[Numpy] add cross product op

* implement - register op, infershape

* implement - cross product 2x2

* implement - cross product 2x3 3x2 3x3

* fix - cudaError: misaligned address

* add - test code

* fix - test axis=None

* add - Register _backward_npi_cross

* implement - get_cross_backward in test

* implement - cross backward 3x3 2x3 3x2

* implement - cross backward 2x2

* fix - sanity pylint

* fix - move the python wrapper to keep consistency

* impl - FFI for cross op

python/mxnet/ndarray/numpy/_op.py

@@ -43,7 +43,7 @@
            'swapaxes', 'clip', 'argmax', 'argmin', 'std', 'var', 'indices', 'copysign', 'ravel', 'unravel_index',
            'diag_indices_from', 'hanning', 'hamming', 'blackman', 'flip', 'flipud', 'fliplr',
            'hypot', 'bitwise_and', 'bitwise_xor', 'bitwise_or', 'rad2deg', 'deg2rad', 'unique', 'lcm',
-           'tril', 'triu', 'identity', 'take', 'ldexp', 'vdot', 'inner', 'outer', 'kron',
+           'tril', 'triu', 'identity', 'take', 'ldexp', 'vdot', 'inner', 'outer', 'cross', 'kron',
            'equal', 'not_equal', 'greater', 'less', 'greater_equal', 'less_equal', 'roll', 'rot90', 'einsum',
            'true_divide', 'nonzero', 'quantile', 'percentile', 'shares_memory', 'may_share_memory', 'interp',
            'diff', 'ediff1d', 'resize', 'polyval', 'nan_to_num', 'isnan', 'isinf', 'isposinf', 'isneginf', 'isfinite',
@@ -6190,6 +6190,46 @@ def ldexp(x1, x2, out=None, **kwargs):
     return _api_internal.ldexp(x1, x2, out)
 
 
+@set_module('mxnet.ndarray.numpy')
+def vdot(a, b):
+    r"""
+    Return the dot product of two vectors.
+    Note that `vdot` handles multidimensional arrays differently than `dot`:
+    it does *not* perform a matrix product, but flattens input arguments
+    to 1-D vectors first. Consequently, it should only be used for vectors.
+
+    Parameters
+    ----------
+    a : ndarray
+        First argument to the dot product.
+    b : ndarray
+        Second argument to the dot product.
+
+    Returns
+    -------
+    output : ndarray
+        Dot product of `a` and `b`.
+
+    See Also
+    --------
+    dot : Return the dot product without using the complex conjugate of the
+        first argument.
+
+    Examples
+    --------
+    Note that higher-dimensional arrays are flattened!
+    >>> a = np.array([[1, 4], [5, 6]])
+    >>> b = np.array([[4, 1], [2, 2]])
+    >>> np.vdot(a, b)
+    30
+    >>> np.vdot(b, a)
+    30
+    >>> 1*4 + 4*1 + 5*2 + 6*2
+    30
+    """
+    return tensordot(a.flatten(), b.flatten(), 1)
+
+
 @set_module('mxnet.ndarray.numpy')
 def inner(a, b):
     r"""
@@ -6297,25 +6337,135 @@ def outer(a, b):
     return tensordot(a.flatten(), b.flatten(), 0)
 
