None:
        roots = numbered_symbols('r', start=1)

    roots = take(roots, n)

    if n != len(roots):
        raise ValueError("required %s roots, got %s" % (n, len(roots)))

    lc, coeffs = f.LC(), f.all_coeffs()
    result, sign = [], -1

    for i, coeff in enumerate(coeffs[1:]):
        poly = symmetric_poly(i + 1, roots)
        coeff = sign*(coeff/lc)
        result.append((poly, coeff))
        sign = -sign

    return result

 def genEval(self):
     text = "        bool evaluate(\n"
     args=[]
     for name,v,lD in self.variables:
         if isinstance(v, Matrix):
             args.append("           const Eigen::Matrix<double, %d, 1> & %s" % (v.rows, name))
         else:
             args.append("           double %s" % name)
             
     args.append("           Eigen::Matrix<double, %d, 1> * F" % self.function.rows)
     
     for name,v,localDim in self.variables:
         if isinstance(v, Matrix):
             args.append("           Eigen::Matrix<double, %d, %d> * J%s" % (self.J.rows, localDim, name))
         else:
             args.append("           Eigen::Matrix<double, %d, 1> * J%s" % (self.J.rows, name))
     text += ",\n".join(args) + ") {\n"
     text += "           if (F) {\n"
     (interm, expr) = cse(self.function,numbered_symbols("__x"));
     for dummy,exp in interm:
         text += "               double %s = %s;\n" % (str(dummy),ccode(exp))
     for i in range(self.function.rows):
         text += "               (*F)(%d) = %s;\n" % (i,ccode(expr[0][i]))
     text += "           }\n"
     text += "           if (%s) {\n" % " && ".join([ "J" + name for name,v,lD in self.variables ])
     (interm, expr) = cse(self.J,numbered_symbols("__x"));
     for dummy,exp in interm:
         text += "               double %s = %s;\n" % (str(dummy),ccode(exp))
     colBase = 0;
     for name,v,localDim in self.variables:
         for i in range(self.J.rows):
             for j in range(0, localDim):
                 text += "               (*J%s)(%d,%d) = %s;\n" % (name, i,j,ccode(expr[0][i,colBase + j]))
         colBase+=localDim
     text += "           }\n"
     text += "           return true;\n"
     text += "       }\n"
     return text

def apart_undetermined_coeffs(P, Q):
    """Partial fractions via method of undetermined coefficients. """
    X = numbered_symbols(cls=Dummy)
    partial, symbols = [], []

    _, factors = Q.factor_list()

    for f, k in factors:
        n, q = f.degree(), Q

        for i in xrange(1, k + 1):
            coeffs, q = take(X, n), q.quo(f)
            partial.append((coeffs, q, f, i))
            symbols.extend(coeffs)

    dom = Q.get_domain().inject(*symbols)
    F = Poly(0, Q.gen, domain=dom)

    for i, (coeffs, q, f, k) in enumerate(partial):
        h = Poly(coeffs, Q.gen, domain=dom)
        partial[i] = (h, f, k)
        q = q.set_domain(dom)
        F += h * q

    system, result = [], S(0)

    for (k,), coeff in F.terms():
        system.append(coeff - P.nth(k))

    from sympy.solvers import solve

    solution = solve(system, symbols)

    for h, f, k in partial:
        h = h.as_expr().subs(solution)
        result += h / f.as_expr() ** k

    return result

 def default(cls):
     return numbered_symbols('s', start=1)

#.........这里部分代码省略.........
    expected = (X(i0), Y(i0), Z(i0), H(i0), X(i1), Y(i1), Z(i1), H(i1))
    exp_map = {i0: Integer(0), i1: Integer(1)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices((x, y, z, h,
                                         x1, y1, z1, h1))
    assert actual == expected
    assert act_map == exp_map

    exp_map = {i0: Integer(1), i1: Integer(0)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(Mul(x1, y1,
                                         z1, h1, x, y, z, h))
    assert actual == expected
    assert act_map == exp_map

    expected = (X(i0), X(i1), Y(i0), Y(i1), Z(i0), Z(i1), H(i0), H(i1))
    exp_map = {i0: Integer(0), i1: Integer(1)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(Mul(x, x1,
                                         y, y1, z, z1, h, h1))
    assert actual == expected
    assert act_map == exp_map

