Numpy入门[17]——数组广播机制

参考：

• https://ailearning.apachecn.org/
• NumPy广播机制

使用Jupyter进行练习

NumPy 中的广播机制（Broadcast）旨在解决不同形状数组之间的算术运算问题。我们知道，如果进行运算的两个数组形状完全相同，它们直接可以做相应的运算。

但如果两个形状不同的数组呢？它们之间就不能做算术运算了吗？当然不是！为了保持数组形状相同，NumPy 设计了一种广播机制，这种机制的核心是对形状较小的数组，在横向或纵向上进行一定次数的重复，使其与形状较大的数组拥有相同的维度。

import numpy as np
# 正常的加法
a = np.array([[ 0, 0, 0],
              [10,10,10],
              [20,20,20],
              [30,30,30]])
b = np.array([[ 0, 1, 2],
              [ 0, 1, 2],
              [ 0, 1, 2],
              [ 0, 1, 2]])
a + b

array([[ 0,  1,  2],
       [10, 11, 12],
       [20, 21, 22],
       [30, 31, 32]])

# 将 b 的值变成一维的 [0,1,2] 之后的加法
b = np.array([0,1,2])

a + b

array([[ 0,  1,  2],
       [10, 11, 12],
       [20, 21, 22],
       [30, 31, 32]])

结果一样，虽然两个数组的维数不一样，但是 Numpy 检测到 b 的维度与 a 的维度匹配，所以将 b 扩展为之前的形式，得到相同的形状。

对于更高维度，这样的扩展依然有效。

如果我们再将 a 变成一个列向量呢？

a = np.array([0,10,20,30])
a.shape = 4,1
a

array([[ 0],
       [10],
       [20],
       [30]])

b

array([0, 1, 2])

a + b

array([[ 0,  1,  2],
       [10, 11, 12],
       [20, 21, 22],
       [30, 31, 32]])

可以看到，虽然两者的维度并不相同，但是Numpy还是根据两者的维度，自动将它们进行扩展然后进行计算。

对于 Numpy 来说，维度匹配当且仅当：

• 维度相同
• 有一个的维度是1

匹配会从最后一维开始进行，直到某一个的维度全部匹配为止，因此对于以下情况，Numpy 都会进行相应的匹配：

A	B	Result
3d array: 256 x 256 x 3	1d array: 3	3d array: 256 x 256 x 3
4d array: 8 x 1 x 6 x 1	3d array: 7 x 1 x 5	3d array: 8 x 7 x 6 x 5
3d array: 5 x 4 x 3	1d array: 1	3d array: 5 x 4 x 3
3d array: 15 x 4 x 13	1d array: 15 x 1 x 13	3d array: 15 x 4 x 13
2d array: 4 x 1	1d array: 3	2d array: 4 x 3

匹配成功后，Numpy 会进行运算得到相应的结果。

当然，如果相应的维度不匹配，那么Numpy会报错：

a = np.array([0,10,20,30])
print(a.shape)
print(b.shape)
a+b

(4,)
(3,)



---------------------------------------------------------------------------

ValueError                                Traceback (most recent call last)

Cell In [8], line 4
      2 print(a.shape)
      3 print(b.shape)
----> 4 a+b


ValueError: operands could not be broadcast together with shapes (4,) (3,) 

将 a 转换为列向量，还是可以计算出结果：

# np.newaxis表示增加一个数据宽度为1的维度
a[:,np.newaxis]+b

array([[ 0,  1,  2],
       [10, 11, 12],
       [20, 21, 22],
       [30, 31, 32]])

应用举例：

import matplotlib.pyplot as plt
# 先形成一个 21 乘 21 的网格
x = np.linspace(-.5,.5, 21)
y = x[:, np.newaxis]
# 再计算网格到原点的距离
radius = np.sqrt(x ** 2 + y ** 2)
print(radius)
# 展示一副热度图，将数组表示为一幅图
plt.imshow(radius)

