User guide

NumPy User Guide, Release 1.9.0
Here are some more examples:
A (2d array): 5 x 4
B (1d array): 1
Result (2d array): 5 x 4
A (2d array): 5 x 4
B (1d array): 4
Result (2d array): 5 x 4
A (3d array): 15 x 3 x 5
B (3d array): 15 x 1 x 5
Result (3d array): 15 x 3 x 5
A (3d array): 15 x 3 x 5
B (2d array): 3 x 5
Result (3d array): 15 x 3 x 5
A (3d array): 15 x 3 x 5
B (2d array): 3 x 1
Result (3d array): 15 x 3 x 5
Here are examples of shapes that do not broadcast:
A (1d array): 3
B (1d array): 4 # trailing dimensions do not match
A (2d array): 2 x 1
B (3d array): 8 x 4 x 3 # second from last dimensions mismatched
An example of broadcasting in practice:
>>> x = np.arange(4)
>>> xx = x.reshape(4,1)
>>> y = np.ones(5)
>>> z = np.ones((3,4))
>>> x.shape
(4,)
>>> y.shape
(5,)
>>> x + y
<type ’exceptions.ValueError’>: shape mismatch: objects cannot be broadcast to a single shape
>>> xx.shape
(4, 1)
>>> y.shape
(5,)
>>> (xx + y).shape
(4, 5)
>>> xx + y
array([[ 1., 1., 1., 1., 1.],
[ 2., 2., 2., 2., 2.],
[ 3., 3., 3., 3., 3.],
[ 4., 4., 4., 4., 4.]])
28 Chapter 2. Numpy basics