Homework_00 - Jupyter and Python
Contents
Homework_00 - Jupyter and Python¶
We will be making heavy use of the Python library called NumPy. It is not included by default, so we first need to import it. Go ahead and run the following cell:
import numpy as np
Now, we have access to all NumPy functions via the variable np (this is the convention in the Scientific Python community for referring to NumPy). We can take a look at what this variable actually is, and see that it is in fact the numpy module (remember that you will need to have run the cell above before np will be defined!):
np
<module 'numpy' from '/home/ryan/.conda/envs/newwork/lib/python3.9/site-packages/numpy/__init__.py'>
NumPy is incredibly powerful and has many features, but this can be a bit intimidating when you’re first starting to use it. If you are familiar with other scientific computing languages, the following guides may be of use:
NumPy for Matlab Users: https://numpy.org/devdocs/user/numpy-for-matlab-users.html
NumPy for R (and S-Plus) Users: http://mathesaurus.sourceforge.net/r-numpy.html
If not, don’t worry! Here we’ll go over the most common NumPy features.
Arrays and lists¶
The core component of NumPy is the ndarray, which is pronounced like “N-D array” (i.e., 1-D, 2-D, …, N-D). We’ll use both the terms ndarray and “array” interchangeably. For now, we’re going to stick to just 1-D arrays – we’ll get to multidimensional arrays later.
Arrays are very similar to lists. Let’s first review how lists work. Remember that we can create them using square brackets:
mylist = [3, 6, 1, 0, 10, 3]
mylist
[3, 6, 1, 0, 10, 3]
And we can access an element via its index. To get the first element, we use an index of 0:
print("The first element of 'mylist' is: " + str(mylist[0]))
The first element of 'mylist' is: 3
To get the second element, we use an index of 1:
print("The second element of 'mylist' is: " + str(mylist[1]))
The second element of 'mylist' is: 6
And so on.
Arrays work very similarly. The first way to create an array is from an already existing list:
myarray = np.array(mylist) # equivalent to np.array([3, 6, 1, 0, 10, 3])
myarray
array([ 3, 6, 1, 0, 10, 3])
myarray looks different than mylist -- it actually tells you that it's an array. If we take a look at the types of mylist and myarray, we will also see that one is a list and one is an array. Using type can be a very useful way to verify that your variables contain what you want them to contain:
# look at what type mylist is
type(mylist)
list
# look at what type myarray is
type(myarray)
numpy.ndarray
We can get elements from a NumPy array in exactly the same way as we get elements from a list:
print("The first element of 'myarray' is: " + str(myarray[0]))
print("The second element of 'myarray' is: " + str(myarray[1]))
The first element of 'myarray' is: 3
The second element of 'myarray' is: 6
Array slicing¶
myarray[a:b:c], where a, b, and c are all optional (though you have to specify at least one). a is the index of the beginning of the slice, b is the index of the end of the slice (exclusive), and c is the step size.
Note that the exclusive slice indexing described above is different than some other languages you may be familiar with, like Matlab and R. myarray[1:2] returns only the second elment in myarray in Python, instead of the first and second element.
First, let’s quickly look at what is in our array and list (defined above), for reference:
print("mylist:", mylist)
print("myarray:", myarray)
mylist: [3, 6, 1, 0, 10, 3]
myarray: [ 3 6 1 0 10 3]
Now, to get all elements except the first:
myarray[1:]
array([ 6, 1, 0, 10, 3])
To get all elements except the last:
myarray[:-1]
array([ 3, 6, 1, 0, 10])
To get all elements except the first and the last:
myarray[1:-1]
array([ 6, 1, 0, 10])
To get every other element of the array (beginning from the first element):
myarray[::2]
array([ 3, 1, 10])
To get every element of the array (beginning from the second element):
myarray[1::2]
array([6, 0, 3])
And to reverse the array:
myarray[::-1]
array([ 3, 10, 0, 1, 6, 3])
Array computations¶
So far, NumPy arrays seem basically the same as regular lists. What’s the big deal about them?