    exp_map = {i0: Integer(1), i1: Integer(0)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices((x1, x, y1, y,
                                         z1, z, h1, h))
    assert actual == expected
    assert act_map == exp_map

    cnot_10 = CNOT(1, 0)
    cnot_01 = CNOT(0, 1)
    cgate_z_10 = CGate(1, Z(0))
    cgate_z_01 = CGate(0, Z(1))

    expected = (X(i0), X(i1), Y(i0), Y(i1), Z(i0), Z(i1),
                H(i0), H(i1), CNOT(i1, i0), CNOT(i0, i1),
                CGate(i1, Z(i0)), CGate(i0, Z(i1)))
    exp_map = {i0: Integer(0), i1: Integer(1)}
    args = (x, x1, y, y1, z, z1, h, h1, cnot_10, cnot_01,
            cgate_z_10, cgate_z_01)
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

    args = (x1, x, y1, y, z1, z, h1, h, cnot_10, cnot_01,
            cgate_z_10, cgate_z_01)
    expected = (X(i0), X(i1), Y(i0), Y(i1), Z(i0), Z(i1),
                H(i0), H(i1), CNOT(i0, i1), CNOT(i1, i0),
                CGate(i0, Z(i1)), CGate(i1, Z(i0)))
    exp_map = {i0: Integer(1), i1: Integer(0)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

    args = (cnot_10, h, cgate_z_01, h)
    expected = (CNOT(i0, i1), H(i1), CGate(i1, Z(i0)), H(i1))
    exp_map = {i0: Integer(1), i1: Integer(0)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

    args = (cnot_01, h1, cgate_z_10, h1)
    exp_map = {i0: Integer(0), i1: Integer(1)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

    args = (cnot_10, h1, cgate_z_01, h1)
    expected = (CNOT(i0, i1), H(i0), CGate(i1, Z(i0)), H(i0))
    exp_map = {i0: Integer(1), i1: Integer(0)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

    i2 = Symbol('i2')
    ccgate_z = CGate(0, CGate(1, Z(2)))
    ccgate_x = CGate(1, CGate(2, X(0)))
    args = (ccgate_z, ccgate_x)

    expected = (CGate(i0, CGate(i1, Z(i2))), CGate(i1, CGate(i2, X(i0))))
    exp_map = {i0: Integer(0), i1: Integer(1), i2: Integer(2)}
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

    ndx_map = {i0: Integer(0)}
    index_gen = numbered_symbols(prefix='i', start=1)
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args,
                                         qubit_map=ndx_map,
                                         start=i0,
                                         gen=index_gen)
    assert actual == expected
    assert act_map == exp_map

    i3 = Symbol('i3')
    cgate_x0_c321 = CGate((3, 2, 1), X(0))
    exp_map = {i0: Integer(3), i1: Integer(2),
               i2: Integer(1), i3: Integer(0)}
    expected = (CGate((i0, i1, i2), X(i3)),)
    args = (cgate_x0_c321,)
    actual, act_map, sndx, gen = convert_to_symbolic_indices(args)
    assert actual == expected
    assert act_map == exp_map

def convert_to_symbolic_indices(seq, start=None, gen=None, qubit_map=None):
    """Returns the circuit with symbolic indices and the
    dictionary mapping symbolic indices to real indices.

    The mapping is 1 to 1 and onto (bijective).

    Parameters
    ==========
    seq : tuple, Gate/Integer/tuple or Mul
        A tuple of Gate, Integer, or tuple objects, or a Mul
    start : Symbol
        An optional starting symbolic index
    gen : object
        An optional numbered symbol generator
    qubit_map : dict
        An existing mapping of symbolic indices to real indices

    All symbolic indices have the format 'i#', where # is
    some number >= 0.
    """

    if isinstance(seq, Mul):
        seq = seq.args

    # A numbered symbol generator
    index_gen = numbered_symbols(prefix='i', start=-1)
    cur_ndx = index_gen.next()

    # keys are symbolic indices; values are real indices
    ndx_map = {}

    def create_inverse_map(symb_to_real_map):
        rev_items = lambda item: tuple([item[1], item[0]])
        return dict(map(rev_items, symb_to_real_map.items()))

    if start is not None:
        if not isinstance(start, Symbol):
            msg = 'Expected Symbol for starting index, got %r.' % start
            raise TypeError(msg)
        cur_ndx = start

