说明

文章主要介绍了 Numpy 中几个乘法函数的使用方法，包含较多的代码示例，建议点击目录直达所需内容。在以下示例代码中，均已事前导入Numpy：import numpy as np

版本

• Python 3.7.0
• Numpy 1.18.1

np.multiply(x1, x2)

函数注释

该函数是逐元素乘法，如果两个参数x1和x2的shape不同，将会进行 broadcastable

Multiply arguments element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays to be multiplied. If x1.shape != x2.shape, they must be broadcastable to a common shape (which becomes the shape of the output).
Returns
-------
y : ndarray
The product of x1 and x2, element-wise.
This is a scalar if both x1 and x2 are scalars.

用法示例

【注】该函数的两个参数的位置可以替换，不影响最后的运算结果。

1. x1 x2都是 scalar，返回两者的乘积，也是 scalar

   print(np.multiply(2, 4))
   print(np.multiply(2, 3.8))
   ---output---
   8
   7.6
   

2. x1 为 scalar，x2 为 array_like，此时为矩阵与数的乘法，该数与矩阵中每一元素相乘，最后返回同shape的 array。按照文档中的描述，应该是先将scalar 广播（broadcast）为与另一参数同shape的矩阵，再进行逐元素的乘法。

   print(np.multiply(2, [3, 4, 8]))
   print(np.multiply(2, np.arange(0, 8).reshape((2, 2, 2))))
   ---output---
   [ 6  8 16]
   [[[ 0  2][ 4  6]][[ 8 10][12 14]]]
   

3. x1 x2都是 array_like, 如果两个的shape相同，直接逐元素相乘再返回即可，否则需要进行广播（broadcast）。可以广播的最基本的条件就是两者shape的最后一个维度必须是相同的，具体的广播策略比较复杂，通过几个例子来试着理解一下。
   【例1】 (2, 2, 3) , (2, 3) —>(2, 2, 3)

   x1 = np.arange(0, 12).reshape((2, 2, 3))
   x2 = np.arange(0, 6).reshape(2, 3)
   print("-----x1------", x1.shape, "\n", x1)
   print("-----x2------", x2.shape, "\n", x2)
   print("-----multiply(x1, x2)------", np.multiply(x1, x2).shape, "\n", np.multiply(x1, x2))
   ---output---
   -----x1------ (2, 2, 3) [[[ 0  1  2][ 3  4  5]][[ 6  7  8][ 9 10 11]]]
   -----x2------ (2, 3) [[0 1 2][3 4 5]]
   -----multiply(x1, x2)------ (2, 2, 3) [[[ 0  1  4][ 9 16 25]][[ 0  7 16][27 40 55]]]
   

   【例2】(2, 2, 1, 3) ，(2, 3) —>(2, 2, 2, 3)

   x1 = np.arange(0, 12).reshape((2, 2, 1, 3))
   x2 = np.arange(0, 6).reshape(2, 3)
   print("-----x1------", x1.shape, "\n", x1)
   print("-----x2------", x2.shape, "\n", x2)
   print("-----multiply(x1, x2)------", np.multiply(x1, x2).shape, "\n", np.multiply(x1, x2))
   ---output---
   -----x1------ (2, 2, 1, 3) [[[[ 0  1  2]][[ 3  4  5]]][[[ 6  7  8]][[ 9 10 11]]]]
   -----x2------ (2, 3) [[0 1 2][3 4 5]]
   -----multiply(x1, x2)------ (2, 2, 2, 3) [[[[ 0  1  4][ 0  4 10]][[ 0  4 10][ 9 16 25]]][[[ 0  7 16][18 28 40]][[ 0 10 22][27 40 55]]]]
   

   【例3】 (2, 3, 1, 2) , (3, 2) —>(2, 3, 3, 2)

   x1 = np.arange(0, 12).reshape((2, 3, 1, 2))
   x2 = np.arange(0, 6).reshape(3, 2)
   print("-----x1------", x1.shape, "\n", x1)
   print("-----x2------", x2.shape, "\n", x2)
   print("-----multiply(x1, x2)------", np.multiply(x1, x2).shape, "\n", np.multiply(x1, x2))
   -----x1------ (2, 3, 1, 2) [[[[ 0  1]][[ 2  3]][[ 4  5]]][[[ 6  7]][[ 8  9]][[10 11]]]]
   -----x2------ (3, 2) [[0 1][2 3][4 5]]
   -----multiply(x1, x2)------ (2, 3, 3, 2) [[[[ 0  1][ 0  3][ 0  5]][[ 0  3][ 4  9][ 8 15]][[ 0  5][ 8 15][16 25]]][[[ 0  7][12 21][24 35]][[ 0  9][16 27][32 45]][[ 0 11][20 33][40 55]]]]
   

np.matmul(x1, x2)

函数注释

该函数就是数学中的矩阵乘法，要求两个参数必须为矩阵，不能是 scalar，而且参数顺序不能替换。维度变化：(n,k),(k,m)->(n,m)，即使当 n=1, m=1 时，结果为shape= (1, 1) 的矩阵。

Matrix product of two arrays.
Parameters
----------
x1, x2 : array_like
Input arrays, scalars not allowed.
Returns
-------
y : ndarray
The matrix product of the inputs.
This is a scalar only when both x1, x2 are 1-d vectors.

