快速入门教程¶

Prerequisites¶

在阅读本教程之前，您应该了解一些Python。如果您想刷新内存，请查看Python教程。

如果您希望使用本教程中的示例，则还必须在计算机上安装一些软件。有关说明，请参阅 https://scipy.org/install.html。

基础¶

NumPy的主要对象是齐次多维数组。它是由非负整数元组索引的所有类型相同的元素（通常为数字）表。在NumPy中，尺寸称为轴。

例如，3D空间中的点的坐标[1, 2, 1]只有一个轴。该轴上有3个元素，所以我们说它的长度为3。在下图所示的示例中，数组有2个轴。第一轴的长度为2，第二轴的长度为3。

[[ 1., 0., 0.],
[ 0., 1., 2.]]

NumPy的数组类称为ndarray。也被称为别名 array。请注意，numpy.array这与标准Python库类不同array.array，后者仅处理一维数组且功能较少。ndarray对象的更重要属性是：

ndarray.ndim

数组的轴数（尺寸）。

ndarray.shape

数组的尺寸。这是一个整数元组，指示每个维度中数组的大小。对于具有n行和m列的矩阵，shape将为(n,m)。shape因此，元组的长度 为轴数ndim。

ndarray.size

数组元素的总数。这等于的元素的乘积shape。

ndarray.dtype

一个对象，描述数组中元素的类型。可以使用标准Python类型创建或指定dtype。另外，NumPy提供了自己的类型。numpy.int32，numpy.int16和numpy.float64是一些示例。

ndarray.itemsize

数组中每个元素的大小（以字节为单位）。例如，类型为元素的数组float64具有itemsize8（= 64/8），而类型complex32中的一个元素具有itemsize4（= 32/8）。等同于ndarray.dtype.itemsize。

ndarray.data

包含数组实际元素的缓冲区。通常，我们不需要使用此属性，因为我们将使用索引工具访问数组中的元素。

一个例子¶

import numpy as np
a = np.arange(15).reshape(3, 5)
a
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])

a.shape
(3, 5)

a.ndim
2

a.dtype.name
‘int64’

a.itemsize
8

a.size
15

type(a)

b = np.array([6, 7, 8])
b
array([6, 7, 8])

type(b)

阵列创建¶

有几种创建数组的方法。

例如，您可以使用array函数从常规Python列表或元组创建数组。根据序列中元素的类型推导所得数组的类型。

import numpy as np
a = np.array([2,3,4])
a
array([2, 3, 4])

a.dtype
dtype(‘int64’)

b = np.array([1.2, 3.5, 5.1])
b.dtype
dtype(‘float64’)

A frequent error consists in calling array with multiple arguments, rather than providing a single sequence as an argument.

a = np.array(1,2,3,4) # WRONG
Traceback (most recent call last):
…
ValueError: only 2 non-keyword arguments accepted

a = np.array([1,2,3,4]) # RIGHT

array transforms sequences of sequences into two-dimensional arrays, sequences of sequences of sequences into three-dimensional arrays, and so on.

b = np.array([(1.5,2,3), (4,5,6)])
b
array([[1.5, 2. , 3. ],
[4. , 5. , 6. ]])

The type of the array can also be explicitly specified at creation time:

c = np.array( [ [1,2], [3,4] ], dtype=complex )
c
array([[1.+0.j, 2.+0.j],
[3.+0.j, 4.+0.j]])

Often, the elements of an array are originally unknown, but its size is known. Hence, NumPy offers several functions to create arrays with initial placeholder content. These minimize the necessity of growing arrays, an expensive operation.

The function zeros creates an array full of zeros, the function ones creates an array full of ones, and the function empty creates an array whose initial content is random and depends on the state of the memory. By default, the dtype of the created array is float64.

np.zeros((3, 4))
array([[0., 0., 0., 0.],
[0., 0., 0., 0.],
[0., 0., 0., 0.]])

np.ones( (2,3,4), dtype=np.int16 ) # dtype can also be specified
array([[[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1]],

[[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1]]], dtype=int16)

np.empty( (2,3) ) # uninitialized
array([[ 3.73603959e-262, 6.02658058e-154, 6.55490914e-260], # may vary
[ 5.30498948e-313, 3.14673309e-307, 1.00000000e+000]])

To create sequences of numbers, NumPy provides the arange function which is analogous to the Python built-in range, but returns an array.

np.arange( 10, 30, 5 )
array([10, 15, 20, 25])

np.arange( 0, 2, 0.3 ) # it accepts float arguments
array([0. , 0.3, 0.6, 0.9, 1.2, 1.5, 1.8])

