Python Poly.as_expr方法代码示例

# 需要导入模块: from sympy.polys import Poly [as 别名]
# 或者: from sympy.polys.Poly import as_expr [as 别名]
def ratint_ratpart(f, g, x):
    """Horowitz-Ostrogradsky algorithm.

       Given a field K and polynomials f and g in K[x], such that f and g
       are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
       such that f/g = A' + B and B has square-free denominator.

    """
    f = Poly(f, x)
    g = Poly(g, x)

    u, v, _ = g.cofactors(g.diff())

    n = u.degree()
    m = v.degree()

    A_coeffs = [ Dummy('a' + str(n-i)) for i in xrange(0, n) ]
    B_coeffs = [ Dummy('b' + str(m-i)) for i in xrange(0, m) ]

    C_coeffs = A_coeffs + B_coeffs

    A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
    B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])

    H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u

    result = solve(H.coeffs(), C_coeffs)

    A = A.as_expr().subs(result)
    B = B.as_expr().subs(result)

    rat_part = cancel(A/u.as_expr(), x)
    log_part = cancel(B/v.as_expr(), x)

    return rat_part, log_part

# 需要导入模块: from sympy.polys import Poly [as 别名]
# 或者: from sympy.polys.Poly import as_expr [as 别名]
def ratint_ratpart(f, g, x):
    """
    Horowitz-Ostrogradsky algorithm.

    Given a field K and polynomials f and g in K[x], such that f and g
    are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
    such that f/g = A' + B and B has square-free denominator.

    Examples
    ========

        >>> from sympy.integrals.rationaltools import ratint_ratpart
        >>> from sympy.abc import x, y
        >>> from sympy import Poly
        >>> ratint_ratpart(Poly(1, x, domain='ZZ'),
        ... Poly(x + 1, x, domain='ZZ'), x)
        (0, 1/(x + 1))
        >>> ratint_ratpart(Poly(1, x, domain='EX'),
        ... Poly(x**2 + y**2, x, domain='EX'), x)
        (0, 1/(x**2 + y**2))
        >>> ratint_ratpart(Poly(36, x, domain='ZZ'),
        ... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
        ((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))

    See Also
    ========

    ratint, ratint_logpart
    """
    from sympy import solve

    f = Poly(f, x)
    g = Poly(g, x)

    u, v, _ = g.cofactors(g.diff())

    n = u.degree()
    m = v.degree()

    A_coeffs = [ Dummy('a' + str(n - i)) for i in range(0, n) ]
    B_coeffs = [ Dummy('b' + str(m - i)) for i in range(0, m) ]

    C_coeffs = A_coeffs + B_coeffs

    A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
    B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])

    H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u

    result = solve(H.coeffs(), C_coeffs)

    A = A.as_expr().subs(result)
    B = B.as_expr().subs(result)

    rat_part = cancel(A/u.as_expr(), x)
    log_part = cancel(B/v.as_expr(), x)

    return rat_part, log_part

# 需要导入模块: from sympy.polys import Poly [as 别名]
# 或者: from sympy.polys.Poly import as_expr [as 别名]
def _solve_as_poly(f, symbol, solveset_solver, invert_func):
    """
    Solve the equation using polynomial techniques if it already is a
    polynomial equation or, with a change of variables, can be made so.
    """
    result = None
    if f.is_polynomial(symbol):

        solns = roots(f, symbol, cubics=True, quartics=True,
                      quintics=True, domain='EX')
        num_roots = sum(solns.values())
        if degree(f, symbol) <= num_roots:
            result = FiniteSet(*solns.keys())
        else:
            poly = Poly(f, symbol)
            solns = poly.all_roots()
            if poly.degree() <= len(solns):
                result = FiniteSet(*solns)
            else:
                result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
    else:
        poly = Poly(f)
        if poly is None:
            result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
        gens = [g for g in poly.gens if g.has(symbol)]

        if len(gens) == 1:
            poly = Poly(poly, gens[0])
            gen = poly.gen
            deg = poly.degree()
            poly = Poly(poly.as_expr(), poly.gen, composite=True)
            poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
                                          quintics=True).keys())

            if len(poly_solns) < deg:
                result = ConditionSet(symbol, Eq(f, 0), S.Complexes)

            if gen != symbol:
                y = Dummy('y')
                lhs, rhs_s = invert_func(gen, y, symbol)
                if lhs is symbol:
                    result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
                else:
                    result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
        else:
            result = ConditionSet(symbol, Eq(f, 0), S.Complexes)

    if result is not None:
        if isinstance(result, FiniteSet):
            # this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
            # - sqrt(2)*I/2. We are not expanding for solution with free
            # variables because that makes the solution more complicated. For
            # example expand_complex(a) returns re(a) + I*im(a)
            if all([s.free_symbols == set() and not isinstance(s, RootOf)
                    for s in result]):
                s = Dummy('s')
                result = imageset(Lambda(s, expand_complex(s)), result)
        return result
    else:
        return ConditionSet(symbol, Eq(f, 0), S.Complexes)

# 需要导入模块: from sympy.polys import Poly [as 别名]
# 或者: from sympy.polys.Poly import as_expr [as 别名]
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

# 需要导入模块: from sympy.polys import Poly [as 别名]
# 或者: from sympy.polys.Poly import as_expr [as 别名]
def gosper_term(f, n):
    r"""
    Compute Gosper's hypergeometric term for ``f``.

