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# APS360 Tutorial 1 - Review of Python and Numpy¶

Adapted by Andrew Jung from the Stanford CS231n Python tutorial by Justin Johnson and MIE324 tutorial by Harris Chan

## Introduction¶

Python is a general-purpose, high-level, dynamically typed (no pre-defined type like in C++ or Java), interpreted language (no compiler). Additionally, Python has very easy to read syntax and can achieve high computational performance by extending Python with C or C++ (e.g. significant portion of Numpy and PyTorch is written in C and C++)

These features helped Python becoming one of the most popular language in scientific community, especially in machine learning. Since so many scientists and engineers use Python, there are many popular libraries that helps tasks commonly performed in scientific computing, such as matrix operations, data visualization, etc... In this course, we will be using Numpy and Matplotlib.

## Basics of Python¶

### Python versions¶

There are currently two different supported versions of Python, 2.7 and 3. (latest one being 3.7). The two versions of Python have a few differences and incompatibilities, and in this class, we will be using Python 3.6. In general, it is recommended to start using 3. versions because Python 2.7 won't be maintained past 2020.

You can check your Python version at the command line by running python --version.

### Environment Setup¶

Follow the instructions to install Python 3.6, PyCharm, and Jupyter Notebook: https://www.cs.toronto.edu/~lczhang/360/files/install.pdf

For simple tasks, Jupyter Notebook might be a bit quicker and easier to use than PyCharm. Also, try to start a habit of using virtual environment (like Virtualenv or Anaconda) all the time. Encapsulating each project's environment becomes very crucial when you have many projects or wish to collaborate with others. Docker is another step in this direction as it encapsulates in OS-level and I recommend learning about it if you have time (refer to the installation guide above).

### Basic data types¶

#### Numbers¶

Integers and floats work as you would expect from other languages:

In [1]:
x = 3
print(x, type(x))

y = 3.5
print(y, type(y))

3 <class 'int'>
3.5 <class 'float'>

In [2]:
print(x + 1)      # Addition;
print(x - 1)      # Subtraction;
print(x * 2)      # Multiplication;
print(x ** 2)     # Exponentiation;
print(x / 2)      # Division; always float division in Python 3.x (unlike Python 2.7)
print(x // 2)     # Floor division; rounding down the result into int
print(int(x / 2)) # Another way to do floor division

4
2
6
9
1.5
1
1

In [3]:
x += 1
print(x)  # Prints "4"
x *= 2
print(x)  # Prints "8"
x **= 2
print(x)  # Prints "64"

4
8
64

In [4]:
y = 2.5
print(type(y)) # Prints "<type 'float'>"
print(y, y + 1, y * 2, y ** 2) # Prints "2.5 3.5 5.0 6.25"

<class 'float'>
2.5 3.5 5.0 6.25

Note that unlike many languages, Python does not have unary increment (x++) or decrement (x--) operators. Python also has built-in types for long integers and complex numbers; you can find all of the details in the documentation.

Most commonly used number type for neural network is 32-bit float (float32), but sometimes 16-bit float (float16) is used for a few applications to achieve better performance.

#### Booleans¶

Python implements all of the usual operators for Boolean logic, but uses English words rather than symbols (&&, ||, etc.):

In [5]:
t, f = True, False
print(type(t)) # Prints "<type 'bool'>"

<class 'bool'>

Now we let's look at the operations:

In [6]:
print(t and f) # Logical AND;
print(t or f)  # Logical OR;
print(not t)   # Logical NOT;
print(t != f)  # Logical XOR;

False
True
False
True

#### Strings¶

In [7]:
hello = 'hello'   # String literals can use single quotes
world = "world"   # or double quotes; it does not matter.
print(hello, len(hello))

hello 5

In [8]:
hw = hello + ' ' + world  # String concatenation
print(hw)  # prints "hello world"

hello world

In [9]:
hw12 = '%s %s %d' % (hello, world, 12)  # sprintf style string formatting
print(hw12)  # prints "hello world 12"

hello world 12

This type of string formatting is very useful, especially for logging or debugging.