When arange is used with floating point arguments, it is generally not possible to predict the number of elements obtained, due to the finite floating point precision. For this reason, it is usually better to use the function linspace that receives as an argument the number of elements that we want, instead of the step:

from numpy import pi
np.linspace( 0, 2, 9 ) # 9 numbers from 0 to 2
array([0. , 0.25, 0.5 , 0.75, 1. , 1.25, 1.5 , 1.75, 2. ])

x = np.linspace( 0, 2*pi, 100 ) # useful to evaluate function at lots of points
f = np.sin(x)

See also

array, zeros, zeros_like, ones, ones_like, empty, empty_like, arange, linspace, numpy.random.Generator.rand, numpy.random.Generator.randn, fromfunction, fromfile

Printing Arrays¶

When you print an array, NumPy displays it in a similar way to nested lists, but with the following layout:

• the last axis is printed from left to right,
• the second-to-last is printed from top to bottom,
• the rest are also printed from top to bottom, with each slice separated from the next by an empty line.

One-dimensional arrays are then printed as rows, bidimensionals as matrices and tridimensionals as lists of matrices.

a = np.arange(6) # 1d array
print(a)
[0 1 2 3 4 5]

b = np.arange(12).reshape(4,3) # 2d array
print(b)
[[ 0 1 2]
[ 3 4 5]
[ 6 7 8]
[ 9 10 11]]

c = np.arange(24).reshape(2,3,4) # 3d array
print(c)
[[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]]

[[12 13 14 15]
[16 17 18 19]
[20 21 22 23]]]

See below to get more details on reshape.

If an array is too large to be printed, NumPy automatically skips the central part of the array and only prints the corners:

print(np.arange(10000))
[ 0 1 2 … 9997 9998 9999]

print(np.arange(10000).reshape(100,100))
[[ 0 1 2 … 97 98 99]
[ 100 101 102 … 197 198 199]
[ 200 201 202 … 297 298 299]
…
[9700 9701 9702 … 9797 9798 9799]
[9800 9801 9802 … 9897 9898 9899]
[9900 9901 9902 … 9997 9998 9999]]

To disable this behaviour and force NumPy to print the entire array, you can change the printing options using set_printoptions.

np.set_printoptions(threshold=sys.maxsize) # sys module should be imported

Basic Operations¶

Arithmetic operators on arrays apply elementwise. A new array is created and filled with the result.

a = np.array( [20,30,40,50] )
b = np.arange( 4 )
b
array([0, 1, 2, 3])

c = a-b
c
array([20, 29, 38, 47])

b**2
array([0, 1, 4, 9])

10*np.sin(a)
array([ 9.12945251, -9.88031624, 7.4511316 , -2.62374854])

a<35
array([ True, True, False, False])

Unlike in many matrix languages, the product operator * operates elementwise in NumPy arrays. The matrix product can be performed using the @ operator (in python >=3.5) or the dot function or method:

A = np.array( [[1,1],
… [0,1]] )

B = np.array( [[2,0],
… [3,4]] )

A * B # elementwise product
array([[2, 0],
[0, 4]])

A @ B # matrix product
array([[5, 4],
[3, 4]])

A.dot(B) # another matrix product
array([[5, 4],
[3, 4]])

Some operations, such as += and *=, act in place to modify an existing array rather than create a new one.

a = np.ones((2,3), dtype=int)
b = rg.random((2,3))
a *= 3
a
array([[3, 3, 3],
[3, 3, 3]])

b += a
b
array([[3.51182162, 3.9504637 , 3.14415961],
[3.94864945, 3.31183145, 3.42332645]])

a += b # b is not automatically converted to integer type
Traceback (most recent call last):
…
numpy.core._exceptions.UFuncTypeError: Cannot cast ufunc ‘add’ output from dtype(‘float64’) to dtype(‘int64’) with casting rule ‘same_kind’

When operating with arrays of different types, the type of the resulting array corresponds to the more general or precise one (a behavior known as upcasting).

a = np.ones(3, dtype=np.int32)
b = np.linspace(0,pi,3)
b.dtype.name
‘float64’

c = a+b
c
array([1. , 2.57079633, 4.14159265])

c.dtype.name
‘float64’

d = np.exp(c*1j)
d
array([ 0.54030231+0.84147098j, -0.84147098+0.54030231j,
-0.54030231-0.84147098j])

d.dtype.name
‘complex128’

Many unary operations, such as computing the sum of all the elements in the array, are implemented as methods of the ndarray class.

