introduction to problem solving python

Python Practice for Beginners: 15 Hands-On Problems

• online practice

Want to put your Python skills to the test? Challenge yourself with these 15 Python practice exercises taken directly from our Python courses!

There’s no denying that solving Python exercises is one of the best ways to practice and improve your Python skills . Hands-on engagement with the language is essential for effective learning. This is exactly what this article will help you with: we've curated a diverse set of Python practice exercises tailored specifically for beginners seeking to test their programming skills.

These Python practice exercises cover a spectrum of fundamental concepts, all of which are covered in our Python Data Structures in Practice and Built-in Algorithms in Python courses. Together, both courses add up to 39 hours of content. They contain over 180 exercises for you to hone your Python skills. In fact, the exercises in this article were taken directly from these courses!

In these Python practice exercises, we will use a variety of data structures, including lists, dictionaries, and sets. We’ll also practice basic programming features like functions, loops, and conditionals. Every exercise is followed by a solution and explanation. The proposed solution is not necessarily the only possible answer, so try to find your own alternative solutions. Let’s get right into it!

Python Practice Problem 1: Average Expenses for Each Semester

John has a list of his monthly expenses from last year:

He wants to know his average expenses for each semester. Using a for loop, calculate John’s average expenses for the first semester (January to June) and the second semester (July to December).

Explanation

We initialize two variables, first_semester_total and second_semester_total , to store the total expenses for each semester. Then, we iterate through the monthly_spending list using enumerate() , which provides both the index and the corresponding value in each iteration. If you have never heard of enumerate() before – or if you are unsure about how for loops in Python work – take a look at our article How to Write a for Loop in Python .

Within the loop, we check if the index is less than 6 (January to June); if so, we add the expense to first_semester_total . If the index is greater than 6, we add the expense to second_semester_total .

After iterating through all the months, we calculate the average expenses for each semester by dividing the total expenses by 6 (the number of months in each semester). Finally, we print out the average expenses for each semester.

Python Practice Problem 2: Who Spent More?

John has a friend, Sam, who also kept a list of his expenses from last year:

They want to find out how many months John spent more money than Sam. Use a for loop to compare their expenses for each month. Keep track of the number of months where John spent more money.

We initialize the variable months_john_spent_more with the value zero. Then we use a for loop with range(len()) to iterate over the indices of the john_monthly_spending list.

Within the loop, we compare John's expenses with Sam's expenses for the corresponding month using the index i . If John's expenses are greater than Sam's for a particular month, we increment the months_john_spent_more variable. Finally, we print out the total number of months where John spent more money than Sam.

Python Practice Problem 3: All of Our Friends

Paul and Tina each have a list of their respective friends:

Combine both lists into a single list that contains all of their friends. Don’t include duplicate entries in the resulting list.

There are a few different ways to solve this problem. One option is to use the + operator to concatenate Paul and Tina's friend lists ( paul_friends and tina_friends ). Afterwards, we convert the combined list to a set using set() , and then convert it back to a list using list() . Since sets cannot have duplicate entries, this process guarantees that the resulting list does not hold any duplicates. Finally, we print the resulting combined list of friends.

If you need a refresher on Python sets, check out our in-depth guide to working with sets in Python or find out the difference between Python sets, lists, and tuples .

Python Practice Problem 4: Find the Common Friends

Now, let’s try a different operation. We will start from the same lists of Paul’s and Tina’s friends:

In this exercise, we’ll use a for loop to get a list of their common friends.

For this problem, we use a for loop to iterate through each friend in Paul's list ( paul_friends ). Inside the loop, we check if the current friend is also present in Tina's list ( tina_friends ). If it is, it is added to the common_friends list. This approach guarantees that we test each one of Paul’s friends against each one of Tina’s friends. Finally, we print the resulting list of friends that are common to both Paul and Tina.

Python Practice Problem 5: Find the Basketball Players

You work at a sports club. The following sets contain the names of players registered to play different sports:

How can you obtain a set that includes the players that are only registered to play basketball (i.e. not registered for football or volleyball)?

This type of scenario is exactly where set operations shine. Don’t worry if you never heard about them: we have an article on Python set operations with examples to help get you up to speed.

