Python mul._keep_coeff函数代码示例

    def as_numer_denom(self):

        # clear rational denominator
        content, expr = self.primitive()
        ncon, dcon = content.as_numer_denom()

        # collect numerators and denominators of the terms
        nd = defaultdict(list)
        for f in expr.args:
            ni, di = f.as_numer_denom()
            nd[di].append(ni)
        # put infinity in the numerator
        if S.Zero in nd:
            n = nd.pop(S.Zero)
            assert len(n) == 1
            n = n[0]
            nd[S.One].append(n / S.Zero)

        # check for quick exit
        if len(nd) == 1:
            d, n = nd.popitem()
            return Add(*[_keep_coeff(ncon, ni) for ni in n]), _keep_coeff(dcon, d)

        # sum up the terms having a common denominator
        for d, n in nd.iteritems():
            if len(n) == 1:
                nd[d] = n[0]
            else:
                nd[d] = Add(*n)

        # assemble single numerator and denominator
        denoms, numers = [list(i) for i in zip(*nd.iteritems())]
        n, d = Add(*[Mul(*(denoms[:i] + [numers[i]] + denoms[i + 1 :])) for i in xrange(len(numers))]), Mul(*denoms)

        return _keep_coeff(ncon, n), _keep_coeff(dcon, d)

    def as_content_primitive(self):
        """Return the tuple (R, self/R) where R is the positive Rational
        extracted from self.

        **Examples**

        >>> from sympy import sqrt
        >>> (3 + 3*sqrt(2)).as_content_primitive()
        (3, 1 + sqrt(2))

        Radical content is also factored out of the primitive:

        >>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive()
        (2, sqrt(2)*(1 + 2*sqrt(5)))

        See docstring of Expr.as_content_primitive for more examples.
        """
        con, prim = Add(*[_keep_coeff(*a.as_content_primitive()) for a in self.args]).primitive()
        if prim.is_Add:
            # look for common radicals that can be removed
            args = prim.args
            rads = []
            common_q = None
            for m in args:
                term_rads = defaultdict(list)
                for ai in Mul.make_args(m):
                    if ai.is_Pow:
                        b, e = ai.as_base_exp()
                        if e.is_Rational and b.is_Integer and b > 0:
                            term_rads[e.q].append(int(b)**e.p)
                if not term_rads:
                    break
                if common_q is None:
                    common_q = set(term_rads.keys())
                else:
                    common_q = common_q & set(term_rads.keys())
                    if not common_q:
                        break
                rads.append(term_rads)
            else:
                # process rads
                # keep only those in common_q
                for r in rads:
                    for q in r.keys():
                        if q not in common_q:
                            r.pop(q)
                    for q in r:
                        r[q] = prod(r[q])
                # find the gcd of bases for each q
                G = []
                for q in common_q:
                    g = reduce(igcd, [r[q] for r in rads], 0)
                    if g != 1:
                        G.append(Pow(g, Rational(1, q)))
                if G:
                    G = Mul(*G)
                    args = [ai/G for ai in args]
                    prim = G*Add(*args)

        return con, prim

    def as_content_primitive(self):
        """Return the tuple (R, self/R) where R is the positive Rational
        extracted from self.

        **Example**
        >>> from sympy import sqrt
        >>> (3 + 3*sqrt(2)).as_content_primitive()
        (3, 1 + sqrt(2))

        See docstring of Expr.as_content_primitive for more examples.
        """
        return Add(*[_keep_coeff(*a.as_content_primitive()) for a in self.args]).primitive()

    def primitive(self):
        """
        Return ``(R, self/R)`` where ``R``` is the Rational GCD of ``self```.

        ``R`` is collected only from the leading coefficient of each term.

        Examples
        ========

        >>> from sympy.abc import x, y

        >>> (2*x + 4*y).primitive()
        (2, x + 2*y)

        >>> (2*x/3 + 4*y/9).primitive()
        (2/9, 3*x + 2*y)

        >>> (2*x/3 + 4.2*y).primitive()
        (1/3, 2*x + 12.6*y)

        No subprocessing of term factors is performed:

        >>> ((2 + 2*x)*x + 2).primitive()
        (1, x*(2*x + 2) + 2)

        Recursive subprocessing can be done with the as_content_primitive()
        method:

        >>> ((2 + 2*x)*x + 2).as_content_primitive()
        (2, x*(x + 1) + 1)