a = rg.random((2,3))
a
array([[0.82770259, 0.40919914, 0.54959369],
[0.02755911, 0.75351311, 0.53814331]])

a.sum()
3.1057109529998157

a.min()
0.027559113243068367

a.max()
0.8277025938204418

By default, these operations apply to the array as though it were a list of numbers, regardless of its shape. However, by specifying the axis parameter you can apply an operation along the specified axis of an array:

b = np.arange(12).reshape(3,4)
b
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])

b.sum(axis=0) # sum of each column
array([12, 15, 18, 21])

b.min(axis=1) # min of each row
array([0, 4, 8])

b.cumsum(axis=1) # cumulative sum along each row
array([[ 0, 1, 3, 6],
[ 4, 9, 15, 22],
[ 8, 17, 27, 38]])

Universal Functions¶

NumPy provides familiar mathematical functions such as sin, cos, and exp. In NumPy, these are called “universal functions”(ufunc). Within NumPy, these functions operate elementwise on an array, producing an array as output.

B = np.arange(3)
B
array([0, 1, 2])

np.exp(B)
array([1. , 2.71828183, 7.3890561 ])

np.sqrt(B)
array([0. , 1. , 1.41421356])

C = np.array([2., -1., 4.])
np.add(B, C)
array([2., 0., 6.])

See also

all, any, apply_along_axis, argmax, argmin, argsort, average, bincount, ceil, clip, conj, corrcoef, cov, cross, cumprod, cumsum, diff, dot, floor, inner, invert, lexsort, max, maximum, mean, median, min, minimum, nonzero, outer, prod, re, round, sort, std, sum, trace, transpose, var, vdot, vectorize, where

Indexing, Slicing and Iterating¶

One-dimensional arrays can be indexed, sliced and iterated over, much like lists and other Python sequences.

a = np.arange(10)**3
a
array([ 0, 1, 8, 27, 64, 125, 216, 343, 512, 729])

a[2]
8

a[2:5]
array([ 8, 27, 64])

equivalent to a[0:6:2] = 1000;

from start to position 6, exclusive, set every 2nd element to 1000

a[:6:2] = 1000
a
array([1000, 1, 1000, 27, 1000, 125, 216, 343, 512, 729])

a[ : :-1] # reversed a
array([ 729, 512, 343, 216, 125, 1000, 27, 1000, 1, 1000])

for i in a:
… print(i**(1/3.))
…
9.999999999999998
1.0
9.999999999999998
3.0
9.999999999999998
4.999999999999999
5.999999999999999
6.999999999999999
7.999999999999999
8.999999999999998

Multidimensional arrays can have one index per axis. These indices are given in a tuple separated by commas:

def f(x,y):
… return 10*x+y
…

b = np.fromfunction(f,(5,4),dtype=int)
b
array([[ 0, 1, 2, 3],
[10, 11, 12, 13],
[20, 21, 22, 23],
[30, 31, 32, 33],
[40, 41, 42, 43]])

b[2,3]
23

b[0:5, 1] # each row in the second column of b
array([ 1, 11, 21, 31, 41])

b[ : ,1] # equivalent to the previous example
array([ 1, 11, 21, 31, 41])

b[1:3, : ] # each column in the second and third row of b
array([[10, 11, 12, 13],
[20, 21, 22, 23]])

When fewer indices are provided than the number of axes, the missing indices are considered complete slices:

b[-1] # the last row. Equivalent to b[-1,:]
array([40, 41, 42, 43])

The expression within brackets in b[i] is treated as an i followed by as many instances of : as needed to represent the remaining axes. NumPy also allows you to write this using dots as b[i,...].

The dots (...) represent as many colons as needed to produce a complete indexing tuple. For example, if x is an array with 5 axes, then

• x[1,2,...] is equivalent to x[1,2,:,:,:],
• x[...,3] to x[:,:,:,:,3] and
• x[4,...,5,:] to x[4,:,:,5,:].

c = np.array( [[[ 0, 1, 2], # a 3D array (two stacked 2D arrays)
… [ 10, 12, 13]],
… [[100,101,102],
… [110,112,113]]])

c.shape
(2, 2, 3)

c[1,…] # same as c[1,:,:] or c[1]
array([[100, 101, 102],
[110, 112, 113]])

c[…,2] # same as c[:,:,2]
array([[ 2, 13],
[102, 113]])

Iterating over multidimensional arrays is done with respect to the first axis:

for row in b:
… print(row)
…
[0 1 2 3]
[10 11 12 13]
[20 21 22 23]
[30 31 32 33]
[40 41 42 43]