String objects have a bunch of useful methods; for example:

In [10]:
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"

Hello
HELLO
hello
hello
he(ell)(ell)o
world

In [11]:
sentence = "Hello this is an apple and that is a pen"
print(sentence.split(" "))

['Hello', 'this', 'is', 'an', 'apple', 'and', 'that', 'is', 'a', 'pen']

You can find a list of all string methods in the documentation.

### Containers¶

Python includes several built-in container types: lists, dictionaries, sets, and tuples.

#### Lists¶

A list is the Python equivalent of an array, but is resizeable and can contain elements of different types:

In [12]:
xs = [3, 1, 2]   # Create a list
print(xs)
print(xs[0])
print(xs[-1])     # Negative indices count from the end of the list; prints "2"

[3, 1, 2]
3
2

In [13]:
xs[2] = 'foo'    # Lists can contain elements of different types
print(xs)

[3, 1, 'foo']

In [14]:
xs.append('bar') # Add a new element to the end of the list
print(xs)

[3, 1, 'foo', 'bar']

In [15]:
x = xs.pop()     # Remove and return the last element of the list
print(x, xs)

bar [3, 1, 'foo']

As usual, you can find all the gory details about lists in the documentation.

#### Slicing¶

In addition to accessing list elements one at a time, Python provides concise syntax to access sublists; this is known as slicing:

In [16]:
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]"

[0, 1, 2, 3, 4]
[2, 3]
[2, 3, 4]
[0, 1]
[0, 1, 2, 3, 4]
[0, 1, 2, 3]
[0, 1, 8, 9, 4]

#### Loops¶

You can loop over the elements of a list like this:

In [17]:
animals = ['cat', 'dog', 'monkey']
for animal in animals:
print(animal)

cat
dog
monkey

If you want access to the index of each element within the body of a loop, use the built-in enumerate function:

In [18]:
animals = ['cat', 'dog', 'monkey']
for idx, animal in enumerate(animals):
print('#%d: %s' % (idx + 1, animal))

#1: cat
#2: dog
#3: monkey

In Python, use range to iterate over a range of values. This is equivalent to for (int i = 0; i < 10; i++ in some other languages.

In [19]:
for i in range(10):
print(i)

0
1
2
3
4
5
6
7
8
9

In [20]:
for i in range(2, 26, 4):
print(i)

2
6
10
14
18
22

#### List comprehensions:¶

When programming, frequently we want to transform one type of data into another. As a simple example, consider the following code that computes square numbers:

In [21]:
nums = [0, 1, 2, 3, 4]
squares = []
for x in nums:
squares.append(x ** 2)
print(squares)

[0, 1, 4, 9, 16]

You can make this code simpler using a list comprehension:

In [22]:
nums = [0, 1, 2, 3, 4]
squares = [x ** 2 for x in nums]
print(squares)

[0, 1, 4, 9, 16]

List comprehensions can also contain conditions:

In [23]:
nums = [0, 1, 2, 3, 4]
even_squares = [x ** 2 for x in nums if x % 2 == 0]
print(even_squares)

[0, 4, 16]

#### Dictionaries¶

A dictionary stores (key, value) pairs, similar to a Map in Java or an object in Javascript. You can use it like this:

In [24]:
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"

cute
True

In [25]:
d['fish'] = 'wet'    # Set an entry in a dictionary
print(d['fish'])      # Prints "wet"

wet

In [26]:
print(d['monkey'])  # KeyError: 'monkey' not a key of d

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-26-78fc9745d9cf> in <module>
----> 1 print(d['monkey'])  # KeyError: 'monkey' not a key of d

KeyError: 'monkey'
In [27]:
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"

N/A
wet

In [28]:
del d['fish']        # Remove an element from a dictionary
print(d.get('fish', 'N/A')) # "fish" is no longer a key; prints "N/A"

N/A

You can find all you need to know about dictionaries in the documentation.

