import numpy as np
给定一个向量，让向量中每一个数乘以2
a = (0,1,2)
a*2 = (0,2,4)
普通python中
n = 10
L = [i for i in range(n)]
L
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
2 * L
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
A = []
for e in L:
A.append(2*e)
A
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
n = 1000000
L = [i for i in range(n)]
%%time
A = []
for e in L:
A.append(2*e)
A
CPU times: user 175 ms, sys: 19.8 ms, total: 195 ms
Wall time: 195 ms
%%time
A = [2*e for e in L]
CPU times: user 75.8 ms, sys: 24.4 ms, total: 100 ms
Wall time: 99.8 ms
使用numpy
L = np.arange(n)
%%time
A = np.array(2*e for e in L)
CPU times: user 11.4 ms, sys: 5.27 ms, total: 16.7 ms
Wall time: 17 ms
%%time
A = 2 * L
CPU times: user 3.27 ms, sys: 3.95 ms, total: 7.22 ms
Wall time: 5.48 ms
A
array([ 0, 2, 4, ..., 1999994, 1999996, 1999998])
n = 10
L = np.arange(n)
2 * L
array([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])
Universal Functions
X = np.arange(1,16).reshape((3,5))
X
array([[ 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10],
[11, 12, 13, 14, 15]])
X + 1
array([[ 2, 3, 4, 5, 6],
[ 7, 8, 9, 10, 11],
[12, 13, 14, 15, 16]])
X - 1
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
X * 2
array([[ 2, 4, 6, 8, 10],
[12, 14, 16, 18, 20],
[22, 24, 26, 28, 30]])
# 浮点数除法
X / 2
array([[0.5, 1. , 1.5, 2. , 2.5],
[3. , 3.5, 4. , 4.5, 5. ],
[5.5, 6. , 6.5, 7. , 7.5]])
# 整数除法
X // 2
array([[0, 1, 1, 2, 2],
[3, 3, 4, 4, 5],
[5, 6, 6, 7, 7]])
# 乘方
X ** 2
array([[ 1, 4, 9, 16, 25],
[ 36, 49, 64, 81, 100],
[121, 144, 169, 196, 225]])
# 取余
X % 2
array([[1, 0, 1, 0, 1],
[0, 1, 0, 1, 0],
[1, 0, 1, 0, 1]])
# 取倒数
1 / X
array([[1. , 0.5 , 0.33333333, 0.25 , 0.2 ],
[0.16666667, 0.14285714, 0.125 , 0.11111111, 0.1 ],
[0.09090909, 0.08333333, 0.07692308, 0.07142857, 0.06666667]])
# 取绝对值
np.abs(X)
array([[ 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10],
[11, 12, 13, 14, 15]])
# 取sin
np.sin(X)
array([[ 0.84147098, 0.90929743, 0.14112001, -0.7568025 , -0.95892427],
[-0.2794155 , 0.6569866 , 0.98935825, 0.41211849, -0.54402111],
[-0.99999021, -0.53657292, 0.42016704, 0.99060736, 0.65028784]])
# 取cos
np.cos(X)
array([[ 0.54030231, -0.41614684, -0.9899925 , -0.65364362, 0.28366219],
[ 0.96017029, 0.75390225, -0.14550003, -0.91113026, -0.83907153],
[ 0.0044257 , 0.84385396, 0.90744678, 0.13673722, -0.75968791]])
# 取tan
np.tan(X)
array([[ 1.55740772e+00, -2.18503986e+00, -1.42546543e-01,
1.15782128e+00, -3.38051501e+00],
[-2.91006191e-01, 8.71447983e-01, -6.79971146e+00,
-4.52315659e-01, 6.48360827e-01],
[-2.25950846e+02, -6.35859929e-01, 4.63021133e-01,
7.24460662e+00, -8.55993401e-01]])
# 取 e的x次方
np.exp(X)