 
+@set_module('mxnet.ndarray.numpy')
+def cross(a, b, axisa=-1, axisb=-1, axisc=-1, axis=None): # pylint: disable=too-many-arguments
+    """
+    Return the cross product of two (arrays of) vectors.
+
+    The cross product of `a` and `b` in :math:`R^3` is a vector perpendicular
+    to both `a` and `b`.  If `a` and `b` are arrays of vectors, the vectors
+    are defined by the last axis of `a` and `b` by default, and these axis
+    can have dimensions 2 or 3.  Where the dimension of either `a` or `b` is
+    2, the third component of the input vector is assumed to be zero and the
+    cross product calculated accordingly.  In cases where both input vectors
+    have dimension 2, the z-component of the cross product is returned.
+
+    Parameters
+    ----------
+    a : ndarray
+        Components of the first vector(s).
+    b : ndarray
+        Components of the second vector(s).
+    axisa : int, optional
+        Axis of `a` that defines the vector(s).  By default, the last axis.
+    axisb : int, optional
+        Axis of `b` that defines the vector(s).  By default, the last axis.
+    axisc : int, optional
+        Axis of `c` containing the cross product vector(s).  Ignored if
+        both input vectors have dimension 2, as the return is scalar.
+        By default, the last axis.
+    axis : int, optional
+        If defined, the axis of `a`, `b` and `c` that defines the vector(s)
+        and cross product(s).  Overrides `axisa`, `axisb` and `axisc`.
+
+    Returns
+    -------
+    c : ndarray
+        Vector cross product(s).
+
+    Raises
+    ------
+    ValueError
+        When the dimension of the vector(s) in `a` and/or `b` does not
+        equal 2 or 3.
+
+    Notes
+    -----
+    Supports full broadcasting of the inputs.
+
+    Examples
+    --------
+    Vector cross-product.
+
+    >>> x = np.array([1., 2., 3.])
+    >>> y = np.array([4., 5., 6.])
+    >>> np.cross(x, y)
+    array([-3.,  6., -3.])
+
+    One vector with dimension 2.
+
+    >>> x = np.array([1., 2.])
+    >>> y = np.array([4., 5., 6.])
+    >>> np.cross(x, y)
+    array([12., -6., -3.])
+
+    Equivalently:
+
+    >>> x = np.array([1., 2., 0.])
+    >>> y = np.array([4., 5., 6.])
+    >>> np.cross(x, y)
+    array([12., -6., -3.])
+
+    Both vectors with dimension 2.
+
+    >>> x = np.array([1., 2.])
+    >>> y = np.array([4., 5.])
+    >>> np.cross(x, y)
+    array(-3.)
+
+    Multiple vector cross-products. Note that the direction of the cross
+    product vector is defined by the `right-hand rule`.
+
+    >>> x = np.array([[1., 2., 3.], [4., 5., 6.]])
+    >>> y = np.array([[4., 5., 6.], [1., 2., 3.]])
+    >>> np.cross(x, y)
+    array([[-3.,  6., -3.],
+           [ 3., -6.,  3.]])
+
+    The orientation of `c` can be changed using the `axisc` keyword.
+
+    >>> np.cross(x, y, axisc=0)
+    array([[-3.,  3.],
+           [ 6., -6.],
+           [-3.,  3.]])
+
+    Change the vector definition of `x` and `y` using `axisa` and `axisb`.
+
+    >>> x = np.array([[1., 2., 3.], [4., 5., 6.], [7., 8., 9.]])
+    >>> y = np.array([[7., 8., 9.], [4., 5., 6.], [1., 2., 3.]])
+    >>> np.cross(x, y)
+    array([[ -6.,  12.,  -6.],
+           [  0.,   0.,   0.],
+           [  6., -12.,   6.]])
+    >>> np.cross(x, y, axisa=0, axisb=0)
+    array([[-24.,  48., -24.],
+           [-30.,  60., -30.],
+           [-36.,  72., -36.]])
+    """
+    if axis is not None:
+        axisa, axisb, axisc = (axis,) * 3
+
+    if isinstance(a, NDArray) and isinstance(b, NDArray):
+        return _api_internal.cross(a, b, axisa, axisb, axisc)
+    else:
+        raise TypeError("Input data should be NDarray")
+
+
 @set_module('mxnet.ndarray.numpy')
 def kron(a, b):
     r"""
     Kronecker product of two arrays.
     Computes the Kronecker product, a composite array made of blocks of the
     second array scaled by the first.
-
     Parameters
     ----------
     a, b : ndarray
-
     Returns
     -------
     out : ndarray
-
     See Also
     --------
     outer : The outer product
-
     Notes
     -----
     The function assumes that the number of dimensions of `a` and `b`
@@ -6331,7 +6481,6 @@ def kron(a, b):
         [[ a[0,0]*b,   a[0,1]*b,  ... , a[0,-1]*b  ],
         [  ...                              ...   ],
         [ a[-1,0]*b,  a[-1,1]*b, ... , a[-1,-1]*b ]]
-
     Examples
     --------
     >>> np.kron([1,10,100], [5,6,7])
@@ -6342,46 +6491,6 @@ def kron(a, b):
     return _api_internal.kron(a, b)
 
 
-@set_module('mxnet.ndarray.numpy')
-def vdot(a, b):
-    r"""
-    Return the dot product of two vectors.
-    Note that `vdot` handles multidimensional arrays differently than `dot`:
-    it does *not* perform a matrix product, but flattens input arguments
-    to 1-D vectors first. Consequently, it should only be used for vectors.
-
-    Parameters
-    ----------
-    a : ndarray
-        First argument to the dot product.
-    b : ndarray
-        Second argument to the dot product.
-
-    Returns
-    -------
-    output : ndarray
-        Dot product of `a` and `b`.
-
-    See Also
-    --------
-    dot : Return the dot product without using the complex conjugate of the
-        first argument.
-
-    Examples
-    --------
-    Note that higher-dimensional arrays are flattened!
-    >>> a = np.array([[1, 4], [5, 6]])
-    >>> b = np.array([[4, 1], [2, 2]])
-    >>> np.vdot(a, b)
-    30
-    >>> np.vdot(b, a)
-    30
-    >>> 1*4 + 4*1 + 5*2 + 6*2
-    30
-    """
-    return tensordot(a.flatten(), b.flatten(), 1)
-
-
 @set_module('mxnet.ndarray.numpy')
 def equal(x1, x2, out=None):
     """