It is easy to iterate over the keys in a dictionary:

In [29]:
d = {'person': 2, 'cat': 4, 'spider': 8}
for animal in d:
legs = d[animal]
print('A %s has %d legs' % (animal, legs))

A person has 2 legs
A cat has 4 legs
A spider has 8 legs

If you want access to keys and their corresponding values, use the items method:

In [30]:
d = {'person': 2, 'cat': 4, 'spider': 8}
for animal, legs in d.items():
print('A %s has %d legs' % (animal, legs))

A person has 2 legs
A cat has 4 legs
A spider has 8 legs

Dictionary comprehensions: These are similar to list comprehensions, but allow you to easily construct dictionaries. For example:

In [31]:
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)

{0: 0, 2: 4, 4: 16}

#### Sets¶

A set is an unordered collection of distinct elements. As a simple example, consider the following:

In [32]:
animals = {'cat', 'dog'}
print('cat' in animals)   # Check if an element is in a set; prints "True"
print('fish' in animals)  # prints "False"

True
False

In [33]:
animals.add('fish')      # Add an element to a set
print('fish' in animals)
print(len(animals))       # Number of elements in a set;

True
3

In [34]:
animals.add('cat')       # Adding an element that is already in the set does nothing
print(len(animals))
animals.remove('cat')    # Remove an element from a set
print(len(animals))

3
2

Loops: Iterating over a set has the same syntax as iterating over a list; however since sets are unordered, you cannot make assumptions about the order in which you visit the elements of the set:

In [35]:
animals = {'cat', 'dog', 'fish'}
for idx, animal in enumerate(animals):
print('#%d: %s' % (idx + 1, animal))
# Prints "#1: fish", "#2: dog", "#3: cat"

#1: fish
#2: cat
#3: dog

Set comprehensions: Like lists and dictionaries, we can easily construct sets using set comprehensions:

In [36]:
from math import sqrt
print({int(sqrt(x)) for x in range(30)})

{0, 1, 2, 3, 4, 5}

#### Tuples¶

A tuple is an (immutable) ordered list of values. A tuple is in many ways similar to a list; one of the most important differences is that tuples can be used as keys in dictionaries and as elements of sets, while lists cannot. Here is a trivial example:

In [37]:
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))
print(d[t])
print(d[(1, 2)])

<class 'tuple'>
5
1

In [38]:
t[0] = 1

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-38-c8aeb8cd20ae> in <module>
----> 1 t[0] = 1

TypeError: 'tuple' object does not support item assignment

### Functions¶

Python functions are defined using the def keyword.

Keep in mind that in Python, instead of using curly brackets like in some other languages, whitespace is used to indicate blocks. Thus, for function definitions, loops, and if statements, make sure to have the right indentation.

For example:

In [40]:
def sign(x):
if x > 0:
return 'positive'
elif x < 0:
return 'negative'
else:
return 'zero'

for x in [-1, 0, 1]:
print(sign(x))

negative
zero
positive

We will often define functions to take optional keyword arguments, like this:

In [41]:
def hello(name, loud=False): # False would be loud's default value
if loud:
print('HELLO, %s' % name.upper())
else:
print('Hello, %s!' % name)

hello('Bob')
hello('Fred', loud=True)

Hello, Bob!
HELLO, FRED

### Classes¶

The syntax for defining classes in Python is straightforward:

In [42]:
class Student: # equivalent to 'class Student(object)'

# A class attribute. It is shared by all instances of this class
university = "University of Toronto"

# Constructor
def __init__(self, name, year, program):
# Assign the argument of the constructor to instance attributes
self.name = name
self.year = year
self.program = program

# Initialize an instance attribute
self.done_pey = False

# An instance method. All instance methods take "self" as the first argument
def get_details(self):
""" Describe the student's identity """
print("{} studies {} at {} and will graduate in {}"
.format(self.name, self.program,
Student.university, self.year))

