DM561 - Linear Algebra with Applications

Sheet 1

Task 1 - Functions

The volume of a sphere with radius $r$ is $V = \frac{4}{3}\pi r^3$. In a Python file, define a function called sphere_volume() that accepts a single parameter $r$. Return the volume of the sphere of radius $r$, using $3.14159$ as an approximation for $\pi$ (for now). Also write an appropriate docstring for your function. Try keeping the body of the function down to a single line of code.

To test your function, call it under the if __name__ == "__main__" clause and print the returned value. Run your file to see if your answer is what you expect it to be.

Task 2 - Printing

The built-in print() function has the useful keyword arguments sep and end. It accepts any number of positional arguments and prints them out with sep inserted between values (defaulting to a space), then prints end (defaulting to the newline character '\n').

>>> print(1, 2, 3, sep=', ', end='!!\n')
1, 2, 3!!

Write a function called isolate() that accepts five arguments. Print the first three separated by 5 spaces, then print the rest with a single space between each output. For example,

>>> isolate(1, 2, 3, 4, 5)
1     2     3 4 5

Task 3 - Slicing strings

Write two new functions, called first_half() and backward().

  1. first_half() should accept a parameter and return the first half of it, excluding the middle character if there is an odd number of characters. (Hint: the built-in function len() returns the length of the input.)
  2. The backward() function should accept a parameter and reverse the order of its characters using slicing, then return the reversed string. (Hint: The step parameter used in slicing can be negative.)

Use IPython to quickly test your syntax for each function.

Task 4* - Lists

Write a function called list_ops(). Define a list with the entries "bear", "ant", "cat", and "dog", in that order. Then perform the following operations on the list:

  1. Append "eagle".
  2. Replace the entry at index 2 with "fox".
  3. Remove (or pop) the entry at index 1.
  4. Sort the list in reverse alphabetical order.
  5. Replace "eagle" with "hawk". (Hint: the list’s index() method may be helpful.)
  6. Add the string "hunter" to the last entry in the list.

Return the resulting list.

Work out (on paper) what the result should be, then check that your function returns the correct list. Consider printing the list at each step to see the intermediate results.

Task 5 - Strings

Write a function called pig_latin(). Accept a parameter word, translate it into Pig Latin, then return the translation. Specifically, if word starts with a vowel, add “hay” to the end; if word starts with a consonant, take the first character of word, move it to the end, and add “ay”. (Hint: use the in operator to check if the first letter is a vowel.)

Task 6

This problem originates from https://projecteuler.net, an excellent resource for math-related coding problems.

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is $9009 = 91 \times 99.$ Write a function called palindrome() that finds and returns the largest palindromic number made from the product of two 3-digit numbers.

Task 6b

The alternating harmonic series is defined as follows.

\[\sum_{n=1}^\infty \frac{(-1)^{(n+1)}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \ldots = \ln(2)\]

Write a function called alt_harmonic() that accepts an integer $n$. Use a list comprehension to quickly compute the first $n$ terms of this series (be careful not to compute only $n-1$ terms). The sum of the first 500,000 terms of this series approximates $\ln(2)$ to five decimal places. (Hint: consider using Python’s built-in sum() function.)

Task 7 - Lists

Write a function that accepts a list $L$ and returns the minimum, maximum, and average of the entries of $L$ (in that order). Can you implement this function in a single line?

Task 8* - Mutable vs Immutable Objects

Determine which of Python’s object types are mutable and which are immutable by repeating the following experiment for an int, str, list, tuple, and set.

  1. Create an object of the given type and assign a name to it.
  2. Assign a new name to the first name.
  3. Alter the object via only one of the names (for tuples, use my_tuple += (1,)).
  4. Check to see if the two names are equal (compare also the address of the object they refer to via the function id()). If they are, then since changing one name changed the other, the names refer to the same object and the object type is mutable. Otherwise, the names refer to different objects—meaning a new object was created in step 2—and therefore the object type is immutable.

For example, the following experiment shows that dict is a mutable type.

