Broadcasting — NumPy V1.23 Manual

General broadcasting rules#

When operating on two arrays, NumPy compares their shapes element-wise. It starts with the trailing (i.e. rightmost) dimension and works its way left. Two dimensions are compatible when

1. they are equal, or

2. one of them is 1.

If these conditions are not met, a ValueError: operands could not be broadcast together exception is thrown, indicating that the arrays have incompatible shapes.

Input arrays do not need to have the same number of dimensions. The resulting array will have the same number of dimensions as the input array with the greatest number of dimensions, where the size of each dimension is the largest size of the corresponding dimension among the input arrays. Note that missing dimensions are assumed to have size one.

For example, if you have a 256x256x3 array of RGB values, and you want to scale each color in the image by a different value, you can multiply the image by a one-dimensional array with 3 values. Lining up the sizes of the trailing axes of these arrays according to the broadcast rules, shows that they are compatible:

Image (3d array): 256 x 256 x 3 Scale (1d array): 3 Result (3d array): 256 x 256 x 3

When either of the dimensions compared is one, the other is used. In other words, dimensions with size 1 are stretched or “copied” to match the other.

In the following example, both the A and B arrays have axes with length one that are expanded to a larger size during the broadcast operation:

A (4d array): 8 x 1 x 6 x 1 B (3d array): 7 x 1 x 5 Result (4d array): 8 x 7 x 6 x 5