本课程中所有作业将使用Python来完成。Python本身就是一种很棒的通用编程语言，现在在一些流行的库（numpy，scipy，matplotlib）的帮助下，它为科学计算提供强大的环境。
我们希望课程中的大部分人都有一些Python和numpy的经验；对于其他人来说，本教程将作为Python用于科学计算的速成课程。
基本数据类型
与大多数语言一样，Python有许多基本类型，包括整数，浮点数，布尔值和字符串。这些数据类型的行为方式与其他编程语言相似。
数字： 整数和浮点数的工作方式与其他语言相同：
x = 3
print(type(x)) # Prints "<class 'int'>"
print(x) # Prints "3"
print(x + 1) # Addition; prints "4"
print(x - 1) # Subtraction; prints "2"
print(x * 2) # Multiplication; prints "6"
print(x ** 2) # Exponentiation; prints "9"
x += 1
print(x) # Prints "4"
x *= 2
print(x) # Prints "8"
y = 2.5
print(type(y)) # Prints "<class 'float'>"
print(y, y + 1, y * 2, y ** 2) # Prints "2.5 3.5 5.0 6.25"
提示：与其他语言不同，Python没有一元增量(x++)和减量(x--)。
布尔： Python中实现所有通常的布尔逻辑，但是经常使用英文词语（and，or等）：
t = True
f = False
print(type(t)) # Prints "<class 'bool'>"
print(t and f) # Logical AND; prints "False"
print(t or f) # Logical OR; prints "True"
print(not t) # Logical NOT; prints "False"
print(t != f) # Logical XOR; prints "True"
字符串： Python对字符串有很好的支持：
hello = 'hello' # String literals can use single quotes
world = "world" # or double quotes; it does not matter.
print(hello) # Prints "hello"
print(len(hello)) # String length; prints "5"
hw = hello + ' ' + world # String concatenation
print(hw) # prints "hello world"
hw12 = '%s %s %d' % (hello, world, 12) # sprintf style string formatting
print(hw12) # prints "hello world 12"
String对象有很多有用的方法; 例如：
s = "hello"
print(s.capitalize()) # Capitalize a string; prints "Hello"
print(s.upper()) # Convert a string to uppercase; prints "HELLO"
print(s.rjust(7)) # Right-justify a string, padding with spaces; prints " hello"
print(s.center(7)) # Center a string, padding with spaces; prints " hello "
print(s.replace('l', '(ell)')) # Replace all instances of one substring with another;
# prints "he(ell)(ell)o"
print(' world '.strip()) # Strip leading and trailing whitespace; prints "world"
容器
Python包含几种内置容器类型: lists, dictionaries, sets, and tuples。
Lists
List是Python的等效数组，但是可以调整大小并且可以包含不同类型的元素：
xs = [3, 1, 2] # Create a list
print(xs, xs[2]) # Prints "[3, 1, 2] 2"
print(xs[-1]) # Negative indices count from the end of the list; prints "2"
xs[2] = 'foo' # Lists can contain elements of different types
print(xs) # Prints "[3, 1, 'foo']"
xs.append('bar') # Add a new element to the end of the list
print(xs) # Prints "[3, 1, 'foo', 'bar']"
x = xs.pop() # Remove and return the last element of the list
print(x, xs) # Prints "bar [3, 1, 'foo']"
Slicing: 除了一次访问一个列表元素外，Python还提供了访问子列表的简明语法; 这被称为 slicing:
nums = list(range(5)) # range is a built-in function that creates a list of integers
print(nums) # Prints "[0, 1, 2, 3, 4]"
print(nums[2:4]) # Get a slice from index 2 to 4 (exclusive); prints "[2, 3]"
print(nums[2:]) # Get a slice from index 2 to the end; prints "[2, 3, 4]"
print(nums[:2]) # Get a slice from the start to index 2 (exclusive); prints "[0, 1]"
print(nums[:]) # Get a slice of the whole list; prints "[0, 1, 2, 3, 4]"
print(nums[:-1]) # Slice indices can be negative; prints "[0, 1, 2, 3]"
nums[2:4] = [8, 9] # Assign a new sublist to a slice
print(nums) # Prints "[0, 1, 8, 9, 4]"