However, if one wants to perform an operation on each element in the array, one can use the flat attribute which is an iterator over all the elements of the array:

for element in b.flat:
… print(element)
…
0
1
2
3
10
11
12
13
20
21
22
23
30
31
32
33
40
41
42
43

See also

Indexing, Indexing (reference), newaxis, ndenumerate, indices

Shape Manipulation¶

Changing the shape of an array¶

An array has a shape given by the number of elements along each axis:

a = np.floor(10*rg.random((3,4)))
a
array([[3., 7., 3., 4.],
[1., 4., 2., 2.],
[7., 2., 4., 9.]])

a.shape
(3, 4)

The shape of an array can be changed with various commands. Note that the following three commands all return a modified array, but do not change the original array:

a.ravel() # returns the array, flattened
array([3., 7., 3., 4., 1., 4., 2., 2., 7., 2., 4., 9.])

a.reshape(6,2) # returns the array with a modified shape
array([[3., 7.],
[3., 4.],
[1., 4.],
[2., 2.],
[7., 2.],
[4., 9.]])

a.T # returns the array, transposed
array([[3., 1., 7.],
[7., 4., 2.],
[3., 2., 4.],
[4., 2., 9.]])

a.T.shape
(4, 3)

a.shape
(3, 4)

The order of the elements in the array resulting from ravel() is normally “C-style”, that is, the rightmost index “changes the fastest”, so the element after a[0,0] is a[0,1]. If the array is reshaped to some other shape, again the array is treated as “C-style”. NumPy normally creates arrays stored in this order, so ravel() will usually not need to copy its argument, but if the array was made by taking slices of another array or created with unusual options, it may need to be copied. The functions ravel() and reshape() can also be instructed, using an optional argument, to use FORTRAN-style arrays, in which the leftmost index changes the fastest.

The reshape function returns its argument with a modified shape, whereas the ndarray.resize method modifies the array itself:

a
array([[3., 7., 3., 4.],
[1., 4., 2., 2.],
[7., 2., 4., 9.]])

a.resize((2,6))
a
array([[3., 7., 3., 4., 1., 4.],
[2., 2., 7., 2., 4., 9.]])

If a dimension is given as -1 in a reshaping operation, the other dimensions are automatically calculated:

a.reshape(3,-1)
array([[3., 7., 3., 4.],
[1., 4., 2., 2.],
[7., 2., 4., 9.]])

See also

ndarray.shape, reshape, resize, ravel

Stacking together different arrays¶

Several arrays can be stacked together along different axes:

a = np.floor(10*rg.random((2,2)))
a
array([[9., 7.],
[5., 2.]])

b = np.floor(10*rg.random((2,2)))
b
array([[1., 9.],
[5., 1.]])

np.vstack((a,b))
array([[9., 7.],
[5., 2.],
[1., 9.],
[5., 1.]])

np.hstack((a,b))
array([[9., 7., 1., 9.],
[5., 2., 5., 1.]])

The function column_stack stacks 1D arrays as columns into a 2D array. It is equivalent to hstack only for 2D arrays:

from numpy import newaxis
np.column_stack((a,b)) # with 2D arrays
array([[9., 7., 1., 9.],
[5., 2., 5., 1.]])

a = np.array([4.,2.])
b = np.array([3.,8.])
np.column_stack((a,b)) # returns a 2D array
array([[4., 3.],
[2., 8.]])

np.hstack((a,b)) # the result is different
array([4., 2., 3., 8.])

a[:,newaxis] # this allows to have a 2D columns vector
array([[4.],
[2.]])

np.column_stack((a[:,newaxis],b[:,newaxis]))
array([[4., 3.],
[2., 8.]])

np.hstack((a[:,newaxis],b[:,newaxis])) # the result is the same
array([[4., 3.],
[2., 8.]])

On the other hand, the function row_stack is equivalent to vstack for any input arrays. In fact, row_stack is an alias for vstack:

np.column_stack is np.hstack
False

np.row_stack is np.vstack
True

In general, for arrays with more than two dimensions, hstack stacks along their second axes, vstack stacks along their first axes, and concatenate allows for an optional arguments giving the number of the axis along which the concatenation should happen.

Note

In complex cases, r_ and c_ are useful for creating arrays by stacking numbers along one axis. They allow the use of range literals (“:”)

np.r_[1:4,0,4]
array([1, 2, 3, 0, 4])

When used with arrays as arguments, r_ and c_ are similar to vstack and hstack in their default behavior, but allow for an optional argument giving the number of the axis along which to concatenate.

