def indicator(X):
     L = np.logical_and(float_cmp(X[:, 0], dd.domain[0, 0]), dd.left == bt)
     R = np.logical_and(float_cmp(X[:, 0], dd.domain[1, 0]), dd.right == bt)
     T = np.logical_and(float_cmp(X[:, 1], dd.domain[1, 1]), dd.top == bt)
     B = np.logical_and(float_cmp(X[:, 1], dd.domain[0, 1]), dd.bottom == bt)
     LR = np.logical_or(L, R)
     TB = np.logical_or(T, B)
     return np.logical_or(LR, TB)

 def test_other_functions(self):
     order = GaussQuadratures.orders[-1]
     for name, function, integral in FUNCTIONS:
         Q = GaussQuadratures.iter_quadrature(order)
         ret = sum([function(p) * w for (p, w) in Q])
         assert float_cmp(ret, integral), \
             f'{name} integral wrong: {integral} vs {ret} (quadrature order {order})'

 def test_other_functions(self):
     order = GaussQuadratures.orders[-1]
     for name, function, integral in FUNCTIONS:
         Q = GaussQuadratures.iter_quadrature(order)
         ret = sum([function(p) * w for (p, w) in Q])
         assert float_cmp(ret, integral), '{} integral wrong: {} vs {} (quadrature order {})'.format(
             name, integral, ret, order)

 def test_polynomials(self):
     for n, function, _, integral in polynomials(GaussQuadratures.orders[-1]):
         name = 'x^{}'.format(n)
         for order in GaussQuadratures.orders:
             if n > order / 2:
                 continue
             Q = GaussQuadratures.iter_quadrature(order)
             ret = sum([function(p) * w for (p, w) in Q])
             assert float_cmp(ret, integral), '{} integral wrong: {} vs {} (quadrature order {})'.format(
                 name, integral, ret, order)

 def test_polynomials(self):
     for n, function, _, integral in polynomials(GaussQuadratures.orders[-1]):
         name = f'x^{n}'
         for order in GaussQuadratures.orders:
             if n > order / 2:
                 continue
             Q = GaussQuadratures.iter_quadrature(order)
             ret = sum([function(p) * w for (p, w) in Q])
             assert float_cmp(ret, integral), \
                 f'{name} integral wrong: {integral} vs {ret} (quadrature order {order})'

    def test_props(self):
        tol_range = [0.0, 1e-8, 1]
        nan = float('nan')
        inf = float('inf')
        for (rtol, atol) in itertools.product(tol_range, tol_range):
            msg = 'rtol: {} | atol {}'.format(rtol, atol)
            assert float_cmp(0., 0., rtol, atol), msg
            assert float_cmp(-0., -0., rtol, atol), msg
            assert float_cmp(-1., -1., rtol, atol), msg
            assert float_cmp(0., -0., rtol, atol), msg
            assert not float_cmp(2., -2., rtol, atol), msg

            assert not float_cmp(nan, nan, rtol, atol), msg
            assert nan != nan
            assert not (nan == nan)
            assert not float_cmp(-nan, nan, rtol, atol), msg

            assert not float_cmp(inf, inf, rtol, atol), msg
            assert not (inf != inf)
            assert inf == inf
            if rtol > 0:
                assert float_cmp(-inf, inf, rtol, atol), msg
            else:
                assert not float_cmp(-inf, inf, rtol, atol), msg

    def test_props(self):
        tol_range = [None, 0.0, 1]
        nan = float('nan')
        inf = float('inf')
        for (rtol, atol) in itertools.product(tol_range, tol_range):
            msg = 'rtol: {} | atol {}'.format(rtol, atol)
            self.assertTrue(float_cmp(0, 0, rtol, atol), msg)
            self.assertTrue(float_cmp(-0, -0, rtol, atol), msg)
            self.assertTrue(float_cmp(-1, -1, rtol, atol), msg)
            self.assertTrue(float_cmp(0, -0, rtol, atol), msg)
            self.assertFalse(float_cmp(2, -2, rtol, atol), msg)

            self.assertFalse(float_cmp(nan, nan, rtol, atol), msg)
            self.assertTrue(nan != nan)
            self.assertFalse(nan == nan)
            self.assertFalse(float_cmp(-nan, nan, rtol, atol), msg)

            self.assertFalse(float_cmp(inf, inf, rtol, atol), msg)
            self.assertFalse(inf != inf)
            self.assertTrue(inf == inf)
            self.assertTrue(float_cmp(-inf, inf, rtol, atol), msg)

    def almost_equal(self, other, ind=None, o_ind=None, rtol=None, atol=None):
        assert self.check_ind(ind)
        assert other.check_ind(o_ind)
        assert self.dim == other.dim

        if NUMPY_INDEX_QUIRK:
            if self._len == 0 and hasattr(ind, '__len__'):
                ind = None
            if other._len == 0 and hasattr(o_ind, '__len__'):
                o_ind = None