我们将在numpy arrays的部分中再次看到slicing。
Loops: 您可以循环遍历列表的元素，如下所示：
animals = ['cat', 'dog', 'monkey']
for animal in animals:
print(animal)
# Prints "cat", "dog", "monkey", each on its own line
如果要访问循环体内每个元素的索引，请使用内置enumerate函数：
animals = ['cat', 'dog', 'monkey']
for idx, animal in enumerate(animals):
print('#%d: %s' % (idx + 1, animal))
# Prints "#1: cat", "#2: dog", "#3: monkey", each on its own line
List comprehensions:
编程时，我们经常想要将一种数据转换为另一种数据。举个简单的例子，考虑以下计算平方数的代码：
nums = [0, 1, 2, 3, 4]
squares = []
for x in nums:
squares.append(x ** 2)
print(squares) # Prints [0, 1, 4, 9, 16]
您可以使用list comprehension使此代码更简单：
nums = [0, 1, 2, 3, 4]
squares = [x ** 2 for x in nums]
print(squares) # Prints [0, 1, 4, 9, 16]
list comprehension还可以包含条件：
nums = [0, 1, 2, 3, 4]
even_squares = [x ** 2 for x in nums if x % 2 == 0]
print(even_squares) # Prints "[0, 4, 16]"
Dictionaries
字典存储（键，值）对，类似于Java或Javascript中的Map对象。你可以像这样使用它：
d = {'cat': 'cute', 'dog': 'furry'} # Create a new dictionary with some data
print(d['cat']) # Get an entry from a dictionary; prints "cute"
print('cat' in d) # Check if a dictionary has a given key; prints "True"
d['fish'] = 'wet' # Set an entry in a dictionary
print(d['fish']) # Prints "wet"
# print(d['monkey']) # KeyError: 'monkey' not a key of d
print(d.get('monkey', 'N/A')) # Get an element with a default; prints "N/A"
print(d.get('fish', 'N/A')) # Get an element with a default; prints "wet"
del d['fish'] # Remove an element from a dictionary
print(d.get('fish', 'N/A')) # "fish" is no longer a key; prints "N/A"
Loops: 很容易迭代字典中的键：
d = {'person': 2, 'cat': 4, 'spider': 8}
for animal in d:
legs = d[animal]
print('A %s has %d legs' % (animal, legs))
# Prints "A person has 2 legs", "A cat has 4 legs", "A spider has 8 legs"
如果要访问Key及其对应的值，请使用以下items方法：
d = {'person': 2, 'cat': 4, 'spider': 8}
for animal, legs in d.items():
print('A %s has %d legs' % (animal, legs))
# Prints "A person has 2 legs", "A cat has 4 legs", "A spider has 8 legs"
Dictionary comprehensions:
这些与列表理解类似，但允许您轻松构建字典。例如：
nums = [0, 1, 2, 3, 4]
even_num_to_square = {x: x ** 2 for x in nums if x % 2 == 0}
print(even_num_to_square) # Prints "{0: 0, 2: 4, 4: 16}"
Sets
集合是不同元素的无序集合。举个简单的例子，请考虑以下事项：
animals = {'cat', 'dog'}
print('cat' in animals) # Check if an element is in a set; prints "True"
print('fish' in animals) # prints "False"
animals.add('fish') # Add an element to a set
print('fish' in animals) # Prints "True"
print(len(animals)) # Number of elements in a set; prints "3"
animals.add('cat') # Adding an element that is already in the set does nothing
print(len(animals)) # Prints "3"
animals.remove('cat') # Remove an element from a set
print(len(animals)) # Prints "2"
Loops: 对集合进行迭代与迭代列表具有相同的语法; 但是由于集合是无序的，因此您无法对访问集合元素的顺序进行假设：
animals = {'cat', 'dog', 'fish'}
for idx, animal in enumerate(animals):
print('#%d: %s' % (idx + 1, animal))
# Prints "#1: fish", "#2: dog", "#3: cat"
Set comprehensions:
像列表和词典一样，我们可以使用Set comprehensions轻松构建集合：
from math import sqrt
nums = {int(sqrt(x)) for x in range(30)}
print(nums) # Prints "{0, 1, 2, 3, 4, 5}"
Tuples
Tuple是（不可变的）有序值列表。
