def quicksort(arr, depth=0, pos="middle", verbose=False):
    if len(arr) <= 1:
        if verbose:
            print("pos:", pos)
            print("depth:", depth)
            print("###")
        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]
    if verbose:
        print("pivot:", pivot)
        print("left:", left)
        print("middle:", middle)
        print("right:", right)
        print("pos:", pos)
        print("depth:", depth)
        print("###")
    return quicksort(left, depth+1, "left") + middle + quicksort(right, depth+1, "right")

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

"""
Explanation: CS228 Python Tutorial
Adapted by Volodymyr Kuleshov and Isaac Caswell from the CS231n Python tutorial by Justin Johnson (http://cs231n.github.io/python-numpy-tutorial/).
Commented and Ported to Python 3 by Jonathan DEKHTIAR
Introduction
Python is a great general-purpose programming language on its own, but with the help of a few popular libraries (numpy, scipy, matplotlib) it becomes a powerful environment for scientific computing.
We 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).
In this tutorial, we will cover:
* Basic Python: Basic data types (Containers, Lists, Dictionaries, Sets, Tuples), Functions, Classes
* Numpy: Arrays, Array indexing, Datatypes, Array math, Broadcasting
* Matplotlib: Plotting, Subplots, Images
* IPython: Creating notebooks, Typical workflows
Basics of Python
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. As an example, here is an implementation of the classic quicksort algorithm in Python:
End of explanation
"""


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

print ("Addition:", x + 1)   # Addition;
print ("Subtraction:", x - 1)   # Subtraction;
print ("Multiplication:", x * 2)   # Multiplication;
print ("Exponentiation:", x ** 2)  # Exponentiation;

x += 1
print ("Incrementing:", x)  # Prints "4"
x *= 2
print ("Exponentiating:", x)  # Prints "8"

y = 2.5
print ("Type of y:", type(y)) # Prints "<type 'float'>"
print ("Many values:", y, y + 1, y * 2, y ** 2) # Prints "2.5 3.5 5.0 6.25"

"""
Explanation: Python versions
There are currently two different supported versions of Python, 2.7 and 3.4. Somewhat confusingly, Python 3.0 introduced many backwards-incompatible changes to the language, so code written for 2.7 may not work under 3.4 and vice versa. For this class all code will use Python 2.7.
You can check your Python version at the command line by running python --version.
Basic data types
Numbers
Integers and floats work as you would expect from other languages:
End of explanation
"""


t, f = True, False
print (type(t)) # Prints "<type 'bool'>"

print ("True AND False:", t and f) # Logical AND;
print ("True OR False:", t or f)  # Logical OR;
print ("NOT True:", not t)   # Logical NOT;
print ("True XOR False:", t != f)  # Logical XOR;

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


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

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"

"""
Explanation: Strings
End of explanation
"""


s = "hello"
print ("Capitalized String:", s.capitalize())  # Capitalize a string; prints "Hello"
print ("Uppercase String:", s.upper())       # Convert a string to uppercase; prints "HELLO"
print ("Right justified String with padding of '7':", s.rjust(7))      # Right-justify a string, padding with spaces; prints "  hello"
print ("Centered String with padding of '7':", s.center(7))     # Center a string, padding with spaces; prints " hello "
print ("Replace 'l' with '(ell)':", s.replace('l', '(ell)'))  # Replace all instances of one substring with another;
                               # prints "he(ell)(ell)o"
print ("Stripped String:", '  world '.strip())  # Strip leading and trailing whitespace; prints "world"

"""
Explanation: String objects have a bunch of useful methods; for example:
End of explanation
"""


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"

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

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

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

"""
Explanation: 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:
End of explanation
"""


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, 8, 4]"

"""
Explanation: 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:
End of explanation
"""


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

"""
Explanation: Loops
You can loop over the elements of a list like this:
End of explanation
"""


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

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


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

"""
Explanation: 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:
End of explanation
"""


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

"""
Explanation: You can make this code simpler using a list comprehension:
End of explanation
"""


