Python Data Science Handbook, 2nd Edition

Chapter 8. Computation on Arrays: Broadcasting

We saw in Chapter 6 how NumPy’s universal functions can be used to vectorize operations and thereby remove slow Python loops. This chapter discusses broadcasting: a set of rules by which NumPy lets you apply binary operations (e.g., addition, subtraction, multiplication, etc.) between arrays of different sizes and shapes.

Introducing Broadcasting

Recall that for arrays of the same size, binary operations are performed on an element-by-element basis:

In [1]: import numpy as np

In [2]: a = np.array([0, 1, 2])
        b = np.array([5, 5, 5])
        a + b
Out[2]: array([5, 6, 7])

Broadcasting allows these types of binary operations to be performed on arrays of different sizes—for example, we can just as easily add a scalar (think of it as a zero-dimensional array) to an array:

In [3]: a + 5
Out[3]: array([5, 6, 7])

We can think of this as an operation that stretches or duplicates the value 5 into the array [5, 5, 5], and adds the results.

We can similarly extend this idea to arrays of higher dimension. Observe the result when we add a one-dimensional array to a two-dimensional array:

In [4]: M = np.ones((3, 3))
        M
Out[4]: array([[1., 1., 1.],
               [1., 1., 1.],
               [1., 1., 1.]])

In [5]: M + a
Out[5]: array([[1., 2., 3.],
               [1., 2., 3.],
               [1., 2., 3.]])

Here the one-dimensional array a is stretched, or broadcasted, across the second dimension in order to match the shape of M.

While these examples are relatively easy to understand, more complicated cases can involve ...