1. Introduction

1.1 Review of Programming

We will mostly make use of the python programming language in this introduction and for much of the course (but not always as we will also look at Mathematica).

Since a basic course in programming is a pre-requisite for this course, some level of familiarity with programming is assumed.

The first step is to make sure that you have the proper environment setup on your computer. Make sure that you have python 3 installed; also install the numpy package. One way to make sure that these two packages are installed is to install the Anaconda Python Distribution. Alternatively, for quick running of code on the web, you could make use of Google Colab, which will open a notebook interface in which you can run python code. Google Colab uses a modified versin of Jupyter Notebook; you can try to install/run jupyter notebook on your computer if you find the notebook interface useful. Alternatively,

Once you have the proper programming environment setup, let's begin with the following basic tasks that most programming languages can do - output, loop and condition.

1.1.1 Printing Output

import numpy as np                  # (1)

print (np.log10(10), np.log(10))    # (2)

1.0 2.302585092994046

You should expect two numbers as the output: one for log base 10 i.e. \(\log_{\rm 10}\) and the other for natural logarithm i.e. \(\ln(10)\). For the above code to run, you may save the code in a file with name such as filename.py and on the command line enter:

python filename.py

pip install numpy

1.1.2 The for loop

Let us look at one more example python code to illustrate the use of variables and for loop.

total = 0                           # (1)

for i in range(1, 101):             # (2)
    total = total + i               # (3)

print (total)                       # (4)

5050

1.1.3 The if condition

Consider the following python code that checks whether an integer is odd or even.

def check_even(n):
    """Checks if the given integer n is even

    Args:
        n: integer

    Returns:
        True if n is even, False else
    """

    if (n%2==0):
        return True

    return False

print (check_even(9), check_even(10))

False True

In the code above we have also used def to define a new function. Inside the def function block, we have used three double-quote """ format for docstring.

1.1.4 The while loop

The while loop runs as long as the condition given in its syntax is satisfied. The while loop can be used instead of the for loop to get the sum of integers; see example below:

total = 0
i = 0
while (i<=100):
    total = total + i
    i = i + 1

print (total)

5050

1.2 Lists and Arrays

1.2.1 In-built Lists in Python

List1 = [1, 2, 3]
List2 = ["red", "blue", "green"]
List3 = [-1, 0, "orange"]

print (List1[0], List2[1], List3[2])

List1.append(4)                         # (1)
print (List1)

List1.pop()                             # (2)
print (List1)

1 blue orange
[1, 2, 3, 4]
[1, 2, 3]

A list can contain different types of elements. As you have seen in the example above, List3 consists of both integers and an string. Elements can also be added or removed from a list i.e. a list can grow or shrink during the execution of a program.

1.2.2 Arrays

The array module in python supports an object type called Array that can do more mathematical operations than the in-built python lists. However, the numpy module has support for a similar object array with support for more mathematical operations (including vector and matrix operations). Therefore, we will look at the numpy.array object:

import numpy as np

# first a simple 1D array
arr1 = np.array([1.0, 2.0, 3.0])
print (arr1)

# a simple 2D array
arr2 = np.array([[1.0, 2.0, 3.0], [1.5, 2.5, 3.5]])
print (arr2)

# to access the element of an array we can use indexing as follows:
print (arr2[0, 0])  # outputs 1.0 (the first row, first column)
print (arr2[0, 1])  # outputs 2.0 (the first row, second column)
print (arr1[1]) # outputs 2.0 (the second element of a 1D array)

[[1.  2.  3. ]
[1.5 2.5 3.5]]

1.0
2.0
2.0

Now, let us make use of the numpy.array and numpy.sum to calculate the sum of integers from 1 to 100.

import numpy as np

arr3 = np.array([i for i in range(0, 101)])

print (arr3)

print (np.sum(arr3))

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,  32,  33,  34,  35,  36,  37,  38,
        39,  40,  41,  42,  43,  44,  45,  46,  47,  48,  49,  50,  51,
        52,  53,  54,  55,  56,  57,  58,  59,  60,  61,  62,  63,  64,
        65,  66,  67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,
        78,  79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90,
        91,  92,  93,  94,  95,  96,  97,  98,  99, 100])  

5050

In the example above, instead of arr2 = np.array([i for i in range(0, 101)]) we could have used arr2 = np.array(range(0, 101)) or simply arr2 = range(0, 101) and still obtained the desired result of 5050. However, the way the syntax is written in the example is more powerful than these substitutes. For example, you can obtain the sum of squares of integers from 1 to 100 \(\left(\sum_{i=1}^{100} i^2\right)\) by the following code:

arr3 = np.array([i**2.0 for i in range(0, 101)])
print (np.sum(arr3))

1.2.3 Numpy arrays as vectors and matrices

A numpy array of one dimension can be thought of as a vector, and numpy supports vector operations such as dot product.

import numpy as np
v1 = np.array([1.0, -1.0, 2.0])
v2 = np.array([-1.0, 1.0, 2.0])

v1dotv2 = v1.dot(v2)    # you could also use np.dot(v1, v2)
print ("The dot product is: ", v1dotv2)

The dot product is: 2.0

1.2.4 Initializing a multidimensional array

Sometimes it is useful to initialize a multidimensional array -- for example a \(N\times N\) matrix with all entries as zeros. Such an array can be updated later in the code with appropriate values for each element as needed.

The numpy.zeros(shape) function can be used to generate arrays of given shape filled with zeros.

np.zeros(5)

array([0., 0., 0., 0., 0.])

Multidimensional arrays can be generated by giving the shape -- for example a \(m \times n\) matrix of zeros can be created using np.zeros((m,n))

np.zeros((2,2))

array([[0., 0.],
       [0., 0.]])

Similar to np.zeros, there is also np.ones that fills the array with ones.

1.3 Version Control

Computer codes (and even documents) may require updating at times. At the early stages of developing a computer code, the frequency of updates may be very high. Even for a matured piece of code, there may be maintenance updates that will require you to update the code time and again. Version control tools help us with this process. They can also be useful if we decide to revert a change at a later time and in case that multiple people are collaborating on the same piece of software or document at once.

We will make use of github for version control. The underlying version control software utilized by github is called git.

Please take the following github skills course to begin.

• Introduction to GitHub - Github Skills

1.4 Documentation

The code you write will likely be read by other people that may include friends, collegues, professors, customors and users. Therefore, good documentation of code is very important and perhaps even more so for large projects. Large projects typically also follow a certain style guide so that the code and documentation written by numerous contributors have consistent style.

For example, there is PEP 8 Style Guide for Python Code and PEP 257 Docstring Conventions that are generally followed.

Another popular style guide for python is the Google Python Style Guide.

In this course, we will not be worrying about strictly following a style guide, but it is good to know that these exist and that you can make use of one if necessary.