Tuple在很多方面类似于列表; 其中一个最重要的区别是Tuple可以用作字典中的键和集合的元素，而列表则不能。这是一个简单的例子：
d = {(x, x + 1): x for x in range(10)} # Create a dictionary with tuple keys
t = (5, 6) # Create a tuple
print(type(t)) # Prints "<class 'tuple'>"
print(d[t]) # Prints "5"
print(d[(1, 2)]) # Prints "1"
Numpy
Numpy 是Python中科学计算的核心库。它提供了一个高性能的多维数组对象，以及用于处理这些数组的工具。如果您已经熟悉MATLAB，那么您可能会发现本教程对Numpy入门非常有用。
Arrays
numpy arrays是一个值网格，所有类型都相同，并由非负整数元组索引。维数是array的排名; arrays的形状是一个给出了arrays中的每个维度大小的整数Tuple。
我们可以从嵌套的Python列表初始化numpy array，并使用方括号访问元素：
import numpy as np a = np.array([1, 2, 3]) # Create a rank 1 array
print(type(a)) # Prints "<class 'numpy.ndarray'>"
print(a.shape) # Prints "(3,)"
print(a[0], a[1], a[2]) # Prints "1 2 3"
a[0] = 5 # Change an element of the array
print(a) # Prints "[5, 2, 3]" b = np.array([[1,2,3],[4,5,6]]) # Create a rank 2 array
print(b.shape) # Prints "(2, 3)"
print(b[0, 0], b[0, 1], b[1, 0]) # Prints "1 2 4"
Numpy还提供了许多创建数组的函数：
import numpy as np a = np.zeros((2,2)) # Create an array of all zeros
print(a) # Prints "[[ 0. 0.]
# [ 0. 0.]]" b = np.ones((1,2)) # Create an array of all ones
print(b) # Prints "[[ 1. 1.]]" c = np.full((2,2), 7) # Create a constant array
print(c) # Prints "[[ 7. 7.]
# [ 7. 7.]]" d = np.eye(2) # Create a 2x2 identity matrix
print(d) # Prints "[[ 1. 0.]
# [ 0. 1.]]" e = np.random.random((2,2)) # Create an array filled with random values
print(e) # Might print "[[ 0.91940167 0.08143941]
# [ 0.68744134 0.87236687]]"
Array indexing
Numpy提供了几种索引数组的方法。
Slicing: 与Python列表类似，可以切割numpy数组。由于数组可能是多维的，因此必须为数组的每个维指定一个切片：
import numpy as np # Create the following rank 2 array with shape (3, 4)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) # Use slicing to pull out the subarray consisting of the first 2 rows
# and columns 1 and 2; b is the following array of shape (2, 2):
# [[2 3]
# [6 7]]
b = a[:2, 1:3] # A slice of an array is a view into the same data, so modifying it
# will modify the original array.
print(a[0, 1]) # Prints "2"
b[0, 0] = 77 # b[0, 0] is the same piece of data as a[0, 1]
print(a[0, 1]) # Prints "77"
您还可以将整数索引与切片索引混合使用。但是，这样做会产生比原始数组更低级别的数组。请注意，这与MATLAB处理数组切片的方式完全不同：
import numpy as np # Create the following rank 2 array with shape (3, 4)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) # Two ways of accessing the data in the middle row of the array.
# Mixing integer indexing with slices yields an array of lower rank,
# while using only slices yields an array of the same rank as the
# original array:
row_r1 = a[1, :] # Rank 1 view of the second row of a
row_r2 = a[1:2, :] # Rank 2 view of the second row of a
print(row_r1, row_r1.shape) # Prints "[5 6 7 8] (4,)"
print(row_r2, row_r2.shape) # Prints "[[5 6 7 8]] (1, 4)" # We can make the same distinction when accessing columns of an array:
col_r1 = a[:, 1]
col_r2 = a[:, 1:2]
print(col_r1, col_r1.shape) # Prints "[ 2 6 10] (3,)"
print(col_r2, col_r2.shape) # Prints "[[ 2]
# [ 6]
# [10]] (3, 1)"
Integer array indexing:
使用切片索引到numpy数组时，生成的数组视图将始终是原始数组的子数组。相反，整数数组索引允许您使用另一个数组中的数据构造任意数组。这是一个例子：
import numpy as np a = np.array([[1,2], [3, 4], [5, 6]]) # An example of integer array indexing.
