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- Python is a great general-purpose programming language,...
- ...and with the help of a libraries such as
`numpy`

,`scipy`

and`matplotlib`

(among many others) it becomes a powerful environment for scientific computing.

Python main characteristics:

- dynamic type system
- interpreted (actually: compiled to bytecode,
`*.pyc`

files) - multi-paradigm: imperative, procedural, object-oriented, (functional),
*literate*; do whatever you want **indentation is important!**

I assume that you have some programming experience:

- Java (static type system, compiled, object-oriented, verbose)
- C/C++ (static type system, compiled, multi-paradigm, low-level)
- Matlab? R?

I also expect that many of you will have some experience with Python and numpy; for the rest of you, this section will serve as a quick crash course both on the Python programming language and on the use of Python for scientific computing.

Some of you may have previous knowledge in Matlab, in which case we also recommend the numpy for Matlab users page (https://docs.scipy.org/doc/numpy-dev/user/numpy-for-matlab-users.html).

- Basic Python: Basic data types, containers, loops, functions and classes.
`numpy`

: arrays, array indexing, datatypes, array math and broadcasting.`matplotlib`

: plotting, subplots, images.`Jupyter`

and`IPython`

: Creating notebooks and typical workflows.

- Python is a high-level, dynamically typed multiparadigm programming language.
- Python code is often said to be almost like pseudocode, since it allows you to express very powerful ideas in very few lines of code while being very readable.

Required Python packages (libraries) to run this notebook:

`Python`

`Jupyter`

/`IPython`

- interactive Python shell`numpy`

- linear algebra library`matplotlib`

- plotting library

Scientific Python Distributions (available for almost every platform):

As an example, here is an implementation of the classic *quicksort algorithm* in Python:

In [1]:

```
def quicksort(arr):
if len(arr) <= 1:
return arr
pivot = arr[int(len(arr) / 2)]
left = [x for x in arr if x < pivot]
middle = [x for x in arr if x == pivot]
right = [x for x in arr if x > pivot]
return quicksort(left) + middle + quicksort(right)
```

In [2]:

```
quicksort([3,6,8,10,1,2,1])
```

Out[2]:

- There are currently two different supported versions of Python, 2.x and 3.x.
- Python 3.0 introduced many backwards-incompatible changes to the language, so code written for 2.x may not work under 3.x and vice versa.
- In our class we will be using Python 3.x.

**Note:** You can check your Python version at the command line by running `python --version`

.

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

In [3]:

```
x = 3
print(x, type(x))
```

In [4]:

```
print(x + 1) # Addition;
print(x - 1) # Subtraction;
print(x * 2) # Multiplication;
print(x ** 2) # Exponentiation.
```

Autoincrements:

In [5]:

```
x += 1
print(x) # Prints "4"
x *= 2
print(x) # Prints "8"
```

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.

Real numbers (`float`

)

In [6]:

```
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"
```

Python implements all of the usual operators for Boolean algebra, but uses English words rather than symbols (`&&`

, `||`

, `!`

, etc.):

In [7]:

```
t, f, aa, bb = True, False, True, False # Hmm... yes you can do this in Python
print(t, f, type(t))
```

Now we let's look at the operations:

In [8]:

```
print(t and f) # Logical AND;
print(t or f) # Logical OR;
print(not t) # Logical NOT;
print(t != f) # Logical XOR;
```

- In programming we invariably need some concept of
*conditions*to allow a program to react differently to different situations. - Python uses a combination of
*Boolean*variables, which evaluate to either`True`

or`False`

, and`if`

statements.

In [5]:

```
day = "Sunday"
if day == 'Sunday':
print('Sleep!!!')
else:
print('Go to work')
```

There is no `switch`

or `case`

statements

In [10]:

```
if day == 'Monday':
print('Week end is over!')
elif day == 'Sunday' or day =='Saturday':
print('Sleep!')
else:
print('Meeeh')
```

In [11]:

```
var_1 = 234
if var_1:
print('Do something with', var_1)
else:
print('Nothing to do')
```

In [12]:

```
1 == 2
```

Out[12]:

In [13]:

```
50 == 2*25
```

Out[13]:

There is another boolean operator `is`

, that tests whether two objects are **the same type**:

In [14]:

```
1 is 1
```

Out[14]:

But not...

