目錄
| 函數 | 描述 | 用法 |
|---|---|---|
| abs fabs | 計算 整型/浮點/複數 的絕對值 對於沒有複數的快速版本求絕對值 | np.abs() np.fabs() |
| sqrt | 計算元素的平方根。等價於array ** 0.5 | np.sqrt() |
| square | 計算元素的平方。等價於 array **2 | np.squart() |
| exp | 計算以自然常數e為底的冪次方 | np.exp() |
| log log10 log2 log1p | 自然對數(e) 基於10的對數 基於2的對數 基於log(1+x)的對數 | np.log() np.log10() np.log2() np.log1p() |
| sign | 計算元素的符號：1：正數 0：0 -1：負數 | np.sign() |
| ceil | 計算大於或等於元素的最小整數 | np.ceil() |
| floor | 計算小於或等於元素的最大整數 | np.floor() |
| rint | 對浮點數取整到最近的整數，但不改變浮點數型別 | np.rint() |
| modf | 分別返回浮點數的整數和小數部分的陣列 | np.modf() |
| isnan | 返回布林陣列標識哪些元素是 NaN （不是一個數） | np.isnan() |
| isfinite isinf | 返回布林陣列標識哪些元素是有限的（non-inf, non-NaN）或無限的 | np.isfiniter() np.isinf() |
| cos, cosh, sin sinh, tan, tanh | 三角函數 | |
| arccos, arccosh, arcsin, arcsinh, arctan, arctanh | 反三角函數 | |
| logical_and/or/not/xor | 邏輯與/或/非/互斥或 等價於 ‘&’ ‘|’ ‘!’ ‘^’ | 測試見下方 |
# 邏輯與
>>> np.logical_and(True, False)
False
>>> np.logical_and([True, False], [False, False])
array([False, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_and(x>1, x<4)
array([False, False, True, True, False], dtype=bool)
# 邏輯或
>>> np.logical_or(True, False)
True
>>> np.logical_or([True, False], [False, False])
array([ True, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_or(x < 1, x > 3)
array([ True, False, False, False, True], dtype=bool)
# 邏輯非
>>> np.logical_not(3)
False
>>> np.logical_not([True, False, 0, 1])
array([False, True, True, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_not(x<3)
array([False, False, False, True, True], dtype=bool)
# 邏輯互斥或
>>> np.logical_xor(True, False)
True
>>> np.logical_xor([True, True, False, False], [True, False, True, False])
array([False, True, True, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_xor(x < 1, x > 3)
array([ True, False, False, False, True], dtype=bool)
>>> np.logical_xor(0, np.eye(2))
array([[ True, False],
[False, True]], dtype=bool)
向量化和廣播這兩個概念是 numpy 內部實現的基礎。有了向量化，編寫程式碼時無需使用顯式迴圈。這些迴圈實際上不能省略，只不過是在內部實現，被程式碼中的其他結構代替。向量化的應用使得程式碼更簡潔，可讀性更強，也可以說使用了向量化方法的程式碼看上去更「Pythonic」。
廣播（Broadcasting）機制描述了 numpy 如何在算術運算期間處理具有不同形狀的陣列，讓較小的陣列在較大的陣列上「廣播」，以便它們具有相容的形狀。並不是所有的維度都要彼此相容才符合廣播機制的要求，但它們必須滿足一定的條件。
若兩個陣列的各維度相容，也就是兩個陣列的每一維等長，或其中一個陣列為 一維，那麼廣播機制就適用。如果這兩個條件不滿足，numpy就會丟擲異常，說兩個陣列不相容。
總結來說，廣播的規則有三個：
二維陣列加一維陣列
import numpy as np
x = np.arange(4)
y = np.ones((3, 4))
print(x.shape) # (4,)
print(y.shape) # (3, 4)
print((x + y).shape) # (3, 4)
print(x + y)
# [[1. 2. 3. 4.]
# [1. 2. 3. 4.]
# [1. 2. 3. 4.]]兩個陣列均需要廣播
import numpy as np
x = np.arange(4).reshape(4, 1)
y = np.ones(5)
print(x.shape) # (4, 1)
print(y.shape) # (5,)
print((x + y).shape) # (4, 5)
print(x + y)
# [[1. 1. 1. 1. 1.]
# [2. 2. 2. 2. 2.]
# [3. 3. 3. 3. 3.]
# [4. 4. 4. 4. 4.]]
x = np.array([0.0, 10.0, 20.0, 30.0])
y = np.array([1.0, 2.0, 3.0])
z = x[:, np.newaxis] + y
print(z)