array([[2.71828183e+00, 7.38905610e+00, 2.00855369e+01, 5.45981500e+01,
1.48413159e+02],
[4.03428793e+02, 1.09663316e+03, 2.98095799e+03, 8.10308393e+03,
2.20264658e+04],
[5.98741417e+04, 1.62754791e+05, 4.42413392e+05, 1.20260428e+06,
3.26901737e+06]])
# 3的x次方
np.power(3,X)
array([[ 3, 9, 27, 81, 243],
[ 729, 2187, 6561, 19683, 59049],
[ 177147, 531441, 1594323, 4782969, 14348907]])
3 **X
array([[ 3, 9, 27, 81, 243],
[ 729, 2187, 6561, 19683, 59049],
[ 177147, 531441, 1594323, 4782969, 14348907]])
# 以e为底x的对数
np.log(X)
array([[0. , 0.69314718, 1.09861229, 1.38629436, 1.60943791],
[1.79175947, 1.94591015, 2.07944154, 2.19722458, 2.30258509],
[2.39789527, 2.48490665, 2.56494936, 2.63905733, 2.7080502 ]])
np.log2(X)
array([[0. , 1. , 1.5849625 , 2. , 2.32192809],
[2.5849625 , 2.80735492, 3. , 3.169925 , 3.32192809],
[3.45943162, 3.5849625 , 3.70043972, 3.80735492, 3.9068906 ]])
np.log10(X)
array([[0. , 0.30103 , 0.47712125, 0.60205999, 0.69897 ],
[0.77815125, 0.84509804, 0.90308999, 0.95424251, 1. ],
[1.04139269, 1.07918125, 1.11394335, 1.14612804, 1.17609126]])
矩阵运算
A = np.arange(4).reshape(2,2)
B = np.full((2,2),10)
B
array([[10, 10],
[10, 10]])
A + B
array([[10, 11],
[12, 13]])
A-B
array([[-10, -9],
[ -8, -7]])
# A和B对应元素的相乘，并不是矩阵相乘
A * B
array([[ 0, 10],
[20, 30]])
# 矩阵相乘
# A中的每一行，和B中的每一列相乘再相加
A.dot(B)
array([[10, 10],
[50, 50]])
# 矩阵的转置
# 行转列，列转行
A.T
array([[0, 2],
[1, 3]])
C = np.full((3,3),6)
C
array([[6, 6, 6],
[6, 6, 6],
[6, 6, 6]])
A.dot(C)
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-46-36f3f9c6ed4d> in <module>()
----> 1 A.dot(C)
ValueError: shapes (2,2) and (3,3) not aligned: 2 (dim 1) != 3 (dim 0)
向量和矩阵运算
v = np.array([1,2])
v
array([1, 2])
A
array([[0, 1],
[2, 3]])
# 数学上没有意义
v+A
array([[1, 3],
[3, 5]])
np.vstack([v] * A.shape[0])+A
array([[1, 3],
[3, 5]])
# tile 行堆叠两次，列堆叠一次
np.tile(v,(2,1)) +A
array([[1, 3],
[3, 5]])
v
array([1, 2])
A
array([[0, 1],
[2, 3]])
v * A
array([[0, 2],
[2, 6]])
v.dot(A)
array([4, 7])
A.dot(v)
array([2, 8])
矩阵的逆
A
array([[0, 1],
[2, 3]])
np.linalg.inv(A)
array([[-1.5, 0.5],
[ 1. , 0. ]])
invA = np.linalg.inv(A)
# 矩阵乘以它的逆矩阵得到的是单位矩阵，对角线为1
# 方阵才有逆矩阵
A.dot(invA)
array([[1., 0.],
[0., 1.]])
invA.dot(A)
array([[1., 0.],
[0., 1.]])
# 伪逆矩阵
X = np.arange(16).reshape(2,8)
pinvX = np.linalg.pinv(X)
pinvX
array([[-1.35416667e-01, 5.20833333e-02],
[-1.01190476e-01, 4.16666667e-02],
[-6.69642857e-02, 3.12500000e-02],
[-3.27380952e-02, 2.08333333e-02],
[ 1.48809524e-03, 1.04166667e-02],
[ 3.57142857e-02, -1.04083409e-17],
[ 6.99404762e-02, -1.04166667e-02],
[ 1.04166667e-01, -2.08333333e-02]])
pinvX.shape
(8, 2)
X.dot(pinvX)
array([[ 1.00000000e+00, -2.49800181e-16],
[ 0.00000000e+00, 1.00000000e+00]])