# The returned array will have shape (3,) and
print(a[[0, 1, 2], [0, 1, 0]]) # Prints "[1 4 5]" # The above example of integer array indexing is equivalent to this:
print(np.array([a[0, 0], a[1, 1], a[2, 0]])) # Prints "[1 4 5]" # When using integer array indexing, you can reuse the same
# element from the source array:
print(a[[0, 0], [1, 1]]) # Prints "[2 2]" # Equivalent to the previous integer array indexing example
print(np.array([a[0, 1], a[0, 1]])) # Prints "[2 2]"
integer array indexing的一个有用技巧是从矩阵的每一行中选择或改变一个元素：
import numpy as np # Create a new array from which we will select elements
a = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]]) print(a) # prints "array([[ 1, 2, 3],
# [ 4, 5, 6],
# [ 7, 8, 9],
# [10, 11, 12]])" # Create an array of indices
b = np.array([0, 2, 0, 1]) # Select one element from each row of a using the indices in b
print(a[np.arange(4), b]) # Prints "[ 1 6 7 11]" # Mutate one element from each row of a using the indices in b
a[np.arange(4), b] += 10 print(a) # prints "array([[11, 2, 3],
# [ 4, 5, 16],
# [17, 8, 9],
# [10, 21, 12]])
Boolean array indexing:
布尔数组索引允许您选择数组的任意元素。通常，这种类型的索引用于选择满足某些条件的数组元素。这是一个例子：
import numpy as np a = np.array([[1,2], [3, 4], [5, 6]]) bool_idx = (a > 2) # Find the elements of a that are bigger than 2;
# this returns a numpy array of Booleans of the same
# shape as a, where each slot of bool_idx tells
# whether that element of a is > 2. print(bool_idx) # Prints "[[False False]
# [ True True]
# [ True True]]" # We use boolean array indexing to construct a rank 1 array
# consisting of the elements of a corresponding to the True values
# of bool_idx
print(a[bool_idx]) # Prints "[3 4 5 6]" # We can do all of the above in a single concise statement:
print(a[a > 2]) # Prints "[3 4 5 6]"
Datatypes
每个numpy数组都是相同类型元素的网格。Numpy提供了一组可用于构造数组的大量数值数据类型。Numpy在创建数组时尝试猜测数据类型，但构造数组的函数通常还包含一个可选参数来显式指定数据类型。这是一个例子：
import numpy as np x = np.array([1, 2]) # Let numpy choose the datatype
print(x.dtype) # Prints "int64" x = np.array([1.0, 2.0]) # Let numpy choose the datatype
print(x.dtype) # Prints "float64" x = np.array([1, 2], dtype=np.int64) # Force a particular datatype
print(x.dtype) # Prints "int64"
Array math
基本数学函数在数组上以元素方式运行，既可以作为运算符重载，也可以作为numpy模块中的函数：
import numpy as np x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64) # Elementwise sum; both produce the array
# [[ 6.0 8.0]
# [10.0 12.0]]
print(x + y)
print(np.add(x, y)) # Elementwise difference; both produce the array
# [[-4.0 -4.0]
# [-4.0 -4.0]]
print(x - y)
print(np.subtract(x, y)) # Elementwise product; both produce the array
# [[ 5.0 12.0]
# [21.0 32.0]]
print(x * y)
print(np.multiply(x, y)) # Elementwise division; both produce the array
# [[ 0.2 0.33333333]
# [ 0.42857143 0.5 ]]
print(x / y)
print(np.divide(x, y)) # Elementwise square root; produces the array
# [[ 1. 1.41421356]
# [ 1.73205081 2. ]]
print(np.sqrt(x))
请注意，与MATLAB不同，*是元素乘法，而不是矩阵乘法。我们使用该dot函数来计算向量的内积，将向量乘以矩阵，并乘以矩阵。dot既可以作为numpy模块中的函数，也可以作为数组对象的实例方法：
import numpy as np x = np.array([[1,2],[3,4]])
y = np.array([[5,6],[7,8]]) v = np.array([9,10])
w = np.array([11, 12]) # Inner product of vectors; both produce 219
print(v.dot(w))
print(np.dot(v, w)) # Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v)) # Matrix / matrix product; both produce the rank 2 array
# [[19 22]
# [43 50]]
print(x.dot(y))
print(np.dot(x, y))
Numpy提供了许多用于在数组上执行计算的有用函数; 其中一个最有用的是sum：