Working with single arrays¶
One advantage of using NumPy arrays over lists is the ability to do a computation over the entire array. For example, if you were using lists and wanted to add one to every element of the list, here’s how you would do it:
mylist = [3, 6, 1, 0, 10, 22]
mylist_plus1 = []
for x in mylist:
mylist_plus1.append(x + 1)
mylist_plus1
[4, 7, 2, 1, 11, 23]
Or, you could use a list comprehension:
mylist = [3, 6, 1, 0, 10, 22]
mylist_plus1 = [x + 1 for x in mylist]
mylist_plus1
[4, 7, 2, 1, 11, 23]
In contrast, adding one to every element of a NumPy array is far simpler:
myarray = np.array([3, 6, 1, 0, 10, 22])
myarray_plus1 = myarray + 1
myarray_plus1
array([ 4, 7, 2, 1, 11, 23])
This won’t work with normal lists. For example, if you ran mylist + 1, you’d get an error like this:
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-19-5b3951a16990> in <module>()
----> 1 mylist + 1
TypeError: can only concatenate list (not "int") to list
We can do the same thing for subtraction, multiplication, etc.:
print("Subtraction: \t" + str(myarray - 2))
print("Multiplication:\t" + str(myarray * 10))
print("Squared: \t" + str(myarray ** 2))
print("Square root: \t" + str(np.sqrt(myarray)))
print("Exponential: \t" + str(np.exp(myarray)))
Subtraction: [ 1 4 -1 -2 8 20]
Multiplication: [ 30 60 10 0 100 220]
Squared: [ 9 36 1 0 100 484]
Square root: [1.73205081 2.44948974 1. 0. 3.16227766 4.69041576]
Exponential: [2.00855369e+01 4.03428793e+02 2.71828183e+00 1.00000000e+00
2.20264658e+04 3.58491285e+09]
Working with multiple arrays¶
We can also easily do these operations for multiple arrays. For example, let’s say we want to add the corresponding elements of two lists together. Here’s how we’d do it with regular lists:
list_a = [1, 2, 3, 4, 5]
list_b = [6, 7, 8, 9, 10]
list_c = [list_a[i] + list_b[i] for i in range(len(list_a))]
list_c
[7, 9, 11, 13, 15]
With NumPy arrays, we just have to add the arrays together:
array_a = np.array(list_a) # equivalent to np.array([1, 2, 3, 4, 5])
array_b = np.array(list_b) # equivalent to np.array([6, 7, 8, 9, 10])
array_c = array_a + array_b
array_c
array([ 7, 9, 11, 13, 15])
list_a + list_b
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Just as when we are working with a single array, we can add, subtract, divide, multiply, etc. several arrays together:
print("Subtraction: \t" + str(array_a - array_b))
print("Multiplication:\t" + str(array_a * array_b))
print("Exponent: \t" + str(array_a ** array_b))
print("Division: \t" + str(array_a / array_b))
Subtraction: [-5 -5 -5 -5 -5]
Multiplication: [ 6 14 24 36 50]
Exponent: [ 1 128 6561 262144 9765625]
Division: [0.16666667 0.28571429 0.375 0.44444444 0.5 ]
Creating and modifying arrays¶
One thing that you can do with lists that you cannot do with NumPy arrays is adding and removing elements. For example, I can create a list and then add elements to it with append:
mylist = []
mylist.append(7)
mylist.append(2)
mylist
[7, 2]
However, you cannot do this with NumPy arrays. If you tried to run the following code, for example:
myarray = np.array([])
myarray.append(7)
You’d get an error like this:
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
<ipython-input-25-0017a7f2667c> in <module>()
1 myarray = np.array([])
----> 2 myarray.append(7)
AttributeError: 'numpy.ndarray' object has no attribute 'append'
There are a few ways to create a new array with a particular size:
np.empty(size)– creates an empty array of sizesizenp.zeros(size)– creates an array of sizesizeand sets all the elements to zeronp.ones(size)– creates an array of sizesizeand sets all the elements to one
So the way that we would create an array like the list above is:
myarray = np.empty(2) # create an array of size 2
myarray[0] = 7
myarray[1] = 2
myarray
array([7., 2.])
np.arange, which will create an array containing a sequence of numbers (it is very similar to the built-in range or xrange functions in Python).