# Another instance method
def do_pey(self, company):
""" Student goes on PEY. Delays graduation by a year """
if not self.done_pey:
self.year += 1
self.done_pey = True
self.pey_company = company # creates a new instance property
print("{} went on PEY at {}".format(self.name, company))
else:
print("{} already went on PEY before at {}"
.format(self.name, self.pey_company))
return

In [43]:
student_1 = Student('John', 2020, 'Engineering')
student_1.get_details()

student_2 = Student('Mike', 2021, 'Math')
student_2.get_details()

John studies Engineering at University of Toronto and will graduate in 2020
Mike studies Math at University of Toronto and will graduate in 2021

In [44]:
student_1.do_pey('DeepMind')
student_1.get_details()

John went on PEY at DeepMind
John studies Engineering at University of Toronto and will graduate in 2021

In [45]:
student_1.do_pey('Google')
student_1.get_details()

John already went on PEY before at DeepMind
John studies Engineering at University of Toronto and will graduate in 2021

#### Inheritance¶

Inheritance allows new child classes to inherit methods and attributes from their parent class.

Using the Student class above as the parent class, we can define a child class, EngSci, which inherits the attributes like "name", "year", and "program", as well as methods, like "do_pey" from the Student class.

EngSci class can also have its own unique attributes and methods.

It can override anything from its parent class by redefining the inherited attributes or methods.

In [46]:
class EngSci(Student): # Notice how EngSci uses Student as base class, instead of Object class
# Constructor
def __init__(self, name, year):
# The "super" function lets you access the parent class's methods.
# In this case, we are calling the __init__ method of the parent class.
super().__init__(name, year, "Engineering Science")

# Additional attributes unique to EngSci class
self.option = "Undeclared"

# Override the get_details() method of Student class so that it displays option
def get_details(self):
""" Describe the student's identity"""
print("{} studies {} in {} option at {} and will graduate in {}"
.format(self.name, self.program, self.option, Student.university, self.year))

# New method unique to EngSci class
def declare_option(self, option):
""" The student declares major option"""
self.option = option
print("{} will major in {}".format(self.name, self.option))

In [47]:
you = EngSci("Geoff", 2020)

In [48]:
you.get_details()

Geoff studies Engineering Science in Undeclared option at University of Toronto and will graduate in 2020

In [49]:
you.declare_option("Engineering Physics")

Geoff will major in Engineering Physics

#### Callable Objects¶

A callable object is any object that can be called like a function. Remember how we call normal object method like student_1.get_details(), but if student_1 was callable object, then it can be used like student_1().

In Python, any object whose class has a call method will be callable. For example, we can define an Adder class that is initialized with a value bias. When the object of the Adder class is called with input, it will return the sum of bias and input

In [50]:
class Adder(object):
# Initialize with the bias value which will be added to the input
def __init__(self, bias):
self.bias = bias

# Make an instance of this class callable
def __call__(self, input):
return self.bias + input

# Update the value parameter of this object
def update_parameter(self, delta):
self.bias += delta
return

In [51]:
# Create an instance of the Adder class
my_adder(3) # Equivalent to my_adder.__call__(3)

Out[51]:
13
In [52]:
# Equivalent to my_adder(3)

Out[52]:
13
In [53]:
my_adder.update_parameter(5)

15

In [54]:
my_adder(4)

Out[54]:
19
In [55]:
# Create another adder with a different adding value
input = 5
print(output)

10

## Numpy¶

Numpy is the core library for scientific computing in Python. It provides a high-performance multidimensional array object, and tools for working with these arrays. If you are already familiar with MATLAB, you might find this tutorial useful to get started with Numpy.

To use Numpy, we first need to import the numpy package:

In [56]:
import numpy as np

### Arrays¶

A numpy array is a grid of values, all of the same type, and is indexed by a tuple of nonnegative integers. The number of dimensions is the rank of the array; the shape of an array is a tuple of integers giving the size of the array along each dimension.