See also

hstack, vstack, column_stack, concatenate, c_, r_

Splitting one array into several smaller ones¶

Using hsplit, you can split an array along its horizontal axis, either by specifying the number of equally shaped arrays to return, or by specifying the columns after which the division should occur:

a = np.floor(10*rg.random((2,12)))
a
array([[6., 7., 6., 9., 0., 5., 4., 0., 6., 8., 5., 2.],
[8., 5., 5., 7., 1., 8., 6., 7., 1., 8., 1., 0.]])

Split a into 3

np.hsplit(a,3)
[array([[6., 7., 6., 9.],
[8., 5., 5., 7.]]), array([[0., 5., 4., 0.],
[1., 8., 6., 7.]]), array([[6., 8., 5., 2.],
[1., 8., 1., 0.]])]

Split a after the third and the fourth column

np.hsplit(a,(3,4))
[array([[6., 7., 6.],
[8., 5., 5.]]), array([[9.],
[7.]]), array([[0., 5., 4., 0., 6., 8., 5., 2.],
[1., 8., 6., 7., 1., 8., 1., 0.]])]

vsplit splits along the vertical axis, and array_split allows one to specify along which axis to split.

Copies and Views¶

When operating and manipulating arrays, their data is sometimes copied into a new array and sometimes not. This is often a source of confusion for beginners. There are three cases:

No Copy at All¶

Simple assignments make no copy of objects or their data.

a = np.array([[ 0, 1, 2, 3],
… [ 4, 5, 6, 7],
… [ 8, 9, 10, 11]])

b = a # no new object is created
b is a # a and b are two names for the same ndarray object
True

Python passes mutable objects as references, so function calls make no copy.

def f(x):
… print(id(x))
…

id(a) # id is a unique identifier of an object
148293216 # may vary

f(a)
148293216 # may vary

View or Shallow Copy¶

Different array objects can share the same data. The view method creates a new array object that looks at the same data.

c = a.view()
c is a
False

c.base is a # c is a view of the data owned by a
True

c.flags.owndata
False

c = c.reshape((2, 6)) # a’s shape doesn’t change
a.shape
(3, 4)

c[0, 4] = 1234 # a’s data changes
a
array([[ 0, 1, 2, 3],
[1234, 5, 6, 7],
[ 8, 9, 10, 11]])

Slicing an array returns a view of it:

s = a[ : , 1:3] # spaces added for clarity; could also be written “s = a[:, 1:3]”
s[:] = 10 # s[:] is a view of s. Note the difference between s = 10 and s[:] = 10
a
array([[ 0, 10, 10, 3],
[1234, 10, 10, 7],
[ 8, 10, 10, 11]])

Deep Copy¶

The copy method makes a complete copy of the array and its data.

d = a.copy() # a new array object with new data is created
d is a
False

d.base is a # d doesn’t share anything with a
False

d[0,0] = 9999
a
array([[ 0, 10, 10, 3],
[1234, 10, 10, 7],
[ 8, 10, 10, 11]])

Sometimes copy should be called after slicing if the original array is not required anymore. For example, suppose a is a huge intermediate result and the final result b only contains a small fraction of a, a deep copy should be made when constructing b with slicing:

a = np.arange(int(1e8))
b = a[:100].copy()
del a # the memory of a can be released.

If b = a[:100] is used instead, a is referenced by b and will persist in memory even if del a is executed.

Functions and Methods Overview¶

Here is a list of some useful NumPy functions and methods names ordered in categories. See Routines for the full list.

Array Creation

arange, array, copy, empty, empty_like, eye, fromfile, fromfunction, identity, linspace, logspace, mgrid, ogrid, ones, ones_like, r_, zeros, zeros_like

Conversions

ndarray.astype, atleast_1d, atleast_2d, atleast_3d, mat

Manipulations

array_split, column_stack, concatenate, diagonal, dsplit, dstack, hsplit, hstack, ndarray.item, newaxis, ravel, repeat, reshape, resize, squeeze, swapaxes, take, transpose, vsplit, vstack

Questions

all, any, nonzero, where

Ordering

argmax, argmin, argsort, max, min, ptp, searchsorted, sort

Operations

choose, compress, cumprod, cumsum, inner, ndarray.fill, imag, prod, put, putmask, real, sum

Basic Statistics

cov, mean, std, var

Basic Linear Algebra

cross, dot, outer, linalg.svd, vdot

Less Basic¶

Broadcasting rules¶

Broadcasting allows universal functions to deal in a meaningful way with inputs that do not have exactly the same shape.