In [15]:

```
1 is int
```

Out[15]:

In [16]:

```
print(type(1), type(int))
```

In [17]:

```
1 is 1.0
```

Out[17]:

String literals can use single quotes (`''`

) or or double quotes(`""`

); it does not matter.

In [18]:

```
hello = 'hello'
world = "world"
print(hello, len(hello))
```

In [19]:

```
hw = hello + ' ' + world # String concatenation
print(hw)
```

String formatting

In [20]:

```
hw12 = '%s %s! your number is: %d' % (hello, world, 12) # sprintf style string formatting
print(hw12)
```

**Note:** Checkout https://docs.python.org/3/library/string.html#formatspec for string formatting specs.

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

In [21]:

```
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"
```

Again, you can find a list of all string methods in the documentation.

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

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

In [22]:

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

In [23]:

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

In [24]:

```
xs.append('bar') # Add a new element to the end of the list
print(xs)
```

In [25]:

```
xs = xs + ['thing1', 'thing2'] # Adding lists (the += op works too)
print(xs)
```

In [26]:

```
x = xs.pop() # Remove and return the last element of the list
print(x, xs)
```

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

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

In [27]:

```
nums = list(range(5)) # range is a built-in function (more on this later)
print(nums)
```

In [28]:

```
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]"
```

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

In [29]:

```
animals = ['cat', 'dog', 'monkey']
for animal in animals:
aa = animal + ' :)'
print(aa)
```

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

function:

In [30]:

```
animals = ['cat', 'dog', 'monkey']
for idx, animal in enumerate(animals):
print('Item number %d is a %s' % (idx + 1, animal))
```

In Python we have only a `while`

loop.

In [31]:

```
i = 0
while i < 3:
print(i)
i += 1
```

When programming, frequently we want to transform one type of data into another.

For example, consider the following code that computes square numbers:

In [32]:

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

You can make this code simpler using a **list comprehension**:

In [33]:

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

List comprehensions can also contain conditions:

In [34]:

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

A dictionary stores (key, value) pairs, similar to a `Map`

in Java or an object in Javascript. You can use it like this:

In [35]:

```
d = {'cat': 'cute', 'dog': 'furry'} # Create a new dictionary with some data
```

In [36]:

```
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"
```

In [37]:

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

Handling not found keys:

In [38]:

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

In [39]:

```
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"
```

In [40]:

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

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

In [7]:

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

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

In [42]:

```
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)
```

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

In [43]:

```
animals = {'cat', 'dog'}
print(animals)
```

In [44]:

```
print('cat' in animals) # Check if an element is in a set; prints "True"
print('fish' in animals) # prints "False"
```

In [45]:

```
animals.add('fish') # Add an element to a set
print('fish' in animals)
print(len(animals)) # Number of elements in a set;
```

In [46]:

```
animals.add('cat') # Adding an element that is already in the set does nothing
print(animals, len(animals))
animals.remove('cat') # Remove an element from a set
print(animals,len(animals))
```

- 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 [47]:

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

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

In [48]:

```
from math import sqrt
print('Set:', {int(sqrt(x)) for x in range(30)})
print('List:', [int(sqrt(x)) for x in range(30)])
```

- An (
**immutable**) ordered list of values; - 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,

Here is a trivial example:

In [9]:

```
t = (5, 6)
print(t, type(t))
```

In [50]:

```
d = {(x, x + 1): x for x in range(10)} # Create a dictionary with tuple keys
print(d)
```

In [51]:

```
print(d[t])
print(d[(1, 2)])
```

In [52]:

```
t[0] = 1 # Produces TypeError 'tuple' object does not support item assignment
```

Python functions are defined using the `def`

keyword. For example:

In [53]:

```
def sign(x):
if x > 0:
return 'positive'
elif x < 0:
return 'negative'
else:
return 'zero'
```

In [54]:

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

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

In [55]:

```
def hello(name, loud=False):
if loud:
print('HELLO, %s' % name.upper())
else:
print('Hello, %s!' % name)
```

In [56]:

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

In [57]:

```
func = lambda x: x**x + 2
print(type(func))
```

In [58]:

```
func(3)
```

Out[58]:

In [59]:

```
x = [1, 5, 3, 2, 7, 8, 3, 6]
print([func(elem) for elem in x])
list(map(lambda x: x**x + 2, x)) # note the map() function
```

Out[59]:

- The syntax for defining classes in Python is straightforward.
- Remember to include
`self`

as the first parameter of the class methods.