# [[ 1. 2. 3.]
# [11. 12. 13.]
# [21. 22. 23.]
# [31. 32. 33.]]不匹配報錯的例子
import numpy as np
x = np.arange(4)
y = np.ones(5)
print(x.shape) # (4,)
print(y.shape) # (5,)
print(x + y)
# ValueError: operands could not be broadcast together with shapes (4,) (5,) numpy.add(x1, x2, *args, **kwargs) Add arguments element-wise.numpy.subtract(x1, x2, *args, **kwargs) Subtract arguments element-wise.numpy.multiply(x1, x2, *args, **kwargs) Multiply arguments element-wise.numpy.divide(x1, x2, *args, **kwargs) Returns a true division of the inputs, element-wise.numpy.floor_divide(x1, x2, *args, **kwargs) Return the largest integer smaller or equal to the division of the inputs.numpy.power(x1, x2, *args, **kwargs) First array elements raised to powers from second array, element-wise.在 numpy 中對以上函數進行了運運算元的過載，且運運算元為 元素級。也就是說，它們只用於位置相同的元素之間，所得到的運算結果組成一個新的陣列。
注意 numpy 的廣播規則。
import numpy as np
x = np.array([1, 2, 3, 4, 5, 6, 7, 8])
y = x + 1
print(y)
print(np.add(x, 1))
# [2 3 4 5 6 7 8 9]
y = x - 1
print(y)
print(np.subtract(x, 1))
# [0 1 2 3 4 5 6 7]
y = x * 2
print(y)
print(np.multiply(x, 2))
# [ 2 4 6 8 10 12 14 16]
y = x / 2
print(y)
print(np.divide(x, 2))
# [0.5 1. 1.5 2. 2.5 3. 3.5 4. ]
y = x // 2
print(y)
print(np.floor_divide(x, 2))
# [0 1 1 2 2 3 3 4]
y = x ** 2
print(y)
print(np.power(x, 2))
# [ 1 4 9 16 25 36 49 64]注意 numpy 的廣播規則。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = x + 1
print(y)
print(np.add(x, 1))
# [[12 13 14 15 16]
# [17 18 19 20 21]
# [22 23 24 25 26]
# [27 28 29 30 31]
# [32 33 34 35 36]]
y = x - 1
print(y)
print(np.subtract(x, 1))
# [[10 11 12 13 14]
# [15 16 17 18 19]
# [20 21 22 23 24]
# [25 26 27 28 29]
# [30 31 32 33 34]]
y = x * 2
print(y)
print(np.multiply(x, 2))
# [[22 24 26 28 30]
# [32 34 36 38 40]
# [42 44 46 48 50]
# [52 54 56 58 60]
# [62 64 66 68 70]]
y = x / 2
print(y)
print(np.divide(x, 2))
# [[ 5.5 6. 6.5 7. 7.5]
# [ 8. 8.5 9. 9.5 10. ]
# [10.5 11. 11.5 12. 12.5]
# [13. 13.5 14. 14.5 15. ]
# [15.5 16. 16.5 17. 17.5]]
y = x // 2
print(y)
print(np.floor_divide(x, 2))
# [[ 5 6 6 7 7]
# [ 8 8 9 9 10]
# [10 11 11 12 12]
# [13 13 14 14 15]
# [15 16 16 17 17]]
y = x ** 2
print(y)
print(np.power(x, 2))
# [[ 121 144 169 196 225]
# [ 256 289 324 361 400]
# [ 441 484 529 576 625]
# [ 676 729 784 841 900]
# [ 961 1024 1089 1156 1225]]注意 numpy 的廣播規則。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.arange(1, 6)
print(y)
# [1 2 3 4 5]
z = x + y
print(z)
print(np.add(x, y))
# [[12 14 16 18 20]
# [17 19 21 23 25]
# [22 24 26 28 30]
# [27 29 31 33 35]
# [32 34 36 38 40]]
z = x - y
print(z)
print(np.subtract(x, y))
# [[10 10 10 10 10]
# [15 15 15 15 15]
# [20 20 20 20 20]
# [25 25 25 25 25]
# [30 30 30 30 30]]
z = x * y
print(z)
print(np.multiply(x, y))
# [[ 11 24 39 56 75]
# [ 16 34 54 76 100]
# [ 21 44 69 96 125]
# [ 26 54 84 116 150]
# [ 31 64 99 136 175]]
z = x / y
print(z)
print(np.divide(x, y))
# [[11. 6. 4.33333333 3.5 3. ]
# [16. 8.5 6. 4.75 4. ]
# [21. 11. 7.66666667 6. 5. ]
# [26. 13.5 9.33333333 7.25 6. ]
# [31. 16. 11. 8.5 7. ]]
z = x // y
print(z)
print(np.floor_divide(x, y))
# [[11 6 4 3 3]
# [16 8 6 4 4]
# [21 11 7 6 5]
# [26 13 9 7 6]
# [31 16 11 8 7]]
z = x ** np.full([1, 5], 2)
print(z)
print(np.power(x, np.full([5, 5], 2)))