The first rule of broadcasting is that if all input arrays do not have the same number of dimensions, a “1” will be repeatedly prepended to the shapes of the smaller arrays until all the arrays have the same number of dimensions.

The second rule of broadcasting ensures that arrays with a size of 1 along a particular dimension act as if they had the size of the array with the largest shape along that dimension. The value of the array element is assumed to be the same along that dimension for the “broadcast” array.

After application of the broadcasting rules, the sizes of all arrays must match. More details can be found in Broadcasting.

Advanced indexing and index tricks¶

NumPy offers more indexing facilities than regular Python sequences. In addition to indexing by integers and slices, as we saw before, arrays can be indexed by arrays of integers and arrays of booleans.

Indexing with Arrays of Indices¶

a = np.arange(12)**2 # the first 12 square numbers
i = np.array([1, 1, 3, 8, 5]) # an array of indices
a[i] # the elements of a at the positions i
array([ 1, 1, 9, 64, 25])

j = np.array([[3, 4], [9, 7]]) # a bidimensional array of indices
a[j] # the same shape as j
array([[ 9, 16],
[81, 49]])

When the indexed array a is multidimensional, a single array of indices refers to the first dimension of a. The following example shows this behavior by converting an image of labels into a color image using a palette.

palette = np.array([[0, 0, 0], # black
… [255, 0, 0], # red
… [0, 255, 0], # green
… [0, 0, 255], # blue
… [255, 255, 255]]) # white

image = np.array([[0, 1, 2, 0], # each value corresponds to a color in the palette
… [0, 3, 4, 0]])

palette[image] # the (2, 4, 3) color image
array([[[ 0, 0, 0],
[255, 0, 0],
[ 0, 255, 0],
[ 0, 0, 0]],

[[ 0, 0, 0],
[ 0, 0, 255],
[255, 255, 255],
[ 0, 0, 0]]])

We can also give indexes for more than one dimension. The arrays of indices for each dimension must have the same shape.

a = np.arange(12).reshape(3,4)
a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])

i = np.array([[0, 1], # indices for the first dim of a
… [1, 2]])

j = np.array([[2, 1], # indices for the second dim
… [3, 3]])

a[i, j] # i and j must have equal shape
array([[ 2, 5],
[ 7, 11]])

a[i, 2]
array([[ 2, 6],
[ 6, 10]])

a[:, j] # i.e., a[ : , j]
array([[[ 2, 1],
[ 3, 3]],

[[ 6, 5],
[ 7, 7]],

[[10, 9],
[11, 11]]])

In Python, arr[i, j] is exactly the same as arr[(i, j)]—so we can put i and j in a tuple and then do the indexing with that.

l = (i, j)

equivalent to a[i, j]

a[l]
array([[ 2, 5],
[ 7, 11]])

However, we can not do this by putting i and j into an array, because this array will be interpreted as indexing the first dimension of a.

s = np.array([i, j])

not what we want

a[s]
Traceback (most recent call last): File “”, line 1, in
IndexError: index 3 is out of bounds for axis 0 with size 3

same as a[i, j]

a[tuple(s)]
array([[ 2, 5],
[ 7, 11]])

Another common use of indexing with arrays is the search of the maximum value of time-dependent series:

time = np.linspace(20, 145, 5) # time scale
data = np.sin(np.arange(20)).reshape(5,4) # 4 time-dependent series
time
array([ 20. , 51.25, 82.5 , 113.75, 145. ])

data
array([[ 0. , 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],
[-0.28790332, -0.96139749, -0.75098725, 0.14987721]])

index of the maxima for each series

ind = data.argmax(axis=0)
ind
array([2, 0, 3, 1])

times corresponding to the maxima

time_max = time[ind]

data_max = data[ind, range(data.shape[1])] # => data[ind[0],0], data[ind[1],1]…

time_max
array([ 82.5 , 20. , 113.75, 51.25])

data_max
array([0.98935825, 0.84147098, 0.99060736, 0.6569866 ])

np.all(data_max == data.max(axis=0))
True

You can also use indexing with arrays as a target to assign to:

a = np.arange(5)
a
array([0, 1, 2, 3, 4])

a[[1,3,4]] = 0
a
array([0, 0, 2, 0, 0])

However, when the list of indices contains repetitions, the assignment is done several times, leaving behind the last value:

a = np.arange(5)
a[[0,0,2]]=[1,2,3]
a
array([2, 1, 3, 3, 4])

This is reasonable enough, but watch out if you want to use Python’s += construct, as it may not do what you expect:

a = np.arange(5)
a[[0,0,2]]+=1
a
array([1, 1, 3, 3, 4])

Even though 0 occurs twice in the list of indices, the 0th element is only incremented once. This is because Python requires “a+=1” to be equivalent to “a = a + 1”.

