Broadcasting describes how NumPy automatically brings two arrays with different shapes to a compatible shape during arithmetic operations. Generally, the smaller array is “repeated” multiple times until both arrays have the same shape. Broadcasting is memory-efficient as it doesn’t actually copy the smaller array multiple times.

Here’s a minimal example:

import numpy as np A = np.array([1, 2, 3]) res = A * 3 # scalar is broadcasted to [3 3 3] print(res) # [3 6 9]

Let’s have a more gentle introduction into the motivation and the concept next.

Table of Contents

## A Visual Representation

The following visual representation of two broadcasting operations shows

- how a scalar (0-dimensional) value can be broadcasted to a 1-dimensional array, and
- how a 1-dimensional array can be broadcasted to a 2-dimensional array.

In both cases, you essentially copy the smaller array until you reach the same shape of the larger array.

## Gentle Introduction

Many NumPy operations such as multiplication are performed “element-wise”. For example: say you multiply two NumPy arrays `A`

and `B`

with the same shape: `A * B`

.

In this case, NumPy performs element-wise matrix multiplication by multiplying cell `A[i,j]`

with cell `B[i,j]`

for every cell in the arrays. As the arrays have the same shape, that’s not a problem.

import numpy as np salary = np.array([2000, 4000, 8000]) salary_bump = np.array([1.1, 1.1, 1.1]) print(salary * salary_bump)

**Puzzle: **What’s the output of this code snippet?

The puzzle shows a scenario where the company decides that all employees get a salary bump by 10% after a successful year.

Thus, the result of the puzzle is the NumPy array:

# [2200. 4400. 8800.]

So far so good. But what happens if you multiply two arrays with different shapes?

The answer is again: ** broadcasting**. Here is the same example but we are a bit lazy now and try to save a few bits.

salary = np.array([2000, 4000, 8000]) salary_bump = 1.1 print(salary * salary_bump)

**Puzzle: **What’s the output of this code snippet?

As all three values of the `salary_bump`

are exactly the same numbers, you try to shorten it by multiplying a NumPy array with a number. Thus, it produces exactly the same output.

# [2200. 4400. 8800.]

Internally, NumPy creates a second implicit array that is filled with the salary bump value 1.1. This array looks as follows: `np.array([1.1, 1.1, 1.1])`

. This procedure is called “broadcasting”.

Not only is the code using broadcasting more concise and readable, but it is also more efficient! In the first example, the `salary_bump`

array contains redundancies. But in the second example, NumPy gets rid of these redundancies — it does not really copy the data into a second NumPy array, this is only the conceptual idea but not the implementation.

## How to Apply Broadcasting to Two NumPy Arrays?

Broadcasting is only possible if,

Missing dimensions are not a problem in this scenario. Here is a nice visualization from the documentation of how NumPy arrays will be broadcasted together:

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

An important observation is the following: **For any dimension where the first array has a size of one, NumPy conceptually copies its data until the size of the second array is reached.** Moreover, if the dimension is completely missing for array B, it is simply copied as well along the missing dimension.

**Exercise**: try to understand what is happening here in the first example of the given visualization.

In summary, broadcasting automatically matches two arrays with incompatible shape — a beautiful feature of the NumPy library!

## Related Video Broadcasting

In the following video from my Python One-Liners book, I discuss broadcasting as well. Check it out to increase your understanding of this crucial Python concept:

## Where to Go From Here?

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