# [[ 121 144 169 196 225]
# [ 256 289 324 361 400]
# [ 441 484 529 576 625]
# [ 676 729 784 841 900]
# [ 961 1024 1089 1156 1225]]numpy.sin(x, *args, **kwargs) Trigonometric sine, element-wise.numpy.cos(x, *args, **kwargs) Cosine element-wise.numpy.tan(x, *args, **kwargs) Compute tangent element-wise.numpy.arcsin(x, *args, **kwargs) Inverse sine, element-wise.numpy.arccos(x, *args, **kwargs) Trigonometric inverse cosine, element-wise.numpy.arctan(x, *args, **kwargs) Trigonometric inverse tangent, element-wise.通用函數（universal function）通常叫作ufunc，它對陣列中的各個元素逐一進行操作。這表明，通用函數分別處理輸入陣列的每個元素，生成的結果組成一個新的輸出陣列。輸出陣列的大小跟輸入陣列相同。
三角函數等很多數學運運算元合通用函數的定義，例如，計算平方根的sqrt()函數、用來取對數的log()函數和求正弦值的sin()函數。
import numpy as np
x = np.linspace(start=0, stop=np.pi / 2, num=10)
print(x)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
y = np.sin(x)
print(y)
# [0. 0.17364818 0.34202014 0.5 0.64278761 0.76604444
# 0.8660254 0.93969262 0.98480775 1. ]
z = np.arcsin(y)
print(z)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
y = np.cos(x)
print(y)
# [1.00000000e+00 9.84807753e-01 9.39692621e-01 8.66025404e-01
# 7.66044443e-01 6.42787610e-01 5.00000000e-01 3.42020143e-01
# 1.73648178e-01 6.12323400e-17]
z = np.arccos(y)
print(z)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
y = np.tan(x)
print(y)
# [0.00000000e+00 1.76326981e-01 3.63970234e-01 5.77350269e-01
# 8.39099631e-01 1.19175359e+00 1.73205081e+00 2.74747742e+00
# 5.67128182e+00 1.63312394e+16]
z = np.arctan(y)
print(z)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]numpy.all(a, axis=None, out=None, keepdims=np._NoValue) Test whether all array elements along a given axis evaluate to True.numpy.any(a, axis=None, out=None, keepdims=np._NoValue) Test whether any array element along a given axis evaluates to True.import numpy as np
a = np.array([0, 4, 5])
b = np.copy(a)
print(np.all(a == b)) # True
print(np.any(a == b)) # True
b[0] = 1
print(np.all(a == b)) # False
print(np.any(a == b)) # True
print(np.all([1.0, np.nan])) # True
print(np.any([1.0, np.nan])) # True
a = np.eye(3)
print(np.all(a, axis=0)) # [False False False]
print(np.any(a, axis=0)) # [ True True True]
numpy.isnan(x, *args, **kwargs) Test element-wise for NaN and return result as a boolean array.a=np.array([1,2,np.nan])
print(np.isnan(a))
#[False False True]numpy.logical_not(x, *args, **kwargs)Compute the truth value of NOT x element-wise.numpy.logical_and(x1, x2, *args, **kwargs) Compute the truth value of x1 AND x2 element-wise.numpy.logical_or(x1, x2, *args, **kwargs)Compute the truth value of x1 OR x2 element-wise.numpy.logical_xor(x1, x2, *args, **kwargs)Compute the truth value of x1 XOR x2, element-wise.計算非x元素的真值。
import numpy as np
print(np.logical_not(3))
# False
print(np.logical_not([True, False, 0, 1]))
# [False True True False]
x = np.arange(5)
print(np.logical_not(x < 3))
# [False False False True True]計算x1 AND x2元素的真值。
print(np.logical_and(True, False))
# False
print(np.logical_and([True, False], [True, False]))
# [ True False]
print(np.logical_and(x > 1, x < 4))
# [False False True True False]逐元素計算x1 OR x2的真值。
print(np.logical_or(True, False))
# True
print(np.logical_or([True, False], [False, False]))
# [ True False]
print(np.logical_or(x < 1, x > 3))
# [ True False False False True]計算x1 XOR x2的真值，按元素計算。
print(np.logical_xor(True, False))
# True
print(np.logical_xor([True, True, False, False], [True, False, True, False]))
# [False True True False]
print(np.logical_xor(x < 1, x > 3))
# [ True False False False True]
print(np.logical_xor(0, np.eye(2)))