[[0.70710678 0.6726812  0.64031242 0.61032778 0.58309519 0.55901699
  0.53851648 0.52201533 0.50990195 0.50249378 0.5        0.50249378
  0.50990195 0.52201533 0.53851648 0.55901699 0.58309519 0.61032778
  0.64031242 0.6726812  0.70710678]
 [0.6726812  0.6363961  0.60207973 0.57008771 0.54083269 0.51478151
  0.49244289 0.47434165 0.46097722 0.45276926 0.45       0.45276926
  0.46097722 0.47434165 0.49244289 0.51478151 0.54083269 0.57008771
  0.60207973 0.6363961  0.6726812 ]
 [0.64031242 0.60207973 0.56568542 0.53150729 0.5        0.47169906
  0.4472136  0.42720019 0.41231056 0.40311289 0.4        0.40311289
  0.41231056 0.42720019 0.4472136  0.47169906 0.5        0.53150729
  0.56568542 0.60207973 0.64031242]
 [0.61032778 0.57008771 0.53150729 0.49497475 0.46097722 0.43011626
  0.40311289 0.38078866 0.36400549 0.35355339 0.35       0.35355339
  0.36400549 0.38078866 0.40311289 0.43011626 0.46097722 0.49497475
  0.53150729 0.57008771 0.61032778]
 [0.58309519 0.54083269 0.5        0.46097722 0.42426407 0.39051248
  0.36055513 0.3354102  0.31622777 0.30413813 0.3        0.30413813
  0.31622777 0.3354102  0.36055513 0.39051248 0.42426407 0.46097722
  0.5        0.54083269 0.58309519]
 [0.55901699 0.51478151 0.47169906 0.43011626 0.39051248 0.35355339
  0.32015621 0.29154759 0.26925824 0.25495098 0.25       0.25495098
  0.26925824 0.29154759 0.32015621 0.35355339 0.39051248 0.43011626
  0.47169906 0.51478151 0.55901699]
 [0.53851648 0.49244289 0.4472136  0.40311289 0.36055513 0.32015621
  0.28284271 0.25       0.2236068  0.20615528 0.2        0.20615528
  0.2236068  0.25       0.28284271 0.32015621 0.36055513 0.40311289
  0.4472136  0.49244289 0.53851648]
 [0.52201533 0.47434165 0.42720019 0.38078866 0.3354102  0.29154759
  0.25       0.21213203 0.18027756 0.15811388 0.15       0.15811388
  0.18027756 0.21213203 0.25       0.29154759 0.3354102  0.38078866
  0.42720019 0.47434165 0.52201533]
 [0.50990195 0.46097722 0.41231056 0.36400549 0.31622777 0.26925824
  0.2236068  0.18027756 0.14142136 0.1118034  0.1        0.1118034
  0.14142136 0.18027756 0.2236068  0.26925824 0.31622777 0.36400549
  0.41231056 0.46097722 0.50990195]
 [0.50249378 0.45276926 0.40311289 0.35355339 0.30413813 0.25495098
  0.20615528 0.15811388 0.1118034  0.07071068 0.05       0.07071068
  0.1118034  0.15811388 0.20615528 0.25495098 0.30413813 0.35355339
  0.40311289 0.45276926 0.50249378]
 [0.5        0.45       0.4        0.35       0.3        0.25
  0.2        0.15       0.1        0.05       0.         0.05
  0.1        0.15       0.2        0.25       0.3        0.35
  0.4        0.45       0.5       ]
 [0.50249378 0.45276926 0.40311289 0.35355339 0.30413813 0.25495098
  0.20615528 0.15811388 0.1118034  0.07071068 0.05       0.07071068
  0.1118034  0.15811388 0.20615528 0.25495098 0.30413813 0.35355339
  0.40311289 0.45276926 0.50249378]
 [0.50990195 0.46097722 0.41231056 0.36400549 0.31622777 0.26925824
  0.2236068  0.18027756 0.14142136 0.1118034  0.1        0.1118034
  0.14142136 0.18027756 0.2236068  0.26925824 0.31622777 0.36400549
  0.41231056 0.46097722 0.50990195]
 [0.52201533 0.47434165 0.42720019 0.38078866 0.3354102  0.29154759
  0.25       0.21213203 0.18027756 0.15811388 0.15       0.15811388
  0.18027756 0.21213203 0.25       0.29154759 0.3354102  0.38078866
  0.42720019 0.47434165 0.52201533]
 [0.53851648 0.49244289 0.4472136  0.40311289 0.36055513 0.32015621
  0.28284271 0.25       0.2236068  0.20615528 0.2        0.20615528
  0.2236068  0.25       0.28284271 0.32015621 0.36055513 0.40311289
  0.4472136  0.49244289 0.53851648]
 [0.55901699 0.51478151 0.47169906 0.43011626 0.39051248 0.35355339
  0.32015621 0.29154759 0.26925824 0.25495098 0.25       0.25495098
  0.26925824 0.29154759 0.32015621 0.35355339 0.39051248 0.43011626
  0.47169906 0.51478151 0.55901699]
 [0.58309519 0.54083269 0.5        0.46097722 0.42426407 0.39051248
  0.36055513 0.3354102  0.31622777 0.30413813 0.3        0.30413813
  0.31622777 0.3354102  0.36055513 0.39051248 0.42426407 0.46097722
  0.5        0.54083269 0.58309519]
 [0.61032778 0.57008771 0.53150729 0.49497475 0.46097722 0.43011626
  0.40311289 0.38078866 0.36400549 0.35355339 0.35       0.35355339
  0.36400549 0.38078866 0.40311289 0.43011626 0.46097722 0.49497475
  0.53150729 0.57008771 0.61032778]
 [0.64031242 0.60207973 0.56568542 0.53150729 0.5        0.47169906
  0.4472136  0.42720019 0.41231056 0.40311289 0.4        0.40311289
  0.41231056 0.42720019 0.4472136  0.47169906 0.5        0.53150729
  0.56568542 0.60207973 0.64031242]
 [0.6726812  0.6363961  0.60207973 0.57008771 0.54083269 0.51478151
  0.49244289 0.47434165 0.46097722 0.45276926 0.45       0.45276926
  0.46097722 0.47434165 0.49244289 0.51478151 0.54083269 0.57008771
  0.60207973 0.6363961  0.6726812 ]
 [0.70710678 0.6726812  0.64031242 0.61032778 0.58309519 0.55901699
  0.53851648 0.52201533 0.50990195 0.50249378 0.5        0.50249378
  0.50990195 0.52201533 0.53851648 0.55901699 0.58309519 0.61032778
  0.64031242 0.6726812  0.70710678]]





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