        A = self._array[:self._len] if ind is None else \
            self._array[ind] if hasattr(ind, '__len__') else self._array[ind:ind + 1]
        B = other._array[:other._len] if o_ind is None else \
            other._array[o_ind] if hasattr(o_ind, '__len__') else other._array[o_ind:o_ind + 1]

        R = np.all(float_cmp(A, B, rtol=rtol, atol=atol), axis=1).squeeze()
        if R.ndim == 0:
            R = R[np.newaxis, ...]
        return R

def test_newton(order):
    U, _ = _newton(order)
    assert float_cmp(U.data, 0.0)

def test_newton(order):
    U, _ = _newton(order, atol=1e-15)
    assert float_cmp(U.data, 0.0)

def flatten_grid(grid):
    """This method is used by our visualizers to render n-dimensional grids which cannot
    be embedded into R^n by duplicating vertices which would have to be mapped to multiple
    points at once. (Think of grids on rectangular domains with identified edges.)

    Parameters
    ----------
    grid
        The |Grid| to flatten.

    Returns
    -------
    subentities
        The `subentities(0, grid.dim)` relation for the flattened grid.
    coordinates
        The coordinates of the codim-`grid.dim` entities.
    entity_map
        Maps the indices of the codim-`grid.dim` entities of the flattened
        grid to the indices of the corresponding entities in the original grid.
    """
    # special handling of known flat grids
    if isinstance(grid, (RectGrid, TriaGrid)) and not grid.identify_left_right and not grid.identify_bottom_top:
        subentities = grid.subentities(0, grid.dim)
        coordinates = grid.centers(grid.dim)
        entity_map = np.arange(grid.size(grid.dim), dtype=np.int32)
        return subentities, coordinates, entity_map

    # first we determine which vertices are mapped to different coordinates when using the
    # embeddings of their codim-0 superentities
    dim = grid.dim
    global_coordinates = grid.embeddings(dim)[1]
    subentities = grid.subentities(0, dim)
    super_entities = grid.superentities(dim, 0)
    superentity_indices = grid.superentity_indices(dim, 0)
    A, B = grid.embeddings(0)
    ref_el_coordinates = grid.reference_element.subentity_embedding(dim)[1]
    local_coordinates = np.einsum('eij,vj->evi', A, ref_el_coordinates) + B[:, np.newaxis, :]
    critical_vertices = np.unique(subentities[np.logical_not(np.all(float_cmp(global_coordinates[subentities],
                                                                              local_coordinates), axis=2))])
    del A
    del B

    # when there are critical vertices, we have to create additional vertices
    if len(critical_vertices) > 0:
        subentities = subentities.copy()
        supe = super_entities[critical_vertices]
        supi = superentity_indices[critical_vertices]
        coord = local_coordinates[supe, supi]

        new_points = np.ones_like(supe, dtype=np.int32) * -1
        new_points[:, 0] = critical_vertices
        num_points = grid.size(dim)
        entity_map = np.empty((0,), dtype=np.int32)
        for i in xrange(new_points.shape[1]):
            for j in xrange(i):
                new_points[:, i] = np.where(supe[:, i] == -1, new_points[:, i],
                                            np.where(np.all(float_cmp(coord[:, i], coord[:, j]), axis=1),
                                                     new_points[:, j], new_points[:, i]))
            new_point_inds = np.where(np.logical_and(new_points[:, i] == -1, supe[:, i] != -1))[0]
            new_points[new_point_inds, i] = np.arange(num_points, num_points + len(new_point_inds))
            num_points += len(new_point_inds)
            entity_map = np.hstack((entity_map, critical_vertices[new_point_inds]))

        entity_map = np.hstack((np.arange(grid.size(dim), dtype=np.int32), entity_map))

        # handle -1 entries in supe/supi correctly ...
        ci = np.where(critical_vertices == subentities[-1, -1])[0]
        if len(ci) > 0:
            assert len(ci) == 1
            ci = ci[0]
            i = np.where(supe[ci] == (grid.size(0) - 1))[0]
            if len(i) > 0:
                assert len(i) == 1
                i = i[0]
                new_points[supe == -1] = new_points[ci, i]
            else:
                new_points[supe == -1] = subentities[-1, -1]
        else:
            new_points[supe == -1] = subentities[-1, -1]
        subentities[supe, supi] = new_points
        super_entities, superentity_indices = inverse_relation(subentities, size_rhs=num_points, with_indices=True)
        coordinates = local_coordinates[super_entities[:, 0], superentity_indices[:, 0]]
    else:
        coordinates = global_coordinates
        entity_map = np.arange(grid.size(dim), dtype=np.int32)

    return subentities, coordinates, entity_map

 def test_weights(self):
     for order in GaussQuadratures.orders:
         _, W = GaussQuadratures.quadrature(order)
         assert float_cmp(sum(W), 1)

def test_newton(order):
    U, _ = _newton(order, atol=1e-15)
    assert float_cmp(U.to_numpy(), 0.0)