    if gen is not None:
        if not isinstance(gen, numbered_symbols().__class__):
            msg = 'Expected a generator, got %r.' % gen
            raise TypeError(msg)
        index_gen = gen

    if qubit_map is not None:
        if not isinstance(qubit_map, dict):
            msg = ('Expected dict for existing map, got ' +
                   '%r.' % qubit_map)
            raise TypeError(msg)
        ndx_map = qubit_map

    ndx_map = _sympify_qubit_map(ndx_map)
    # keys are real indices; keys are symbolic indices
    inv_map = create_inverse_map(ndx_map)

    sym_seq = ()
    for item in seq:
        # Nested items, so recurse
        if isinstance(item, Gate):
            result = convert_to_symbolic_indices(item.args,
                                                 qubit_map=ndx_map,
                                                 start=cur_ndx,
                                                 gen=index_gen)
            sym_item, new_map, cur_ndx, index_gen = result
            ndx_map.update(new_map)
            inv_map = create_inverse_map(ndx_map)

        elif isinstance(item, tuple) or isinstance(item, Tuple):
            result = convert_to_symbolic_indices(item,
                                                 qubit_map=ndx_map,
                                                 start=cur_ndx,
                                                 gen=index_gen)
            sym_item, new_map, cur_ndx, index_gen = result
            ndx_map.update(new_map)
            inv_map = create_inverse_map(ndx_map)

        elif item in inv_map:
            sym_item = inv_map[item]

        else:
            cur_ndx = gen.next()
            ndx_map[cur_ndx] = item
            inv_map[item] = cur_ndx
            sym_item = cur_ndx

        if isinstance(item, Gate):
            sym_item = item.__class__(*sym_item)

        sym_seq = sym_seq + (sym_item,)

    return sym_seq, ndx_map, cur_ndx, index_gen

def primitive_element(extension, x=None, **args):
    """Construct a common number field for all extensions. """
    if not extension:
        raise ValueError("can't compute primitive element for empty extension")

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly
    if not args.get('ex', False):
        extension = [ AlgebraicNumber(ext, gen=x) for ext in extension ]

        g, coeffs = extension[0].minpoly.replace(x), [1]

        for ext in extension[1:]:
            s, _, g = sqf_norm(g, x, extension=ext)
            coeffs = [ s*c for c in coeffs ] + [1]

        if not args.get('polys', False):
            return g.as_expr(), coeffs
        else:
            return cls(g), coeffs

    generator = numbered_symbols('y', cls=Dummy)

    F, Y = [], []

    for ext in extension:
        y = next(generator)

        if ext.is_Poly:
            if ext.is_univariate:
                f = ext.as_expr(y)
            else:
                raise ValueError("expected minimal polynomial, got %s" % ext)
        else:
            f = minpoly(ext, y)

        F.append(f)
        Y.append(y)

    coeffs_generator = args.get('coeffs', _coeffs_generator)

    for coeffs in coeffs_generator(len(Y)):
        f = x - sum([ c*y for c, y in zip(coeffs, Y)])
        G = groebner(F + [f], Y + [x], order='lex', field=True)

        H, g = G[:-1], cls(G[-1], x, domain='QQ')

        for i, (h, y) in enumerate(zip(H, Y)):
            try:
                H[i] = Poly(y - h, x,
                            domain='QQ').all_coeffs()  # XXX: composite=False
            except CoercionFailed:  # pragma: no cover
                break  # G is not a triangular set
        else:
            break
    else:  # pragma: no cover
        raise RuntimeError("run out of coefficient configurations")

    _, g = g.clear_denoms()

    if not args.get('polys', False):
        return g.as_expr(), coeffs, H
    else:
        return g, coeffs, H

def _minpoly_groebner(ex, x, cls):
    """
    Computes the minimal polynomial of an algebraic number
    using Groebner bases

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, Rational
    >>> from sympy.abc import x
    >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False)
    x**2 - 2*x - 1

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_multinomial

    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    def update_mapping(ex, exp, base=None):
        a = next(generator)
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base**(-ex.exp)
                if not ex.exp.is_Integer:
                    base, exp = (
                        ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp
                base = bottom_up_scan(base)
                expr = base**exp

                if expr not in mapping:
                    return update_mapping(expr, 1/exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    def simpler_inverse(ex):
        """
        Returns True if it is more likely that the minimal polynomial
        algorithm works better with the inverse
        """
        if ex.is_Pow:
            if (1/ex.exp).is_integer and ex.exp < 0:
                if ex.base.is_Add:
                    return True
        if ex.is_Mul:
            hit = True
            a = []
            for p in ex.args:
                if p.is_Add:
                    return False
                if p.is_Pow:
                    if p.base.is_Add and p.exp > 0:
                        return False

            if hit:
#.........这里部分代码省略.........