import numpy as np x = np.array([[1,2],[3,4]]) print(np.sum(x)) # Compute sum of all elements; prints "10"
print(np.sum(x, axis=0)) # Compute sum of each column; prints "[4 6]"
print(np.sum(x, axis=1)) # Compute sum of each row; prints "[3 7]"
您可以在文档中找到numpy提供的完整数学函数列表。
除了使用数组计算数学函数之外，我们经常需要重新整形或以其他方式操纵数组中的数据。这种操作的最简单的例子是转置矩阵; 要转置矩阵，只需使用T数组对象的属性：
import numpy as np x = np.array([[1,2], [3,4]])
print(x) # Prints "[[1 2]
# [3 4]]"
print(x.T) # Prints "[[1 3]
# [2 4]]" # Note that taking the transpose of a rank 1 array does nothing:
v = np.array([1,2,3])
print(v) # Prints "[1 2 3]"
print(v.T) # Prints "[1 2 3]"
Broadcasting
广播是一种强大的机制，允许numpy在执行算术运算时使用不同形状的数组。我们经常有一个较小的数组和一个较大的数组，我们希望多次使用较小的array来对较大的array执行某些操作。
例如，假设我们想要向矩阵的每一行添加一个常量向量。我们可以这样做：
import numpy as np # We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = np.empty_like(x) # Create an empty matrix with the same shape as x # Add the vector v to each row of the matrix x with an explicit loop
for i in range(4):
y[i, :] = x[i, :] + v # Now y is the following
# [[ 2 2 4]
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]
print(y)
这有效; 但是当矩阵x非常大时，在Python中计算显式循环可能会很慢。请注意，将向量添加v到矩阵的每一行 x等同于vv通过堆叠多个v垂直副本来形成矩阵，然后执行和的元素x和求和vv。我们可以像这样实现这种方法：
import numpy as np # We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
vv = np.tile(v, (4, 1)) # Stack 4 copies of v on top of each other
print(vv) # Prints "[[1 0 1]
# [1 0 1]
# [1 0 1]
# [1 0 1]]"
y = x + vv # Add x and vv elementwise
print(y) # Prints "[[ 2 2 4
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"
Numpy广播允许我们执行此计算而无需实际创建多个副本v。考虑这个版本，使用广播：
import numpy as np # We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = x + v # Add v to each row of x using broadcasting
print(y) # Prints "[[ 2 2 4]
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"
将两个数组一起广播遵循以下规则：
- 如果数组不具有相同的等级，则将较低等级数组的形状添加为1，直到两个形状具有相同的长度。
- 如果两个数组在维度中具有相同的大小，或者如果其中一个数组在该维度中具有大小1，则称这两个数组在维度上是兼容的。
- 如果阵列在所有维度上兼容，则可以一起广播。
- 在广播之后，每个阵列的行为就好像它的形状等于两个输入数组的形状的元素最大值。
- 在一个数组的大小为1且另一个数组的大小大于1的任何维度中，第一个数组的行为就像沿着该维度复制一样。
以下是广播的一些应用：
import numpy as np # Compute outer product of vectors
v = np.array([1,2,3]) # v has shape (3,)
w = np.array([4,5]) # w has shape (2,)
# To compute an outer product, we first reshape v to be a column
# vector of shape (3, 1); we can then broadcast it against w to yield
# an output of shape (3, 2), which is the outer product of v and w:
# [[ 4 5]
# [ 8 10]
# [12 15]]
print(np.reshape(v, (3, 1)) * w) # Add a vector to each row of a matrix
x = np.array([[1,2,3], [4,5,6]])
# x has shape (2, 3) and v has shape (3,) so they broadcast to (2, 3),
# giving the following matrix:
# [[2 4 6]
# [5 7 9]]
print(x + v) # Add a vector to each column of a matrix
# x has shape (2, 3) and w has shape (2,).
# If we transpose x then it has shape (3, 2) and can be broadcast
# against w to yield a result of shape (3, 2); transposing this result
# yields the final result of shape (2, 3) which is the matrix x with
# the vector w added to each column. Gives the following matrix:
# [[ 5 6 7]
# [ 9 10 11]]
print((x.T + w).T)
# Another solution is to reshape w to be a column vector of shape (2, 1);
# we can then broadcast it directly against x to produce the same
# output.
print(x + np.reshape(w, (2, 1))) # Multiply a matrix by a constant:
# x has shape (2, 3). Numpy treats scalars as arrays of shape ();
# these can be broadcast together to shape (2, 3), producing the
# following array:
# [[ 2 4 6]
# [ 8 10 12]]
print(x * 2)