用法示例

1. 最常见的用法：(2, 2),(2, 3) -->(2, 3)

   x1 = np.arange(0, 4).reshape(2, 2)
   x2 = np.arange(0, 6).reshape(2, 3)
   print("-----x1------", x1.shape, "\n", x1)
   print("-----x2------", x2.shape, "\n", x2)
   print("-----matmul(x1, x2)------", np.matmul(x1, x2).shape, "\n", np.matmul(x1, x2))
   ---output---
   -----x1------ (2, 2) [[0 1][2 3]]
   -----x2------ (2, 3) [[0 1 2][3 4 5]]
   -----matmul(x1, x2)------ (2, 3) [[ 3  4  5][ 9 14 19]]
   

   2.计算结果只有一个元素的情形：(1, 3), (3, 1) -->(1, 1)

   x1 = np.arange(0, 3).reshape(1, 3)
   x2 = np.arange(0, 3).reshape(3, 1)
   print("-----x1------", x1.shape, "\n", x1)
   print("-----x2------", x2.shape, "\n", x2)
   print("-----matmul(x1, x2)------", np.matmul(x1, x2).shape, "\n", np.matmul(x1, x2))
   ---output---
   -----x1------ (1, 3) [[0 1 2]]
   -----x2------ (3, 1) [[0][1][2]]
   -----matmul(x1, x2)------ (1, 1) [[5]]
   

   3.多维参数：(2, 3, 2), (2, 3) -->(2, 3, 3), 按照 (n,k),(k,m)->(n,m)， n=3, k=2, m=3

   x1 = np.arange(0, 12).reshape(2, 3, 2)
   x2 = np.arange(0, 6).reshape(2, 3)
   print("-----x1------", x1.shape, "\n", x1)
   print("-----x2------", x2.shape, "\n", x2)
   print("-----matmul(x1, x2)------", np.matmul(x1, x2).shape, "\n", np.matmul(x1, x2))
   ---output---
   -----x1------ (2, 3, 2) [[[ 0  1][ 2  3][ 4  5]][[ 6  7][ 8  9][10 11]]]
   -----x2------ (2, 3) [[0 1 2][3 4 5]]
   -----matmul(x1, x2)------ (2, 3, 3) [[[ 3  4  5][ 9 14 19][15 24 33]][[21 34 47][27 44 61][33 54 75]]]
   

np.dot(a, b)

函数注释

参数的类型可以多样，对应着不同的运算方式和结果，总的来说，还是矩阵乘法用的多。一些冷门的调用方式感觉这辈子都不会用到，就不整理了。

Dot product of two arrays. Specifically,
- If both a and b are 1-D arrays, it is inner product of vectors (without complex conjugation).
- If both a and b are 2-D arrays, it is matrix multiplication, but using :func:matmul or a @ b is preferred.
- If either a or b is 0-D (scalar), it is equivalent to :func: multiply and using numpy.multiply(a, b) or a * b is preferred.
- If a is an N-D array and b is a 1-D array, it is a sum product over the last axis of a and b.
- If a is an N-D array and b is an M-D array (where M>=2), it is a sum product over the last axis of a and the second-to-last axis of b:: dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m])

用法示例

a	b	数学意义	示例
scalar	scalar	两数相乘	np.dot(3, 7) --> 21
scalar	1-D array	向量数乘	print(np.dot(3, [7, 2])) --> [21 6]
1-D array	1-D array	向量内积	np.dot([1, 3], [7, 2]) --> 13 np.dot(np.array([1, 3]), np.array([7, 2]))–>13
2-D array	2-D array	矩阵乘法	如下所示

print("---output---")
print(np.dot(3, 7))
print(np.dot(3, [7, 2]))
print(np.dot(3, np.arange(0, 4).reshape(2, 2)))
print(np.dot([1, 3], [7, 2]))
print(np.dot(np.array([1, 3]), np.array([7, 2])))
print(np.dot(np.array([1, 3]).reshape(1, 2), np.array([7, 2]).reshape(2, 1)))print("---output---")
x1 = np.arange(0, 4).reshape(2, 2)
x2 = np.arange(0, 6).reshape(2, 3)
print("-----x1------", x1.shape, "\n", x1)
print("-----x2------", x2.shape, "\n", x2)
print("-----dot(x1, x2)------", np.dot(x1, x2).shape, "\n", np.dot(x1, x2))
---output---
21
[21  6]
[[0 3][6 9]]
13
13
[[13]]
---output---
-----x1------ (2, 2) [[0 1][2 3]]
-----x2------ (2, 3) [[0 1 2][3 4 5]]
-----dot(x1, x2)------ (2, 3) [[ 3  4  5][ 9 14 19]]

总结

三个函数的功能有所交叉重合，现将常用的数学操作及其对应的函数选择总结如下：

x1	x2	数学操作	函数选择	示例
scalar	scalar	两数相乘	np.dot() 或 *	3 * 7 --> 21 np.dot(3, 7) --> 21
scalar	1-D array	向量数乘	np.dot() 或 np.multiply()	np.dot(3, [1, 4]) --> [ 3 12] np.multiply(3, [1, 4]) --> [ 3 12]
scalar	2-D array	矩阵数乘	np.dot() 或 np.multiply()	np.dot(3, np.arange(0, 4).reshape(2, 2)) --> [[0 3]， [6 9]] np.multiply(3, np.arange(0, 4).reshape(2, 2)) --> 同上
1-D array	1-D array	点积 （内积、数量积）	np.dot() 或 np.matmul()	np.dot([2, 5], [1, 4]) --> 22 np.matmul([2, 5], [1, 4]) --> 22
2-D array	2-D array	矩阵乘法	np.dot() 或 np.matmul()	x1 = np.arange(0, 3).reshape(1, 3) x2 = np.arange(0, 6).reshape(3, 2) np.dot(x1, x2) --> [[10 13]] np.matmul(x1, x2) --> [[10 13]]
2-D array	2-D array	哈达马乘积	np.multiply()	x3 = np.arange(0, 3).reshape(1, 3) x4 = np.arange(0, 3).reshape(1, 3) np.multiply(x3, x4) --> [[0 1 4]] np.multiply([2, 5], [1, 4]) --> [ 2 20]