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    **Example**

    >>> from sympy import minimal_polynomial, sqrt
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1

    """
    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    ex = sympify(ex)

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational and ex.q != 0:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base**(-ex.exp)

                if not ex.exp.is_Integer:
                    base, exp = (ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp

                base = bottom_up_scan(base)
                expr = base**exp

                if expr not in mapping:
                    return update_mapping(expr, 1/exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    polys = args.get('polys', False)

    if ex.is_AlgebraicNumber:
        if not polys:
            return ex.minpoly.as_expr(x)
        else:
            return ex.minpoly.replace(x)
    elif ex.is_Rational and ex.q != 0:
        result = ex.q*x - ex.p
    else:
        F = [x - bottom_up_scan(ex)] + mapping.values()
        G = groebner(F, symbols.values() + [x], order='lex')

        _, factors = factor_list(G[-1])
#.........这里部分代码省略.........

    try:
        f, opt = poly_from_expr(f, *gens, **args)
    except PolificationFailed, exc:
        raise ComputationFailed('viete', 1, exc)

    if f.is_multivariate:
        raise MultivariatePolynomialError("multivariate polynomials are not allowed")

    n = f.degree()

    if n < 1:
        raise ValueError("can't derive Viete's formulas for a constant polynomial")

    if roots is None:
        roots = numbered_symbols('r', start=1)

    roots = take(roots, n)

    if n != len(roots):
        raise ValueError("required %s roots, got %s" % (n, len(roots)))

    lc, coeffs = f.LC(), f.all_coeffs()
    result, sign = [], -1

    for i, coeff in enumerate(coeffs[1:]):
        poly = symmetric_poly(i+1, roots)
        coeff = sign*(coeff/lc)
        result.append((poly, coeff))
        sign = -sign

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    Parameters
    ==========

    ex : algebraic number expression

    x : indipendent variable of the minimal polynomial

    Options
    =======

    compose : if ``True`` _minpoly1`` is used, else the ``groebner`` algorithm

    polys : if ``True`` returns a ``Poly`` object

    Notes
    =====

    By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
    are computed, then the arithmetic operations on them are performed using the resultant
    and factorization.
    If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
    The default algorithm stalls less frequently.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, solve
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1
    >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
    x**3 + x + 3

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_multinomial
    from sympy.core.basic import preorder_traversal

    compose = args.get('compose', True)
    polys = args.get('polys', False)
    ex = sympify(ex)
    for expr in preorder_traversal(ex):
        if expr.is_AlgebraicNumber:
            compose = False
            break

    if ex.is_AlgebraicNumber:
        compose = False

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    if compose:
        result = _minpoly1(ex, x)
        result = result.primitive()[1]
        c = result.coeff(x**degree(result, x))
        if c < 0:
            result = expand_mul(-result)
            c = -c
        return cls(result, x, field=True) if polys else result

    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
#.........这里部分代码省略.........

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_mul, expand_multinomial
    from sympy.simplify.simplify import _is_sum_surds

    generator = numbered_symbols("a", cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    ex = sympify(ex)

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy("x"), PurePoly

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a ** exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[bottom_up_scan(g) for g in ex.args])
        elif ex.is_Mul:
            return Mul(*[bottom_up_scan(g) for g in ex.args])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base ** (-ex.exp)
                if not ex.exp.is_Integer:
                    base, exp = (ex.base ** ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp
                base = bottom_up_scan(base)
                expr = base ** exp

                if expr not in mapping:
                    return update_mapping(expr, 1 / exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    def simpler_inverse(ex):
        """
        Returns True if it is more likely that the minimal polynomial
        algorithm works better with the inverse
        """
        if ex.is_Pow:
            if (1 / ex.exp).is_integer and ex.exp < 0:
                if ex.base.is_Add:
                    return True
        if ex.is_Mul:
            hit = True
#.........这里部分代码省略.........