        See also: primitive() function in polytools.py

        """

        terms = []
        inf = False
        for a in self.args:
            c, m = a.as_coeff_Mul()
            if not c.is_Rational:
                c = S.One
                m = a
            inf = inf or m is S.ComplexInfinity
            terms.append((c.p, c.q, m))

        if not inf:
            ngcd = reduce(igcd, [t[0] for t in terms], 0)
            dlcm = reduce(ilcm, [t[1] for t in terms], 1)
        else:
            ngcd = reduce(igcd, [t[0] for t in terms if t[1]], 0)
            dlcm = reduce(ilcm, [t[1] for t in terms if t[1]], 1)

        if ngcd == dlcm == 1:
            return S.One, self
        if not inf:
            for i, (p, q, term) in enumerate(terms):
                terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term)
        else:
            for i, (p, q, term) in enumerate(terms):
                if q:
                    terms[i] = _keep_coeff(Rational((p//ngcd)*(dlcm//q)), term)
                else:
                    terms[i] = _keep_coeff(Rational(p, q), term)

        # we don't need a complete re-flattening since no new terms will join
        # so we just use the same sort as is used in Add.flatten. When the
        # coefficient changes, the ordering of terms may change, e.g.
        #     (3*x, 6*y) -> (2*y, x)
        #
        # We do need to make sure that term[0] stays in position 0, however.
        #
        if terms[0].is_Number or terms[0] is S.ComplexInfinity:
            c = terms.pop(0)
        else:
            c = None
        terms.sort(key=hash)
        if c:
            terms.insert(0, c)
        return Rational(ngcd, dlcm), self._new_rawargs(*terms)

    def as_content_primitive(self, radical=False):
        """Return the tuple (R, self/R) where R is the positive Rational
        extracted from self.

        Examples
        ========

        >>> from sympy import sqrt
        >>> sqrt(4 + 4*sqrt(2)).as_content_primitive()
        (2, sqrt(1 + sqrt(2)))
        >>> sqrt(3 + 3*sqrt(2)).as_content_primitive()
        (1, sqrt(3)*sqrt(1 + sqrt(2)))

        >>> from sympy import expand_power_base, powsimp, Mul
        >>> from sympy.abc import x, y

        >>> ((2*x + 2)**2).as_content_primitive()
        (4, (x + 1)**2)
        >>> (4**((1 + y)/2)).as_content_primitive()
        (2, 4**(y/2))
        >>> (3**((1 + y)/2)).as_content_primitive()
        (1, 3**((y + 1)/2))
        >>> (3**((5 + y)/2)).as_content_primitive()
        (9, 3**((y + 1)/2))
        >>> eq = 3**(2 + 2*x)
        >>> powsimp(eq) == eq
        True
        >>> eq.as_content_primitive()
        (9, 3**(2*x))
        >>> powsimp(Mul(*_))
        3**(2*x + 2)

        >>> eq = (2 + 2*x)**y
        >>> s = expand_power_base(eq); s.is_Mul, s
        (False, (2*x + 2)**y)
        >>> eq.as_content_primitive()
        (1, (2*(x + 1))**y)
        >>> s = expand_power_base(_[1]); s.is_Mul, s
        (True, 2**y*(x + 1)**y)

        See docstring of Expr.as_content_primitive for more examples.
        """

        b, e = self.as_base_exp()
        b = _keep_coeff(*b.as_content_primitive(radical=radical))
        ce, pe = e.as_content_primitive(radical=radical)
        if b.is_Rational:
            #e
            #= ce*pe
            #= ce*(h + t)
            #= ce*h + ce*t
            #=> self
            #= b**(ce*h)*b**(ce*t)
            #= b**(cehp/cehq)*b**(ce*t)
            #= b**(iceh+r/cehq)*b**(ce*t)
            #= b**(iceh)*b**(r/cehq)*b**(ce*t)
            #= b**(iceh)*b**(ce*t + r/cehq)
            h, t = pe.as_coeff_Add()
            if h.is_Rational:
                ceh = ce*h
                c = Pow(b, ceh)
                r = S.Zero
                if not c.is_Rational:
                    iceh, r = divmod(ceh.p, ceh.q)
                    c = Pow(b, iceh)
                return c, Pow(b, _keep_coeff(ce, t + r/ce/ceh.q))
        e = _keep_coeff(ce, pe)
        # b**e = (h*t)**e = h**e*t**e = c*m*t**e
        if e.is_Rational and b.is_Mul:
            h, t = b.as_content_primitive(radical=radical)  # h is positive
            c, m = Pow(h, e).as_coeff_Mul()  # so c is positive
            m, me = m.as_base_exp()
            if m is S.One or me == e:  # probably always true
                # return the following, not return c, m*Pow(t, e)
                # which would change Pow into Mul; we let sympy
                # decide what to do by using the unevaluated Mul, e.g
                # should it stay as sqrt(2 + 2*sqrt(5)) or become
                # sqrt(2)*sqrt(1 + sqrt(5))
                return c, Pow(_keep_coeff(m, t), e)
        return S.One, Pow(b, e)