Here are a few examples of using np.arange. Try playing around with them and make sure you understand how it works:
# create an array of numbers from 0 to 3
np.arange(3)
array([0, 1, 2])
# create an array of numbers from 1 to 5
np.arange(1, 5)
array([1, 2, 3, 4])
# create an array of every third number between 2 and 10
np.arange(2, 10, 3)
array([2, 5, 8])
# create an array of numbers between 0.1 and 1.1 spaced by 0.1
np.arange(0.1, 1.1, 0.1)
array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. ])
“Vectorized” computations¶
Another very useful thing about NumPy is that it comes with many so-called “vectorized” operations. A vectorized operation (or computation) works across the entire array. For example, let’s say we want to add together all the numbers in a list. In regular Python, we might do it like this:
mylist = [3, 6, 1, 10, 22]
total = 0
for number in mylist:
total += number
total
42
Using NumPy arrays, we can just use the np.sum function:
# you can also just do np.sum(mylist) -- it converts it to an
# array for you!
myarray = np.array(mylist)
np.sum(myarray)
42
np.prod), mean (np.mean), and variance (np.var). They all act essentially the same way as np.sum -- give the function an array, and it computes the relevant function across all the elements in the array.
Exercise: Euclidean distance (2 points)¶
Recall that the Euclidean distance \(d\) is given by the following equation:
In NumPy, this is a fairly simple computation because we can rely on array computations and the np.sum function to do all the heavy lifting for us.
euclidean_distance below to compute $d(a,b)$, as given by the equation above. Note that you can compute the square root using np.sqrt.
def euclidean_distance(a, b):
"""Computes the Euclidean distance between a and b.
Hint: your solution can be done in a single line of code!
Parameters
----------
a, b : numpy arrays or scalars with the same size
Returns
-------
the Euclidean distance between a and b
"""
# YOUR CODE HERE
raise NotImplementedError()
euclidean_distance), and then run the cell below to check your answer. If you make changes to the cell with your answer, you will need to first re-run that cell, and then re-run the test cell to check your answer again.Creating multidimensional arrays¶
Previously, we saw that functions like np.zeros or np.ones could be used to create a 1-D array. We can also use them to create N-D arrays. Rather than passing an integer as the first argument, we pass a list or tuple with the shape of the array that we want. For example, to create a \(3\times 4\) array of zeros:
arr = np.zeros((3, 4))
arr
array([[0., 0., 0., 0.],
[0., 0., 0., 0.],
[0., 0., 0., 0.]])
shape attribute:
arr.shape
(3, 4)
Note that for 1-D arrays, the shape returned by the shape attribute is still a tuple, even though it only has a length of one:
np.zeros(3).shape
(3,)
This also means that we can create 1-D arrays by passing a length one tuple. Thus, the following two arrays are identical:
np.zeros((3,))
array([0., 0., 0.])
np.zeros(3)
array([0., 0., 0.])
(3, 4), we must use np.zeros((3, 4)). The following will not work:np.zeros(3, 4)
It will give an error like this:
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-39-06beb765944a> in <module>()
----> 1 np.zeros(3, 4)
TypeError: data type not understood
This is because the second argument to np.zeros is the data type, so numpy thinks you are trying to create an array of zeros with shape (3,) and datatype 4. It (understandably) doesn’t know what you mean by a datatype of 4, and so throws an error.
size attribute:
arr = np.zeros((3, 4))
arr.size
12
We can also create arrays and then reshape them into any shape, provided the new array has the same size as the old array:
arr = np.arange(32).reshape((8, 4))
arr
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23],
[24, 25, 26, 27],
[28, 29, 30, 31]])
Accessing and modifying multidimensional array elements¶
To access or set individual elements of the array, we can index with a sequence of numbers:
# set the 3rd element in the 1st row to 0
arr[0, 2] = 0
arr
array([[ 0, 1, 0, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23],
[24, 25, 26, 27],
[28, 29, 30, 31]])
We can also access the element on it’s own, without having the equals sign and the stuff to the right of it:
arr[0, 2]
0
We frequently will want to access ranges of elements. In NumPy, the first dimension (or axis) corresponds to the rows of the array, and the second axis corresponds to the columns. For example, to look at the first row of the array:
# the first row
arr[0]
array([0, 1, 0, 3])
To look at columns, we use the following syntax:
# the second column
arr[:, 1]
array([ 1, 5, 9, 13, 17, 21, 25, 29])
The colon in the first position essentially means “select from every row”. So, we can interpret arr[:, 1] as meaning “take the second element of every row”, or simply “take the second column”.