We can initialize numpy arrays from nested Python lists, and access elements using square brackets:

In [57]:
a = np.array([1, 2, 3])  # Create a rank 1 array
print(type(a), a.shape, a[0], a[1], a[2])
a[0] = 5                 # Change an element of the array
print(a)

<class 'numpy.ndarray'> (3,) 1 2 3
[5 2 3]

In [58]:
b = np.array([[1,2,3],[4,5,6]])   # Create a rank 2 array
print(b)

[[1 2 3]
[4 5 6]]

In [59]:
print(b.shape)
print(b[0, 0], b[0, 1], b[1, 0])

(2, 3)
1 2 4

Numpy also provides many functions to create arrays:

In [60]:
a = np.zeros((2,2))  # Create an array of all zeros
print(a)

[[0. 0.]
[0. 0.]]

In [61]:
b = np.ones((1,2))   # Create an array of all ones
print(b)

[[1. 1.]]

In [62]:
c = np.full((2,2), 7) # Create a constant array
print(c)

[[7 7]
[7 7]]

In [63]:
d = np.eye(2)        # Create a 2x2 identity matrix
print(d)

[[1. 0.]
[0. 1.]]

In [64]:
e = np.random.random((2,2)) # Create an array filled with random values
print(e)

[[0.42486478 0.84195689]
[0.18505352 0.66940913]]

### Array indexing¶

Numpy offers several ways to index into arrays.

Slicing: Similar to Python lists, numpy arrays can be sliced. Since arrays may be multidimensional, you must specify a slice for each dimension of the array:

In [65]:
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]
print(b)

[[2 3]
[6 7]]

A slice of an array is a view into the same data (i.e. pointing to same data in memory), so modifying it will modify the original array.

In [66]:
print(a[0, 1])
b[0, 0] = 77    # b[0, 0] is the same piece of data as a[0, 1]
print(a[0, 1])

2
77

You can also mix integer indexing with slice indexing. However, doing so will yield an array of lower rank than the original array. Note that this is quite different from the way that MATLAB handles array slicing:

In [67]:
# Create the following rank 2 array with shape (3, 4)
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
print(a)

[[ 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:

In [68]:
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
row_r3 = a[[1], :]  # Rank 2 view of the second row of a
print(row_r1, row_r1.shape)
print(row_r2, row_r2.shape)
print(row_r3, row_r3.shape)

[5 6 7 8] (4,)
[[5 6 7 8]] (1, 4)
[[5 6 7 8]] (1, 4)

In [69]:
# 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)
print(col_r2, col_r2.shape)

[ 2  6 10] (3,)
[[ 2]
[ 6]
[10]] (3, 1)

#### Integer array indexing¶

When you index into numpy arrays using slicing, the resulting array view will always be a subarray of the original array.

In contrast, integer array indexing allows you to construct arbitrary arrays using the data from another array (i.e. doesn't need to be subarray as long as all the data come from the original array).

In integer array indexing, indices of from each axis are specified in an array. Here is an example:

In [70]:
# a = [[1, 2],
#      [3, 4],
#      [5, 6]

a = np.array([[1,2], [3, 4], [5, 6]])

# An example of integer array indexing selecting 3 points from a
# The returned array will have shape (3,) and
print(a[[0, 1, 2], [0, 1, 0]])

# The above example of integer array indexing is equivalent to this:
print(np.array([a[0, 0], a[1, 1], a[2, 0]]))

[1 4 5]
[1 4 5]

In [71]:
# When using integer array indexing, you can reuse the same
# element from the source array:
print(a[[0, 0], [1, 1]])

# Equivalent to the previous integer array indexing example
print(np.array([a[0, 1], a[0, 1]]))

[2 2]
[2 2]

One useful trick with integer array indexing is selecting or mutating one element from each row of a matrix:

In [72]:
# 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)

[[ 1  2  3]
[ 4  5  6]
[ 7  8  9]
[10 11 12]]

In [73]:
# 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]"

[ 1  6  7 11]

In [74]:
# Mutate one element from each row of a using the indices in b
a[np.arange(4), b] += 10
print(a)

[[11  2  3]
[ 4  5 16]
[17  8  9]
[10 21 12]]

#### Boolean array indexing¶

Boolean array indexing lets you pick out arbitrary elements of an array. Frequently this type of indexing is used to select the elements of an array that satisfy some condition. Here is an example:

In [75]:
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)

[[False False]
[ True  True]
[ True  True]]

In [76]:
# 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])

# We can do all of the above in a single concise statement:
print(a[a > 2])

[3 4 5 6]
[3 4 5 6]

For brevity we have left out a lot of details about numpy array indexing; if you want to know more you should read the documentation.