In [20]:

```
class Greeter():
# Constructor
def __init__(self, name):
self.name = name # Create an instance variable
# Instance method
def greet(self, loud=False):
if loud:
print('HELLO, %s!' % self.name.upper())
else:
print('Hello, %s' % self.name)
```

In [23]:

```
g = Greeter('Fred') # Construct an instance of the Greeter class
g.greet() # Call an instance method; prints "Hello, Fred"
g.greet(loud=True) # Call an instance method; prints "HELLO, FRED!"
```

For more information check the Python docs

As in many other modern languages Python can handle exceptions.

In [25]:

```
def my_func(a):
return a * a
```

In [26]:

```
print(my_func(2))
```

In [27]:

```
my_func('AAAAA')
```

In [64]:

```
try:
print(my_func("A"))
except TypeError as e:
print('Caught exception:', e)
```

`import`

statement¶- We have seen already the
`import`

statement in action. - Python has a huge number of libraries included with the distribution.
- Most of these variables and functions are not accessible from a normal Python interactive session.
- Instead, you have to import them.

- A module allows you to logically organize your Python code;
groups related code into a module makes the code easier to understand and use.

A module is a file consisting of Python code, can define functions, classes and variables.

- A module can also include runnable code.

**Note:** A module is also a Python object with arbitrarily named attributes that you can bind and reference.

Importing whole modules with an additional short name:

In [65]:

```
import scipy.spatial.distance
```

In [66]:

```
scipy.spatial.distance.euclidean((1,1), (2,2))
```

Out[66]:

In [67]:

```
import scipy.spatial.distance as dists
```

In [68]:

```
dists.euclidean((1,1), (2,2))
```

Out[68]:

For example, there is a `math`

module containing many useful functions. To access, say, the square root function, you can

In [69]:

```
from math import sqrt
```

and then

In [70]:

```
sqrt(81)
```

Out[70]:

- Iterators and generators.
- map, reduce, zip
- Decorators.
- Installing packages pip/conda/etc.

`numpy`

: scientific computing with Python¶- 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.

To use `numpy`

, we first need to `import numpy`

package:

In [71]:

```
import numpy as np
```

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.
- useful linear algebra, Fourier transform, random number capabilities and more.
- Complemented by
`scipy`

.

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

In [72]:

```
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)
```

In [73]:

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

In [74]:

```
print(b.shape)
print(b[0, 0], b[0, 1], b[1, 0])
```

Numpy provides many shortcut functions to create arrays:

In [75]:

```
a = np.zeros((2,2)) # Create an array of all zeros
print(a)
```

In [76]:

```
b = np.ones((1,2)) # Create an array of all ones
print(b)
```

In [77]:

```
c = np.full((2,2), 7)
print(c)
```

In [78]:

```
d = np.eye(2) # Create a 2x2 identity matrix
print(d)
```

In [79]:

```
e = np.random.random((2,2)) # Create an array filled with random values
print(e)
```

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 [80]:

```
# 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]])
a
```

Out[80]:

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)`

:

In [81]:

```
b = a[:2, 1:3]
print(b)
```

A slice of an array is a view into the same data, so modifying it will modify the original array.

In [82]:

```
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])
```

You can also mix integer indexing with slice indexing. However, doing so will yield an array of lower rank than the original array.

**Note:** This is quite different from the way that MATLAB handles array slicing.

In [83]:

```
# 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)
```

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 [84]:

```
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)
```

We can make the same distinction when accessing columns of an array:

In [85]:

```
col_r1 = a[:, 1]
col_r2 = a[:, 1:2]
print(col_r1, col_r1.shape)
print(col_r2, col_r2.shape)
```

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.