Using this syntax, we can select whole regions of an array. For example:
# select a rectangular region from the array
arr[2:5, 1:3]
array([[ 9, 10],
[13, 14],
[17, 18]])
For example, if I want to create a second array that mutliples every other value in arr by two, the following code will work but will have unexpected consequences:
arr = np.arange(10)
arr2 = arr
arr2[::2] = arr2[::2] * 2
print("arr: " + str(arr))
print("arr2: " + str(arr2))
arr: [ 0 1 4 3 8 5 12 7 16 9]
arr2: [ 0 1 4 3 8 5 12 7 16 9]
Note that arr and arr2 both have the same values! This is because the line arr2 = arr doesn’t actually copy the array: it just makes another pointer to the same object. To truly copy the array, we need to use the .copy() method:
arr = np.arange(10)
arr2 = arr.copy()
arr2[::2] = arr2[::2] * 2
print("arr: " + str(arr))
print("arr2: " + str(arr2))
arr: [0 1 2 3 4 5 6 7 8 9]
arr2: [ 0 1 4 3 8 5 12 7 16 9]
Problem 2: Border¶
Write a function to create a 2D array of arbitrary shape. This array should have all zero values, except for the elements around the border (i.e., the first and last rows, and the first and last columns), which should have a value of one.
def border(n, m):
"""Creates an array with shape (n, m) that is all zeros
except for the border (i.e., the first and last rows and
columns), which should be filled with ones.
Hint: you should be able to do this in three lines
(including the return statement)
Parameters
----------
n, m: int
Number of rows and number of columns
Returns
-------
numpy array with shape (n, m)
"""
# YOUR CODE HERE
# Show work
Problem 3¶
Below a 2D array, A88 is created that reshapes the integers 1 through 64 into
an \(8\times8\) array. Create a new 1-D array that only contains the last
column of a8 e.g. [8, 16, 24, 32, 40, 48, 56, 64].
A88 = np.arange(1,65).reshape(8,8)
# a8 = your work
Vector algebra with Arrays¶
You can represent physics vectors with 1D arrays. If you have a chosen basis set, you can save the components of the vector. Consider the following example, a force, \(\mathbf{F}\), is applied to a an object and it moves one meter along the x-axis. The force has three components, \(\mathbf{F} = 2\hat{i}+3\hat{j}+0\hat{k}~N\).
You can determine the work done by the force using np.dot as such
F = np.array([2, 3, 0])
dx = np.array([1, 0, 0])
W = np.dot(F,dx)
print('work done by F is {} N-m'.format(W))
work done by F is 2 N-m
You can also animate the motion of the object with the applied force. Here, the motion is assumed to travel at a constant speed from \(x = 0 - 1~m\).
import matplotlib.pyplot as plt
from matplotlib import animation, rc
from IPython.display import HTML
fig, ax = plt.subplots()
ax.set_xlim(( 0, 3))
ax.set_ylim((-0.1, 5))
line1, = ax.plot([], [], 's')
line2, = ax.plot([], [], 'r-v')
def init():
line.set_data([], [])
line2.set_data([],[])
return (line1, line2)
def animate(i):
x = np.linspace(0, 1, 100)
line1.set_data(x[i], 0)
line2.set_data([x[i], x[i] + F[0]], [0, 0 + F[1]])
#plt.quiver(x[i],0,F[0],F[1])
return (line1, line2)
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=100, interval=20, blit=True)
HTML(anim.to_html5_video())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Input In [53], in <cell line: 26>()
23 #plt.quiver(x[i],0,F[0],F[1])
24 return (line1, line2)
---> 26 anim = animation.FuncAnimation(fig, animate, init_func=init,
27 frames=100, interval=20, blit=True)
29 HTML(anim.to_html5_video())
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:1634, in FuncAnimation.__init__(self, fig, func, frames, init_func, fargs, save_count, cache_frame_data, **kwargs)
1631 # Needs to be initialized so the draw functions work without checking
1632 self._save_seq = []
-> 1634 super().__init__(fig, **kwargs)
1636 # Need to reset the saved seq, since right now it will contain data
1637 # for a single frame from init, which is not what we want.