Indexing with Boolean Arrays¶

When we index arrays with arrays of (integer) indices we are providing the list of indices to pick. With boolean indices the approach is different; we explicitly choose which items in the array we want and which ones we don’t.

The most natural way one can think of for boolean indexing is to use boolean arrays that have the same shape as the original array:

a = np.arange(12).reshape(3,4)
b = a > 4
b # b is a boolean with a’s shape
array([[False, False, False, False],
[False, True, True, True],
[ True, True, True, True]])

a[b] # 1d array with the selected elements
array([ 5, 6, 7, 8, 9, 10, 11])

This property can be very useful in assignments:

a[b] = 0 # All elements of ‘a’ higher than 4 become 0
a
array([[0, 1, 2, 3],
[4, 0, 0, 0],
[0, 0, 0, 0]])

You can look at the following example to see how to use boolean indexing to generate an image of the Mandelbrot set:

import numpy as np
import matplotlib.pyplot as plt
def mandelbrot( h,w, maxit=20 ):
… “””Returns an image of the Mandelbrot fractal of size (h,w).”””
… y,x = np.ogrid[ -1.4:1.4:h_1j, -2:0.8:w_1j ]
… c = x+y1j
… z = c
… divtime = maxit + np.zeros(z.shape, dtype=int)
…
… for i in range(maxit):
… z = z**2 + c
… diverge = znp.conj(z) > 2**2 # who is diverging
… div_now = diverge & (divtime==maxit) # who is diverging now
… divtime[div_now] = i # note when
… z[diverge] = 2 # avoid diverging too much
…
… return divtime

plt.imshow(mandelbrot(400,400))

The second way of indexing with booleans is more similar to integer indexing; for each dimension of the array we give a 1D boolean array selecting the slices we want:

a = np.arange(12).reshape(3,4)
b1 = np.array([False,True,True]) # first dim selection
b2 = np.array([True,False,True,False]) # second dim selection

a[b1,:] # selecting rows
array([[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])

a[b1] # same thing
array([[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])

a[:,b2] # selecting columns
array([[ 0, 2],
[ 4, 6],
[ 8, 10]])

a[b1,b2] # a weird thing to do
array([ 4, 10])

Note that the length of the 1D boolean array must coincide with the length of the dimension (or axis) you want to slice. In the previous example, b1 has length 3 (the number of rows in a), and b2 (of length 4) is suitable to index the 2nd axis (columns) of a.

The ix_() function¶

The ix_ function can be used to combine different vectors so as to obtain the result for each n-uplet. For example, if you want to compute all the a+b*c for all the triplets taken from each of the vectors a, b and c:

a = np.array([2,3,4,5])
b = np.array([8,5,4])
c = np.array([5,4,6,8,3])
ax,bx,cx = np.ix_(a,b,c)
ax
array([[[2]],

[[3]],

[[4]],

[[5]]])

bx
array([[[8],
[5],
[4]]])

cx
array([[[5, 4, 6, 8, 3]]])

ax.shape, bx.shape, cx.shape
((4, 1, 1), (1, 3, 1), (1, 1, 5))

result = ax+bx*cx
result
array([[[42, 34, 50, 66, 26],
[27, 22, 32, 42, 17],
[22, 18, 26, 34, 14]],

[[43, 35, 51, 67, 27],
[28, 23, 33, 43, 18],
[23, 19, 27, 35, 15]],

[[44, 36, 52, 68, 28],
[29, 24, 34, 44, 19],
[24, 20, 28, 36, 16]],

[[45, 37, 53, 69, 29],
[30, 25, 35, 45, 20],
[25, 21, 29, 37, 17]]])

result[3,2,4]
17

a[3]+b[2]*c[4]
17

You could also implement the reduce as follows:

def ufuncreduce(ufct, _vectors):
… vs = np.ix(_vectors)
… r = ufct.identity
… for v in vs:
… r = ufct(r,v)
… return r

and then use it as:

ufunc_reduce(np.add,a,b,c)
array([[[15, 14, 16, 18, 13],
[12, 11, 13, 15, 10],
[11, 10, 12, 14, 9]],

[[16, 15, 17, 19, 14],
[13, 12, 14, 16, 11],
[12, 11, 13, 15, 10]],

[[17, 16, 18, 20, 15],
[14, 13, 15, 17, 12],
[13, 12, 14, 16, 11]],

[[18, 17, 19, 21, 16],
[15, 14, 16, 18, 13],
[14, 13, 15, 17, 12]]])

The advantage of this version of reduce compared to the normal ufunc.reduce is that it makes use of the Broadcasting Rules in order to avoid creating an argument array the size of the output times the number of vectors.