# [[ True False]
# [False True]]numpy.greater(x1, x2, *args, **kwargs) Return the truth value of (x1 > x2) element-wise.numpy.greater_equal(x1, x2, *args, **kwargs) Return the truth value of (x1 >= x2) element-wise.numpy.equal(x1, x2, *args, **kwargs) Return (x1 == x2) element-wise.numpy.not_equal(x1, x2, *args, **kwargs) Return (x1 != x2) element-wise.numpy.less(x1, x2, *args, **kwargs) Return the truth value of (x1 < x2) element-wise.numpy.less_equal(x1, x2, *args, **kwargs) Return the truth value of (x1 =< x2) element-wise.import numpy as np
x = np.array([1, 2, 3, 4, 5, 6, 7, 8])
y = x > 2
print(y)
print(np.greater(x, 2))
# [False False True True True True True True]
y = x >= 2
print(y)
print(np.greater_equal(x, 2))
# [False True True True True True True True]
y = x == 2
print(y)
print(np.equal(x, 2))
# [False True False False False False False False]
y = x != 2
print(y)
print(np.not_equal(x, 2))
# [ True False True True True True True True]
y = x < 2
print(y)
print(np.less(x, 2))
# [ True False False False False False False False]
y = x <= 2
print(y)
print(np.less_equal(x, 2))
# [ True True False False False False False False]numpy.isclose(a, b, rtol=1.e-5, atol=1.e-8, equal_nan=False) Returns a boolean array where two arrays are element-wise equal within a tolerance.numpy.allclose(a, b, rtol=1.e-5, atol=1.e-8, equal_nan=False) Returns True if two arrays are element-wise equal within a tolerance.numpy.allclose() 等價於 numpy.all(isclose(a, b, rtol=rtol, atol=atol, equal_nan=equal_nan))。
The tolerance values are positive, typically very small numbers. The relative difference (rtol * abs(b)) and the absolute difference atol are added together to compare against the absolute difference between a and b.
判斷是否為True的計算依據：
np.absolute(a - b) <= (atol + rtol * absolute(b))
- atol：float，絕對公差。
- rtol：float，相對公差。np.absolute(a - b) <= (atol + rtol * absolute(b)) - atol：float，絕對公差。 - rtol：float，相對公差。
NaNs are treated as equal if they are in the same place and if equal_nan=True. Infs are treated as equal if they are in the same place and of the same sign in both arrays.
比較兩個陣列是否可以認為相等。
import numpy as np
x = np.isclose([1e10, 1e-7], [1.00001e10, 1e-8])
print(x) # [ True False]
x = np.allclose([1e10, 1e-7], [1.00001e10, 1e-8])
print(x) # False
x = np.isclose([1e10, 1e-8], [1.00001e10, 1e-9])
print(x) # [ True True]
x = np.allclose([1e10, 1e-8], [1.00001e10, 1e-9])
print(x) # True
x = np.isclose([1e10, 1e-8], [1.0001e10, 1e-9])
print(x) # [False True]
x = np.allclose([1e10, 1e-8], [1.0001e10, 1e-9])
print(x) # False
x = np.isclose([1.0, np.nan], [1.0, np.nan])
print(x) # [ True False]
x = np.allclose([1.0, np.nan], [1.0, np.nan])
print(x) # False
x = np.isclose([1.0, np.nan], [1.0, np.nan], equal_nan=True)
print(x) # [ True True]
x = np.allclose([1.0, np.nan], [1.0, np.nan], equal_nan=True)
print(x) # Truenumpy.exp(x, *args, **kwargs) Calculate the exponential of all elements in the input array.numpy.log(x, *args, **kwargs) Natural logarithm, element-wise.numpy.exp2(x, *args, **kwargs) Calculate 2**p for all p in the input array.numpy.log2(x, *args, **kwargs) Base-2 logarithm of x.numpy.log10(x, *args, **kwargs) Return the base 10 logarithm of the input array, element-wise.The natural logarithm log is the inverse of the exponential function, so that log(exp(x)) = x. The natural logarithm is logarithm in base e.
import numpy as np
x = np.arange(1, 5)
print(x)
# [1 2 3 4]
y = np.exp(x)
print(y)
# [ 2.71828183 7.3890561 20.08553692 54.59815003]
z = np.log(y)
print(z)
# [1. 2. 3. 4.]