>>> dict_1 = {1: 'x', 2: 'b'}           # Create a dictionary.
>>> dict_2 = dict_1                     # Assign it a new name.
>>> dict_2[1] = 'a'                     # Change the 'new' dictionary.
>>> dict_1 == dict_2                    # Compare the two names.
 True                                   # Both names changed!

Print a statement of your conclusions that clearly indicates which object types are mutable and which are immutable.

Task 9* - Implementing Modules

Create a module called calculator.py. Write a function that returns the sum of two arguments and a function that returns the product of two arguments. Also use import to add the sqrt() function from the math module to the namespace. When this file is either run or imported, nothing should be executed.

In your solutions file, import your new custom module. Write a function that accepts two numbers representing the lengths of the sides of a right triangle. Using only the functions from calculator.py, calculate and return the length of the hypotenuse of the triangle.

Task 10 - Modules

In IPython explore the modules:

Task 11 - Module Itertools

The power set of a set $A$, denoted $\mathcal{P}(A)$ or $2^A$, is the set of all subsets of $A$, including the empty set $\emptyset$ and $A$ itself. For example, the power set of the set $A = \{a, b, c\}$ is $2^A = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\} \}$.

Write a function that accepts an iterable $A$. Use an itertools function to compute the power set of $A$ as a list of sets (why couldn’t it be a set of sets in Python?). (Hint: The power set of a set with $n$ elements should have exactly $2^n$ elements.)

Task 12* - Classes

Expand the Backpack class from the file object_oriented.py to match the following specifications.

  1. Modify the constructor so that it accepts three total arguments: name, color, and max_size (in that order). Make max_size a keyword argument that defaults to $5$. Store each input as an attribute.

  2. Modify the put() method to check that the backpack does not go over capacity. If there are already max_size items or more, print “No Room!” and do not add the item to the contents list.

  3. Write a new method called dump() that resets the contents of the backpack to an empty list. This method should not receive any arguments (except self).

  4. Documentation is especially important in classes so that the user knows what an object’s attributes represent and how to use methods appropriately. Update (or write) the docstrings for the __init__(), put(), and dump() methods, as well as the actual class docstring (under class but before __init__()) to reflect the changes from parts 1-3 of this problem.

To ensure that your class works properly, write a test function outside outside of the Backpack class that instantiates and analyzes a Backpack object.

def test_backpack():
    testpack = Backpack("Barry", "black")       # Instantiate the object.
    if testpack.name != "Barry":                # Test an attribute.
        print("Backpack.name assigned incorrectly")
    for item in ["pencil", "pen", "paper", "computer"]:
        testpack.put(item)                      # Test a method.
    print("Contents:", testpack.contents)
    # ...

Task 13 – Inheritance

In object_oriented.py write a Jetpack class that inherits from the Backpack class.

  1. Override the constructor so that in addition to a name, color, and maximum size, it also accepts an amount of fuel. Change the default value of max_size to $2$, and set the default value of fuel to $10$. Store the fuel as an attribute.

  2. Add a fly() method that accepts an amount of fuel to be burned and decrements the fuel attribute by that amount. If the user tries to burn more fuel than remains, print “Not enough fuel!” and do not decrement the fuel.

  3. Override the dump() method so that both the contents and the fuel tank are emptied.

  4. Write clear, detailed docstrings for the class and each of its methods.

Task 14* – Magic Methods

Endow the Backpack class from the file object_oriented.py that you have been developing in the two previous tasks with two additional magic methods:

  1. The __eq__() magic method is used to determine if two objects are equal, and is invoked by the == operator. Implement the __eq__() magic method for the Backpack class so that two Backpack objects are equal if and only if they have the same name, color, and number of contents.

  2. The __str__() magic method returns the string representation of an object. This method is invoked by str() and used by print(). Implement the __str__() method in the Backpack class so that printing a Backpack object yields the following output (that is, construct and return the following string).

    <<Owner:      <name>
    Color:      <color>
    Size:       <number of items in contents>
    Max Size:   <max_size>
    Contents:   [<item1>, <item2>, ...]>>
    

    (Hint: Use the tab and newline characters '\\t' and '\\n' to align output nicely.)