Indexing with strings¶

See Structured arrays.

Linear Algebra¶

Work in progress. Basic linear algebra to be included here.

Simple Array Operations¶

See linalg.py in numpy folder for more.

import numpy as np
a = np.array([[1.0, 2.0], [3.0, 4.0]])
print(a)
[[1. 2.]
[3. 4.]]

a.transpose()
array([[1., 3.],
[2., 4.]])

np.linalg.inv(a)
array([[-2. , 1. ],
[ 1.5, -0.5]])

u = np.eye(2) # unit 2x2 matrix; “eye” represents “I”
u
array([[1., 0.],
[0., 1.]])

j = np.array([[0.0, -1.0], [1.0, 0.0]])

j @ j # matrix product
array([[-1., 0.],
[ 0., -1.]])

np.trace(u) # trace
2.0

y = np.array([[5.], [7.]])
np.linalg.solve(a, y)
array([[-3.],
[ 4.]])

np.linalg.eig(j)
(array([0.+1.j, 0.-1.j]), array([[0.70710678+0.j , 0.70710678-0.j ],
[0. -0.70710678j, 0. +0.70710678j]]))

Parameters:
square matrix
Returns
The eigenvalues, each repeated according to its multiplicity.
The normalized (unit “length”) eigenvectors, such that the
column v[:,i] is the eigenvector corresponding to the
eigenvalue w[i] .

Tricks and Tips¶

Here we give a list of short and useful tips.

“Automatic” Reshaping¶

To change the dimensions of an array, you can omit one of the sizes which will then be deduced automatically:

a = np.arange(30)
b = a.reshape((2, -1, 3)) # -1 means “whatever is needed”
b.shape
(2, 5, 3)

b
array([[[ 0, 1, 2],
[ 3, 4, 5],
[ 6, 7, 8],
[ 9, 10, 11],
[12, 13, 14]],

[[15, 16, 17],
[18, 19, 20],
[21, 22, 23],
[24, 25, 26],
[27, 28, 29]]])

Vector Stacking¶

How do we construct a 2D array from a list of equally-sized row vectors? In MATLAB this is quite easy: if x and y are two vectors of the same length you only need do m=[x;y]. In NumPy this works via the functions column_stack, dstack, hstack and vstack, depending on the dimension in which the stacking is to be done. For example:

x = np.arange(0,10,2)
y = np.arange(5)
m = np.vstack([x,y])
m
array([[0, 2, 4, 6, 8],
[0, 1, 2, 3, 4]])

xy = np.hstack([x,y])
xy
array([0, 2, 4, 6, 8, 0, 1, 2, 3, 4])

The logic behind those functions in more than two dimensions can be strange.

See also

NumPy for Matlab users

Histograms¶

histogram应用于数组的NumPy 函数返回一对向量：数组的直方图和bin边的向量。当心： matplotlib还具有建立直方图的功能（hist在Matlab中称为），该功能不同于NumPy中的直方图。主要区别是pylab.hist自动绘制直方图，而 numpy.histogram仅生成数据。

import numpy as np
rg = np.random.default_rng(1)
import matplotlib.pyplot as plt

Build a vector of 10000 normal deviates with variance 0.5^2 and mean 2

mu, sigma = 2, 0.5
v = rg.normal(mu,sigma,10000)

Plot a normalized histogram with 50 bins

plt.hist(v, bins=50, density=1) # matplotlib version (plot)

Compute the histogram with numpy and then plot it

(n, bins) = np.histogram(v, bins=50, density=True) # NumPy version (no plot)
plt.plot(.5*(bins[1:]+bins[:-1]), n)

进一步阅读¶

• 在Python的教程
• NumPy参考
• 科学教程
• 科学演讲笔记
• 甲MATLAB，R，IDL，NumPy的/ SciPy的字典
• ©版权所有2008-2019，SciPy社区。
• 上次更新时间为2020年3月8日。
• 使用Sphinx 2.3.1 创建。

原文

提供更好的翻译建议