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

"""
Explanation: List comprehensions can also contain conditions:
End of explanation
"""


d = {'cat': 'cute', 'dog': 'furry'}  # Create a new dictionary with some data
print ("Value of the dictionary for the key 'cat':", d['cat'])       # Get an entry from a dictionary; prints "cute"
print ("Is 'cat' is the dictionary d:", 'cat' in d)     # Check if a dictionary has a given key; prints "True"

d['fish'] = 'wet'    # Set an entry in a dictionary
print ("Value of the dictionary for the key 'fish':", d['fish'])      # Prints "wet"

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

print ("Get 'monkey' value or default:", d.get('monkey', 'N/A'))  # Get an element with a default; prints "N/A"
print ("Get 'fish' value or default:", d.get('fish', 'N/A'))      # Get an element with a default; prints "wet"

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

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


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

"""
Explanation: You can find all you need to know about dictionaries in the documentation.
It is easy to iterate over the keys in a dictionary:
End of explanation
"""


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

"""
Explanation: If you want access to keys and their corresponding values, use the iteritems method:
End of explanation
"""


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)

"""
Explanation: Dictionary comprehensions: These are similar to list comprehensions, but allow you to easily construct dictionaries. 
For example:
End of explanation
"""


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

animals.add('fish')        # Add an element to a set
print ("Is 'fish' in the set:", 'fish' in animals)
print ("What is the length of the set:", len(animals))       # Number of elements in a set;

animals.add('cat')       # Adding an element that is already in the set does nothing
print ("What is the length of the set:", len(animals))   

animals.remove('cat')    # Remove an element from a set
print ("What is the length of the set:", len(animals))

"""
Explanation: Sets
A set is an unordered collection of distinct elements. As a simple example, consider the following:
End of explanation
"""


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

"""
Explanation: 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:
End of explanation
"""


from math import sqrt
set_comprehension = {int(sqrt(x)) for x in range(30)}

print (set_comprehension)
print (type(set_comprehension))

"""
Explanation: Set comprehensions: Like lists and dictionaries, we can easily construct sets using set comprehensions:
End of explanation
"""


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)])

print ("Access the 1st value of Tuple:", t[0])
print ("Access the 2nd value of Tuple:", t[1])

t[0] = 1 # This does NOT work !

t = (1, t[1]) # This DOES work !
print (t)

"""
Explanation: 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:
End of explanation
"""


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))

"""
Explanation: Functions
Python functions are defined using the def keyword. For example:
End of explanation
"""


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

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

"""
Explanation: We will often define functions to take optional keyword arguments, like this:
End of explanation
"""


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)

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!"

"""
Explanation: Classes
The syntax for defining classes in Python is straightforward:
End of explanation
"""


import numpy as np
import warnings
warnings.filterwarnings('ignore') # To remove warnings about "deprecated" or "future" features

"""
Explanation: 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:
End of explanation
"""


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)

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

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

"""
Explanation: 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.

Why using Numpy Array over Python Lists ?
NumPy's arrays are more compact than Python lists -- a list of lists as you describe, in Python, would take at least 20 MB or so, while a NumPy 3D array with single-precision floats in the cells would fit in 4 MB. Access in reading and writing items is also faster with NumPy.
Maybe you don't care that much for just a million cells, but you definitely would for a billion cells -- neither approach would fit in a 32-bit architecture, but with 64-bit builds NumPy would get away with 4 GB or so, Python alone would need at least about 12 GB (lots of pointers which double in size) -- a much costlier piece of hardware!
The difference is mostly due to "indirectness" -- a Python list is an array of pointers to Python objects, at least 4 bytes per pointer plus 16 bytes for even the smallest Python object (4 for type pointer, 4 for reference count, 4 for value -- and the memory allocators rounds up to 16). A NumPy array is an array of uniform values -- single-precision numbers takes 4 bytes each, double-precision ones, 8 bytes. Less flexible, but you pay substantially for the flexibility of standard Python lists!
Author: Alex Martelli
Source: StackOverFlow

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


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

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

c = np.full((2,2), 7) # Create a constant array
print (c)

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

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

"""
Explanation: Numpy also provides many functions to create arrays:
End of explanation
"""


# 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)

"""
Explanation: 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:
End of explanation
"""


print ("Original Matrix before modification:", a[0, 1])

b[0, 0] = 77    # b[0, 0] is the same piece of data as a[0, 1]
print ("Original Matrix after modification:", a[0, 1])

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


# 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)

"""
Explanation: 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:
End of explanation
"""