### Datatypes¶

Every numpy array is a grid of elements of the same type. Numpy provides a large set of numeric datatypes that you can use to construct arrays. Numpy tries to guess a datatype when you create an array, but functions that construct arrays usually also include an optional argument to explicitly specify the datatype. Here is an example:

In [77]:
x = np.array([1, 2])  # Let numpy choose the datatype
y = np.array([1.0, 2.0])  # Let numpy choose the datatype
z = np.array([1, 2], dtype=np.int64)  # Force a particular datatype

print(x.dtype, y.dtype, z.dtype)

int64 float64 int64

### Array math¶

Basic mathematical functions operate elementwise on arrays, and are available both as operator overloads (like +, -, *, /) and as functions in the numpy module:

In [78]:
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
print(x + y)

[[ 6.  8.]
[10. 12.]]
[[ 6.  8.]
[10. 12.]]

In [79]:
# Elementwise difference; both produce the array
print(x - y)
print(np.subtract(x, y))

[[-4. -4.]
[-4. -4.]]
[[-4. -4.]
[-4. -4.]]

In [80]:
# Elementwise product; both produce the array
print(x * y)
print(np.multiply(x, y))

[[ 5. 12.]
[21. 32.]]
[[ 5. 12.]
[21. 32.]]

In [81]:
# Elementwise division; both produce the array
# [[ 0.2         0.33333333]
#  [ 0.42857143  0.5       ]]
print(x / y)
print(np.divide(x, y))

[[0.2        0.33333333]
[0.42857143 0.5       ]]
[[0.2        0.33333333]
[0.42857143 0.5       ]]

In [82]:
# Elementwise exponential; produces the array
# [[ 1.   4.]
#  [ 9.  16.]]
print(x ** 2)

[[ 1.  4.]
[ 9. 16.]]

In [83]:
# Elementwise square root; produces the array
# [[ 1.          1.41421356]
#  [ 1.73205081  2.        ]]
print(np.sqrt(x))

[[1.         1.41421356]
[1.73205081 2.        ]]

Note that unlike MATLAB, * is elementwise multiplication, not matrix multiplication.

We instead use the dot function to compute inner products of vectors, to multiply a vector by a matrix, and to multiply matrices. dot is available both as a function in the numpy module and as an instance method of array objects:

In [84]:
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))

219
219

In [85]:
# Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v))

[29 67]
[29 67]

In [86]:
# Matrix / matrix product; both produce the rank 2 array
# [[19 22]
#  [43 50]]
print(x.dot(y))
print(np.dot(x, y))

[[19 22]
[43 50]]
[[19 22]
[43 50]]

Numpy provides many useful functions for performing computations on arrays; one of the most useful is sum:

In [87]:
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]"

10
[4 6]
[3 7]

You can find the full list of mathematical functions provided by numpy in the documentation.

Apart from computing mathematical functions using arrays, we frequently need to reshape or otherwise manipulate data in arrays. The simplest example of this type of operation is transposing a matrix; to transpose a matrix, simply use the T attribute of an array object:

In [88]:
print(x)
print(x.T)

[[1 2]
[3 4]]
[[1 3]
[2 4]]

In [89]:
v = np.array([[1,2,3]])
print(v)
print(v.T)

[[1 2 3]]
[[1]
[2]
[3]]

Broadcasting is a powerful mechanism that allows numpy to work with arrays of different shapes when performing arithmetic operations. Frequently we have a smaller array and a larger array, and we want to use the smaller array multiple times to perform some operation on the larger array.