Here is an example:

In [86]:

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

An example of integer array indexing. The returned array will have shape `(3,)`

.

In [87]:

```
a[[0, 1, 2], [0, 1, 0]]
```

Out[87]:

The above example of integer array indexing is equivalent to this:

In [88]:

```
np.array([a[0, 0], a[1, 1], a[2, 0]])
```

Out[88]:

When using integer array indexing, you can reuse the same element from the source array:

In [89]:

```
a[[0, 0], [1, 1]]
```

Out[89]:

Equivalent to the previous integer array indexing example

In [90]:

```
np.array([a[0, 1], a[0, 1]])
```

Out[90]:

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

Create a new array from which we will select elements

In [91]:

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

Create an array of indices

In [92]:

```
b = np.array([0, 2, 0, 1])
```

Select one element from each row of a using the indices in `b`

.

In [93]:

```
print(a[np.arange(4), b]) # Prints "[ 1 6 7 11]"
```

Mutate one element from each row of a using the indices in `b`

.

In [94]:

```
a[np.arange(4), b] += 10
print(a)
```

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 [95]:

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

Find the elements of a that are bigger than 2:

In [96]:

```
bool_idx = (a > 2)
print(bool_idx)
```

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.

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`

.

In [97]:

```
print(a[bool_idx])
```

We can do all of the above in a single concise statement:

In [98]:

```
print(a[a > 2])
```

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.

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 [30]:

```
import numpy as np
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.float32) # Force a particular datatype
print(x.dtype, y.dtype, z.dtype)
```

You can read all about numpy datatypes in the documentation.

Basic mathematical functions operate elementwise on arrays, and are available both as operator overloads and as functions in the numpy module:

In [100]:

```
x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)
```

In [101]:

```
print(x + y)
```

In [102]:

```
print(np.add(x, y))
```

Elementwise difference

In [103]:

```
print(x - y)
print(np.subtract(x, y))
```

Elementwise product

In [104]:

```
print(x * y)
print(np.multiply(x, y))
```

Elementwise division - both produce the same result.

In [105]:

```
print(x / y)
print(np.divide(x, y))
```

Elementwise square root;

In [106]:

```
print(np.sqrt(x))
```

- Unlike MATLAB,
`*`

is elementwise multiplication, not matrix multiplication. - We instead use the
`numpy.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 [107]:

```
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))
```

In [108]:

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

In [109]:

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

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

:

In [110]:

```
x = np.array([[1,2],[3,4]])
print(x)
```

In [111]:

```
np.sum(x) # Compute sum of all elements; prints "10"
```

Out[111]:

In [112]:

```
np.sum(x, axis=0) # Compute sum of each column; prints "[4 6]"
```

Out[112]:

In [113]:

```
np.sum(x, axis=1) # Compute sum of each row; prints "[3 7]"
```

Out[113]:

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 [114]:

```
print(x)
print(x.T)
```

In [115]:

```
v = np.array([1,2,3])
print(v)
print(v.T)
```

Is `numpy`

actually faster than 'plain' Python?

In [116]:

```
import random
import numpy as np
```

In [117]:

```
%timeit sum([random.random() for _ in range(random.randint(10000,10100))])
```

In [118]:

```
%timeit np.sum(np.random.random(random.randint(10000,10100)))
```

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 [119]:

```
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
x
```

Out[119]:

In [120]:

```
v = np.array([1, 0, 1])
v
```

Out[120]:

In [121]:

```
y = np.empty_like(x) # Creates 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
```

In [122]:

```
y
```

Out[122]:

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 [123]:

```
vv = np.tile(v, (4, 1)) # Stack 4 copies of v on top of each other
vv
```

Out[123]:

In [124]:

```
y = x + vv # Add x and vv elementwise
y
```

Out[124]:

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

In [125]:

```
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
y
```

Out[125]:

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.