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 ("Rank 1 access of the 2nd row:", row_r1, row_r1.shape) 
print ("Rank 2 access of the 2nd row:", row_r2, row_r2.shape)
print ("Rank 2 access of the 2nd row:", row_r3, row_r3.shape)

# We can make the same distinction when accessing columns of an array:
col_r1 = a[:, 1]
col_r2 = a[:, 1:2]
print ("Rank 1 access of the 2nd column:", col_r1, col_r1.shape)
print ()
print ("Rank 2 access of the 2nd column:\n", col_r2, col_r2.shape)

"""
Explanation: 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:
End of explanation
"""


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]])

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

# 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]]))

# 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)

# Create an array of indices
b = np.array([0, 2, 0, 1])
b_range = np.arange(4)
print ("b_range:", b_range)

# Select one element from each row of a using the indices in b
print ("Selected Matrix Values:", a[b_range, b])  # Prints "[ 1  6  7 11]"

# Mutate one element from each row of a using the indices in b
a[b_range, b] += 10 # Only the selected values are modified in the "a" matrix.
print ("Modified 'a' Matrix:\n", a)

"""
Explanation: 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:
End of explanation
"""


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)

# 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])

"""
Explanation: 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:
End of explanation
"""


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)

"""
Explanation: 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:
End of explanation
"""


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

# Elementwise difference; both produce the array
print (x - y)
print ()
print (np.subtract(x, y))

# Elementwise product; both produce the array
print (x * y)
print ()
print (np.multiply(x, y))

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

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

"""
Explanation: You can read all about numpy datatypes in the documentation.
Array math
Basic mathematical functions operate elementwise on arrays, and are available both as operator overloads and as functions in the numpy module:
End of explanation
"""


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.w 'dot' product:", v.dot(w))
print ("numpy 'dot' product (v,w):", np.dot(v, w))

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

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

"""
Explanation: 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:
End of explanation
"""


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

print ("Sum of all element:", np.sum(x))  # Compute sum of all elements; prints "10"
print ("Sum of each column:", np.sum(x, axis=0))  # Compute sum of each column; prints "[4 6]"
print ("Sum of each row:", np.sum(x, axis=1))  # Compute sum of each row; prints "[3 7]"

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


print ("Matrix x:\n", x)
print ()
print ("Matrix x transposed:\n", x.T)

v = np.array([[1,2,3]])
print ("Matrix v:\n", v)
print ()
print ("Matrix v transposed:\n", v.T)

"""
Explanation: 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:
End of explanation
"""



# 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)

"""
Explanation: Broadcasting
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:
End of explanation
"""


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)

# 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)

"""
Explanation: 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:
End of explanation
"""


# 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:

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:

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:

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.
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:
print (x * 2)

"""
Explanation: 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.
Here are some applications of broadcasting:
End of explanation
"""


import matplotlib.pyplot as plt

"""
Explanation: 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.
Matplotlib
Matplotlib is a plotting library. In this section give a brief introduction to the matplotlib.pyplot module, which provides a plotting system similar to that of MATLAB.
End of explanation
"""


%matplotlib inline

"""
Explanation: By running this special iPython command, we will be displaying plots inline:
End of explanation
"""


# Compute the x and y coordinates for points on a sine curve
x = np.arange(0, 3 * np.pi, 0.1)
y = np.sin(x)

# Plot the points using matplotlib
plt.plot(x, y)

"""
Explanation: Plotting
The most important function in matplotlib is plot, which allows you to plot 2D data. Here is a simple example:
End of explanation
"""


y_sin = np.sin(x)
y_cos = np.cos(x)

# Plot the points using matplotlib
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'])

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


# Compute the x and y coordinates for points on sine and cosine curves
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)
y_cos = np.cos(x)

# Set up a subplot grid that has height 2 and width 1,
# and set the first such subplot as active.
plt.subplot(2, 1, 1)

# Make the first plot
plt.plot(x, y_sin)
plt.title('Sine')

# Set the second subplot as active, and make the second plot.
plt.subplot(2, 1, 2)
plt.plot(x, y_cos)
plt.title('Cosine')

# Show the figure.
plt.show()

"""
Explanation: Subplots
You can plot different things in the same figure using the subplot function. Here is an example:
End of explanation
"""