For example, suppose that we want to add a constant vector to each row of a matrix. We could do it like this:

In [90]:
# 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

print(y)

[[ 2  2  4]
[ 5  5  7]
[ 8  8 10]
[11 11 13]]

This works; however when the matrix x is very large, computing an explicit loop in Python could be slow. Note that adding the vector v to each row of the matrix x is equivalent to forming a matrix vv by stacking multiple copies of v vertically, then performing elementwise summation of x and vv. We could implement this approach like this:

In [91]:
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]]"

[[1 0 1]
[1 0 1]
[1 0 1]
[1 0 1]]

In [92]:
y = x + vv  # Add x and vv elementwise
print(y)

[[ 2  2  4]
[ 5  5  7]
[ 8  8 10]
[11 11 13]]

Numpy broadcasting allows us to perform this computation without actually creating multiple copies of v. Consider this version, using broadcasting:

In [93]:
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)

[[ 2  2  4]
[ 5  5  7]
[ 8  8 10]
[11 11 13]]

The line y = x + v works even though x has shape (4, 3) and v has shape (3,) due to broadcasting; this line works as if v actually had shape (4, 3), where each row was a copy of v, and the sum was performed elementwise.

In broad casting, smaller array is duplicated along the last mismatched dimension.

Broadcasting two arrays together follows these rules:

1. If the arrays do not have the same rank, prepend the shape of the lower rank array with 1s until both shapes have the same length.

e.g. (2, 3) (3, ) becomes (2, 3) (1, 3)

1. The two arrays are said to be compatible in a dimension if they have the same size in the dimension, or if one of the arrays has size 1 in that dimension.

e.g. (2, 4, 6) * (2, 1, 6) is compatible

1. The arrays can be broadcast together if they are compatible in all dimensions.

e.g. (2, 3) (2, ) becomes (2, 3) (1, 2) => incompatible

1. After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.

e.g. For (2, 4, 6) * (2, 1, 6), the second array behaves as if (2, 4, 6)

1. In any dimension where one array had size 1 and the other array had size greater than 1, the smaller array behaves as if it were copied along that dimension

e.g. For (2, 4, 6) * (2, 1, 6) from the example above, the second array behaves as if (2, 4, 6) by copying itself 4 times along the second axis

If this explanation does not make sense, try reading the explanation from the documentation or this explanation.

Functions that support broadcasting are known as universal functions. You can find the list of all universal functions in the documentation.

Here are some applications of broadcasting:

In [94]:
# 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:

# So basically (3, 1) * (2, ) => (3, 1) * (1, 2)

print(np.reshape(v, (3, 1)) * w)

[[ 4  5]
[ 8 10]
[12 15]]

In [95]:
# 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:

print(x + v)

[[2 4 6]
[5 7 9]]

In [96]:
# 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:

print((x.T + w).T)

[[ 5  6  7]
[ 9 10 11]]

In [97]:
# Another solution is to reshape w to be a row 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)))

[[ 5  6  7]
[ 9 10 11]]

In [98]:
# 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:
print(x * 2)

[[ 2  4  6]
[ 8 10 12]]

Broadcasting typically makes your code more concise and faster, so you should strive to use it where possible.

This brief overview has touched on many of the important things that you need to know about numpy, but is far from complete. Check out the numpy reference to find out much more about numpy.


text:
How do you factor  12a^2 - 24a^3?

Jul 12, 2016

$12 {a}^{2} \left(1 - 2 a\right)$

Explanation:

Let's take out common factors. First, we need to list out the factors.

$12 {a}^{2} = 12 \times a \times a$

$24 {a}^{3} = 24 \times a \times a \times a$

Let look at what we can take out. Well $12$ is a factor of both $12$ and $24$, so let's pull out a $12$. We also see that both terms have at least two $a$s, so let's pull out two $a$s.

$12 {a}^{2} \left(1 - 2 a\right)$ or $12 \left(a\right) \left(a\right) \left(1 - 2 a\right)$

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