Broadcasting two arrays together follows these rules:

- 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.
- 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.
- The arrays can be broadcast together if they are compatible in all dimensions.
- After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.
- In any dimension where one array had size 1 and the other array had size greater than 1, the first array behaves as if it were copied along that dimension

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.

**Compute outer product of vectors**

In [126]:

```
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

- reshape
`v`

to be a column vector of shape`(3, 1)`

, and - broadcast it against
`w`

to yield an output of shape`(3, 2)`

, which is the outer product of`v`

and`w`

:

In [127]:

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

**Add a vector to each row of a matrix**

In [128]:

```
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)`

, producing the following matrix:

In [129]:

```
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.

This gives the following matrix:

In [130]:

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

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.

In [131]:

```
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`()`

;- they can be broadcasted together to shape
`(2, 3)`

, - producing the following array:

In [132]:

```
print(x * 2)
```

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

So far, we 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.
- If you are already familiar with MATLAB, you might find this tutorial useful to get started with
`numpy`

. - Nicolas P. Rougier's Numpy tutorial
.*highly recommended*

- Matplotlib is a plotting library.
- Provides a plotting system similar -and in my opinion better- to that of MATLAB.

In [133]:

```
import matplotlib.pyplot as plt
```

By running this special IPython notebook *magic* command, we will be displaying plots inline and with appropiate resolutions for *retina* displays. Check the docs for more options.

In [134]:

```
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
```

The most important function in `matplotlib`

is `plot`

, which allows you to plot 2D data.
Here is a simple example:

In [135]:

```
# Compute the x and y coordinates for points on a sine curve
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)
# Plot the points using matplotlib
plt.plot(x, y_sin);
```

With just a little bit of extra work we can easily plot multiple lines at once, and add a title, legend, and axis labels:

In [136]:

```
y_cos = np.cos(x)
plt.plot(x, y_sin)
plt.plot(x, y_cos)
plt.xlabel('x axis label')
plt.ylabel('y axis label')
plt.title('Sine and Cosine')
plt.legend(['Sine', 'Cosine']);
```

You can plot different things in the same figure using the subplot function. Here is an example:

In [137]:

```
plt.subplot(2, 1, 1)
plt.plot(x, y_sin)
plt.title('Sine')
plt.subplot(2, 1, 2)
plt.plot(x, y_cos)
plt.title('Cosine')
plt.tight_layout()
```

Check out Nicolas P. Rougier's Matplotlib tutorial.

- Install Python on your own computer (remember to install version 3.x)
- I recommend you to use Anaconda

- Retrive this notebook from the class GitHub repository: http://github.com/lmarti/machine-learning.
- If you are not familiar with the Git version control system you can use GitHub desktop.

- Try running this notebook.

- Official Python 3 Docs are here.
- IPython tutorial.
- Learn Python The Hard Way
- Dive Into Python, in particular if you're interested in Python 3.
- Invent With Python, probably best for kids.
- Python Functional Programming HOWTO
- The Structure and Interpretation of Computer Programs, written in Scheme, a Lisp dialect, but one of the best books on computer programming ever written.
- Generator Tricks for Systems Programmers Beazley's slides on just what generators can do for you.
- Python Module of the Week is a series going through in-depth analysis of the Python standard library in a very easy to understand way.

- Rob Johansson's excellent notebooks, including Scientific Computing with Python and Computational Quantum Physics with QuTiP lectures;
- XKCD style graphs in matplotlib;
- A collection of Notebooks for using IPython effectively
- A gallery of interesting IPython Notebooks
- Cross-disciplinary computational analysis IPython Notebooks From Hadoop World 2012
- Quantites Units in Python.

- Part of this material has been adapted from the
`CS231n`

Python tutorial by Justin Johnson (http://cs231n.github.io/python-numpy-tutorial/).

In [138]:

```
%load_ext version_information
%version_information scipy, numpy, matplotlib
```

Out[138]:

In [1]:

```
# this code is here for cosmetic reasons
from IPython.core.display import HTML
from urllib.request import urlopen
HTML(urlopen('https://raw.githubusercontent.com/lmarti/jupyter_custom/master/custom.include').read().decode('utf-8'))
```

Out[1]: