def METHOD_NAME(self, **options):
    r"""
    Test the antipode.
    An *antipode* `S` of a (super) Hopf algebra is a linear
    endomorphism of the Hopf algebra that satisfies the
    following conditions (see :wikipedia:`HopfAlgebra`).
    - If `\mu` and `\Delta` denote the product and coproduct of the
      Hopf algebra, respectively, then `S` satisfies
      .. MATH::
          \mu \circ (S \tensor 1) \circ \Delta = unit \circ counit
          \mu \circ (1 \tensor S) \circ \Delta = unit \circ counit
    - `S` is an *anti*-homomorphism:
       .. MATH::
           S(ab) = (-1)^{\deg a \deg b} S(b) S(a)
       for homogeneous `a` and `b`.
    These properties are tested on :meth:`some_elements`.
    TESTS::
        sage: A = SteenrodAlgebra(7)                                            # needs sage.combinat sage.modules
        sage: A._test_antipode()        # long time                             # needs sage.combinat sage.modules
    """
    tester = self._tester(**options)
    S = self.antipode
    IS = lambda x: self.sum(c * self.monomial(t1) * S(self.monomial(t2))
                        for ((t1, t2), c) in x.coproduct())
    SI = lambda x: self.sum(c * S(self.monomial(t1)) * self.monomial(t2)
                        for ((t1, t2), c) in x.coproduct())
    for x in tester.some_elements():
        x_even = x.even_component()
        x_odd = x.odd_component()
        for y in tester.some_elements():
            y_even = y.even_component()
            y_odd = y.odd_component()
            # The antipode is a graded anti-homomorphism.
            tester.assertEqual(S(x_even) * S(y_even),
                               S(y_even * x_even))
            tester.assertEqual(S(x_even) * S(y_odd),
                               S(y_odd * x_even))
            tester.assertEqual(S(x_odd) * S(y_even),
                               S(y_even * x_odd))
            tester.assertEqual(S(x_odd) * S(y_odd),
                               -S(y_odd * x_odd))
        # mu * (S # I) * delta == counit * unit
        tester.assertEqual(SI(x), self.counit(x) * self.one())
        # mu * (I # S) * delta == counit * unit
        tester.assertEqual(IS(x), self.counit(x) * self.one())