1638 self._save_seq = []
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:1396, in TimedAnimation.__init__(self, fig, interval, repeat_delay, repeat, event_source, *args, **kwargs)
1394 if event_source is None:
1395 event_source = fig.canvas.new_timer(interval=self._interval)
-> 1396 super().__init__(fig, event_source=event_source, *args, **kwargs)
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:883, in Animation.__init__(self, fig, event_source, blit)
880 self._close_id = self._fig.canvas.mpl_connect('close_event',
881 self._stop)
882 if self._blit:
--> 883 self._setup_blit()
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:1197, in Animation._setup_blit(self)
1194 self._drawn_artists = []
1195 self._resize_id = self._fig.canvas.mpl_connect('resize_event',
1196 self._on_resize)
-> 1197 self._post_draw(None, self._blit)
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:1150, in Animation._post_draw(self, framedata, blit)
1148 self._blit_draw(self._drawn_artists)
1149 else:
-> 1150 self._fig.canvas.draw_idle()
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/backend_bases.py:2060, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
2058 if not self._is_idle_drawing:
2059 with self._idle_draw_cntx():
-> 2060 self.draw(*args, **kwargs)
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:436, in FigureCanvasAgg.draw(self)
432 # Acquire a lock on the shared font cache.
433 with RendererAgg.lock, \
434 (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
435 else nullcontext()):
--> 436 self.figure.draw(self.renderer)
437 # A GUI class may be need to update a window using this draw, so
438 # don't forget to call the superclass.
439 super().draw()
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/artist.py:73, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
71 @wraps(draw)
72 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 73 result = draw(artist, renderer, *args, **kwargs)
74 if renderer._rasterizing:
75 renderer.stop_rasterizing()
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/artist.py:50, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
47 if artist.get_agg_filter() is not None:
48 renderer.start_filter()
---> 50 return draw(artist, renderer)
51 finally:
52 if artist.get_agg_filter() is not None:
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/figure.py:2847, in Figure.draw(self, renderer)
2844 finally:
2845 self.stale = False
-> 2847 self.canvas.draw_event(renderer)
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/backend_bases.py:1779, in FigureCanvasBase.draw_event(self, renderer)
1777 s = 'draw_event'
1778 event = DrawEvent(s, self, renderer)
-> 1779 self.callbacks.process(s, event)
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/cbook/__init__.py:292, in CallbackRegistry.process(self, s, *args, **kwargs)
290 except Exception as exc:
291 if self.exception_handler is not None:
--> 292 self.exception_handler(exc)
293 else:
294 raise
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/cbook/__init__.py:96, in _exception_printer(exc)
94 def _exception_printer(exc):
95 if _get_running_interactive_framework() in ["headless", None]:
---> 96 raise exc
97 else:
98 traceback.print_exc()
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/cbook/__init__.py:287, in CallbackRegistry.process(self, s, *args, **kwargs)
285 if func is not None:
286 try:
--> 287 func(*args, **kwargs)
288 # this does not capture KeyboardInterrupt, SystemExit,
289 # and GeneratorExit
290 except Exception as exc:
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:907, in Animation._start(self, *args)
904 self._fig.canvas.mpl_disconnect(self._first_draw_id)
906 # Now do any initial draw
--> 907 self._init_draw()
909 # Add our callback for stepping the animation and
910 # actually start the event_source.
911 self.event_source.add_callback(self._step)
File ~/.conda/envs/newwork/lib/python3.9/site-packages/matplotlib/animation.py:1698, in FuncAnimation._init_draw(self)
1696 self._draw_frame(frame_data)
1697 else:
-> 1698 self._drawn_artists = self._init_func()
1699 if self._blit:
1700 if self._drawn_artists is None:
Input In [53], in init()
14 def init():
---> 15 line.set_data([], [])
16 line2.set_data([],[])
17 return (line1, line2)
NameError: name 'line' is not defined