    Suppose ``f`` is a hypergeometric term such that:

    .. math::
        s_n = \sum_{k=0}^{n-1} f_k

    and `f_k` doesn't depend on `n`. Returns a hypergeometric
    term `g_n` such that `g_{n+1} - g_n = f_n`.

    **Examples**

    >>> from sympy.concrete.gosper import gosper_term
    >>> from sympy.functions import factorial
    >>> from sympy.abc import n

    >>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)
    (-n - 1/2)/(n + 1/4)

    """
    r = hypersimp(f, n)

    if r is None:
        return None  # 'f' is *not* a hypergeometric term

    p, q = r.as_numer_denom()

    A, B, C = gosper_normal(p, q, n)
    B = B.shift(-1)

    N = S(A.degree())
    M = S(B.degree())
    K = S(C.degree())

    if (N != M) or (A.LC() != B.LC()):
        D = set([K - max(N, M)])
    elif not N:
        D = set([K - N + 1, S(0)])
    else:
        D = set([K - N + 1, (B.nth(N - 1) - A.nth(N - 1)) / A.LC()])

    for d in set(D):
        if not d.is_Integer or d < 0:
            D.remove(d)

    if not D:
        return None  # 'f(n)' is *not* Gosper-summable

    d = max(D)

    coeffs = symbols("c:%s" % (d + 1), cls=Dummy)
    domain = A.get_domain().inject(*coeffs)

    x = Poly(coeffs, n, domain=domain)
    H = A * x.shift(1) - B * x - C

    solution = solve(H.coeffs(), coeffs)

    if solution is None:
        return None  # 'f(n)' is *not* Gosper-summable

    x = x.as_expr().subs(solution)

    for coeff in coeffs:
        if coeff not in solution:
            x = x.subs(coeff, 0)

    if x is S.Zero:
        return None  # 'f(n)' is *not* Gosper-summable
    else:
        return B.as_expr() * x / C.as_expr()

# 需要导入模块: from sympy.polys import Poly [as 别名]
# 或者: from sympy.polys.Poly import as_expr [as 别名]
def _solve_inequality(ie, s, linear=False):
    """Return the inequality with s isolated on the left, if possible.
    If the relationship is non-linear, a solution involving And or Or
    may be returned. False or True are returned if the relationship
    is never True or always True, respectively.

    If `linear` is True (default is False) an `s`-dependent expression
    will be isoloated on the left, if possible
    but it will not be solved for `s` unless the expression is linear
    in `s`. Furthermore, only "safe" operations which don't change the
    sense of the relationship are applied: no division by an unsigned
    value is attempted unless the relationship involves Eq or Ne and
    no division by a value not known to be nonzero is ever attempted.

    Examples
    ========

    >>> from sympy import Eq, Symbol
    >>> from sympy.solvers.inequalities import _solve_inequality as f
    >>> from sympy.abc import x, y

    For linear expressions, the symbol can be isolated:

    >>> f(x - 2 < 0, x)
    x < 2
    >>> f(-x - 6 < x, x)
    x > -3

    Sometimes nonlinear relationships will be False

    >>> f(x**2 + 4 < 0, x)
    False

    Or they may involve more than one region of values:

    >>> f(x**2 - 4 < 0, x)
    (-2 < x) & (x < 2)

    To restrict the solution to a relational, set linear=True
    and only the x-dependent portion will be isolated on the left:

    >>> f(x**2 - 4 < 0, x, linear=True)
    x**2 < 4

    Division of only nonzero quantities is allowed, so x cannot
    be isolated by dividing by y:

    >>> y.is_nonzero is None  # it is unknown whether it is 0 or not
    True
    >>> f(x*y < 1, x)
    x*y < 1

    And while an equality (or unequality) still holds after dividing by a
    non-zero quantity

    >>> nz = Symbol('nz', nonzero=True)
    >>> f(Eq(x*nz, 1), x)
    Eq(x, 1/nz)

    the sign must be known for other inequalities involving > or <:

    >>> f(x*nz <= 1, x)
    nz*x <= 1
    >>> p = Symbol('p', positive=True)
    >>> f(x*p <= 1, x)
    x <= 1/p

    When there are denominators in the original expression that
    are removed by expansion, conditions for them will be returned
    as part of the result:

    >>> f(x < x*(2/x - 1), x)
    (x < 1) & Ne(x, 0)
    """
    from sympy.solvers.solvers import denoms
    if s not in ie.free_symbols:
        return ie
    if ie.rhs == s:
        ie = ie.reversed
    if ie.lhs == s and s not in ie.rhs.free_symbols:
        return ie
    expr = ie.lhs - ie.rhs
    rv = None
    try:
        p = Poly(expr, s)
        if p.degree() == 0:
            rv = ie.func(p.as_expr(), 0)
        elif not linear and p.degree() > 1:
            # handle in except clause
            raise NotImplementedError
    except (PolynomialError, NotImplementedError):
        if not linear:
            try:
                return reduce_rational_inequalities([[ie]], s)
            except PolynomialError:
                return solve_univariate_inequality(ie, s)
        else:
            p = Poly(expr)

    e = expanded = p.as_expr()  # this is in exanded form
#.........这里部分代码省略.........