Broadcasting describes how numpy treats arrays with different shapes during arithmetic operations.

# Overview

Broadcasting provides a means of vectorizing array operations so that looping occurs in C instead of Python. It does this without making needless copies of data and usually leads to efficient algorithm implementations.

NumPy’s broadcasting rule relaxes this constraint when the arrays’ shapes meet certain constraints.

# Simple Broadcasting: Array * Scalar

1 | 1.0, 2.0, 3.0]) a = np.array([ |

We can think of the scalar `b`

being *stretched* during the arithmetic operation into an array with the same shape as `a`

. And then, element-wise multiplication is performed.

The stretching analogy is only conceptual. NumPy is smart enough to use the original scalar value without actually making copies, so that broadcasting operations are as **memory and computationally efficient** as possible.

# General Broadcasting Rules: Array * Array

总体思想：先看shape的对应，画图，再计算。

## Comparing shape

When operating on two arrays, NumPy **compares their shapes** element-wise. It **starts with the last** dimensions, and works its way forward.

Two dimensions are compatible when

- they are equal, or
- one of them is 1 (or None)

If these conditions are not met, a`ValueError: frames are not aligned`

exception is thrown, indicating that the arrays have incompatible shapes.

The size of the resulting array is the**maximum size along each dimension**of the input arrays.

## Multiplication

**Example1**

Arrays do not need to have the same *number* of dimensions. 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:

1 | Image (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.

总结：有三个256X256的2Darray, 各个 array 分别用一个1d array的三个scalar乘以其全部element

**Example2**

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:

1 | A (4d array): 8 x 1 x 6 x 1 |

**More examples**

1 | A (3d array): 15 x 3 x 5 |

**An example of broadcasting in practice**

1 | >>> x = np.arange(4) |

## Application: outer operation

Broadcasting provides a convenient way of taking the outer product (or any other outer operation) of two arrays.

The following example shows an outer addition operation of two 1-d arrays:

1 | >>> a = np.array([0.0, 10.0, 20.0, 30.0]) |

Here the `newaxis`

index operator inserts a new axis into `a`

, making it a two-dimensional `4x1`

array. Combining the `4x1`

array with `b`

, which has shape `(3,)`

, yields a `4x3`

array.