Task 15* - Handling Exceptions

A random walk is a path created by a sequence of random steps. The following function simulates a random walk by repeatedly adding or subtracting $1$ to a running total.

from random import choice

def random_walk(max_iters=1e12):
    walk = 0
    directions = [1, -1]
    for i in range(int(max_iters)):
        walk += choice(directions)
    return walk

A KeyboardInterrupt is a special exception that can be triggered at any time by entering ctrl+c (on most systems) in the keyboard. Modify random_walk() so that if the user raises a KeyboardInterrupt by pressing ctrl+c while the program is running, the function catches the exception and prints “Process interrupted at iteration $i$”. If no KeyboardInterrupt is raised, print “Process completed”. In both cases, return walk as before.

Task 16* - File Input/Output

Define a class called ContentFilter. Implement the constructor so that it accepts the name of a file to be read.

  1. If the file name is invalid in any way, prompt the user for another filename using input(). Continue prompting the user until they provide a valid filename.

    >>> cf1 = ContentFilter("hello_world.txt")  # File exists.
    >>> cf2 = ContentFilter("not-a-file.txt")   # File doesn't exist.
    <<Please enter a valid file name: >><r<still-not-a-file.txt>r>
    <<Please enter a valid file name: >><r<hello_world.txt>r>
    >>> cf3 = ContentFilter([1, 2, 3])          # Not even a string.
    <<Please enter a valid file name: >><r<hello_world.txt>r>
    

    (Hint: open() might raise a FileNotFoundError, a TypeError, or an OSError.)

  2. Read the file and store its name and contents as attributes (store the contents as a single string). Make sure the file is securely closed.

Task 17* - Matrix Multiplication in Numpy

Matrix Multiplication in Numpy There are two main ways to perform matrix multiplication in NumPy: with NumPy’s dot() function (np.dot(A, B)), or with the @ operator (A @ B). Write a function that defines the following matrices as NumPy arrays.

Return the matrix product $AB$. For examples of array initialization and matrix multiplication, use object introspection in IPython to look up the documentation for np.ndarray, np.array() and np.dot().

In [1]: import numpy as np

In [2]: np.array?       # press 'enter'

Task 18 - Matrix Multiplication

Write a function that defines the following matrix as a NumPy array.

Return the matrix $-A^3 + 9A^2 - 15A$.

In this context, $A^2 = AA$ (the matrix product, not the component-wise square). The somewhat surprising result is a demonstration of the Cayley-Hamilton theorem.

Task 19* - Array Creation

Write a function that defines the following matrices as NumPy arrays using the functions to create arrays in the slides (that is, without hard writing all the numbers). Calculate the matrix product $ABA$. Change the data type of the resulting matrix to np.int64, then return it.

Task 20* - Arrays

Write a function that accepts a single array as input. Make a copy of the array, then use fancy indexing to set all negative entries of the copy to $0$. Return the copy.

Task 21* - Arrays

Write a function that defines the following matrices as NumPy arrays.

Use NumPy’s stacking functions to create and return the block matrix:

where $I$ is the $3\times 3$ identity matrix and each $\mathbf{0}$ is a matrix of all zeros of appropriate size.

A block matrix of this form is used in the interior point method for linear optimization.

Task 22* - Arrays

A matrix is called row-stochastic if its rows each sum to $1$. Stochastic matrices are fundamentally important for finite discrete random processes and some machine learning algorithms.

Write a function than accepts a matrix (as a 2-D array). Divide each row of the matrix by the row sum and return the new row-stochastic matrix. Use array broadcasting and the axis argument instead of a loop.

(Similarly, a matrix is called column-stochastic if its columns each sum to $1$.)

Task 23 - Polynomials

Use NumPy’s polynomial objects to approximate the following series.

This series converges on $(-1, 1)$. Use your series approximation to approximate $\pi$. Use the function np.allclose to verify that your approximation is close. Hint: think of the powers of $x$ that are not included in the series as having zero coefficients.