First, we use the | (union) operator to combine the sets of football and volleyball players into a single set. In the same line, we use the - (difference) operator to subtract this combined set from the set of basketball players. The result is a set containing only the players registered for basketball and not for football or volleyball.

If you prefer, you can also reach the same answer using set methods instead of the operators:

It’s essentially the same operation, so use whichever you think is more readable.

Python Practice Problem 6: Count the Votes

Let’s try counting the number of occurrences in a list. The list below represent the results of a poll where students were asked for their favorite programming language:

Use a dictionary to tally up the votes in the poll.

In this exercise, we utilize a dictionary ( vote_tally ) to count the occurrences of each programming language in the poll results. We iterate through the poll_results list using a for loop; for each language, we check if it already is in the dictionary. If it is, we increment the count; otherwise, we add the language to the dictionary with a starting count of 1. This approach effectively tallies up the votes for each programming language.

If you want to learn more about other ways to work with dictionaries in Python, check out our article on 13 dictionary examples for beginners .

Python Practice Problem 7: Sum the Scores

Three friends are playing a game, where each player has three rounds to score. At the end, the player whose total score (i.e. the sum of each round) is the highest wins. Consider the scores below (formatted as a list of tuples):

Create a dictionary where each player is represented by the dictionary key and the corresponding total score is the dictionary value.

This solution is similar to the previous one. We use a dictionary ( total_scores ) to store the total scores for each player in the game. We iterate through the list of scores using a for loop, extracting the player's name and score from each tuple. For each player, we check if they already exist as a key in the dictionary. If they do, we add the current score to the existing total; otherwise, we create a new key in the dictionary with the initial score. At the end of the for loop, the total score of each player will be stored in the total_scores dictionary, which we at last print.

Python Practice Problem 8: Calculate the Statistics

Given any list of numbers in Python, such as …

 … write a function that returns a tuple containing the list’s maximum value, sum of values, and mean value.

We create a function called calculate_statistics to calculate the required statistics from a list of numbers. This function utilizes a combination of max() , sum() , and len() to obtain these statistics. The results are then returned as a tuple containing the maximum value, the sum of values, and the mean value.

The function is called with the provided list and the results are printed individually.

Python Practice Problem 9: Longest and Shortest Words

Given the list of words below ..

… find the longest and the shortest word in the list.

To find the longest and shortest word in the list, we initialize the variables longest_word and shortest_word as the first word in the list. Then we use a for loop to iterate through the word list. Within the loop, we compare the length of each word with the length of the current longest and shortest words. If a word is longer than the current longest word, it becomes the new longest word; on the other hand, if it's shorter than the current shortest word, it becomes the new shortest word. After iterating through the entire list, the variables longest_word and shortest_word will hold the corresponding words.

There’s a catch, though: what happens if two or more words are the shortest? In that case, since the logic used is to overwrite the shortest_word only if the current word is shorter – but not of equal length – then shortest_word is set to whichever shortest word appears first. The same logic applies to longest_word , too. If you want to set these variables to the shortest/longest word that appears last in the list, you only need to change the comparisons to <= (less or equal than) and >= (greater or equal than), respectively.

If you want to learn more about Python strings and what you can do with them, be sure to check out this overview on Python string methods .

Python Practice Problem 10: Filter a List by Frequency

Given a list of numbers …

… create a new list containing only the numbers that occur at least three times in the list.

Here, we use a for loop to iterate through the number_list . In the loop, we use the count() method to check if the current number occurs at least three times in the number_list . If the condition is met, the number is appended to the filtered_list .

After the loop, the filtered_list contains only numbers that appear three or more times in the original list.

Python Practice Problem 11: The Second-Best Score

You’re given a list of students’ scores in no particular order:

Find the second-highest score in the list.

This one is a breeze if we know about the sort() method for Python lists – we use it here to sort the list of exam results in ascending order. This way, the highest scores come last. Then we only need to access the second to last element in the list (using the index -2 ) to get the second-highest score.

Python Practice Problem 12: Check If a List Is Symmetrical

Given the lists of numbers below …

… create a function that returns whether a list is symmetrical. In this case, a symmetrical list is a list that remains the same after it is reversed – i.e. it’s the same backwards and forwards.

Reversing a list can be achieved by using the reverse() method. In this solution, this is done inside the is_symmetrical function.

To avoid modifying the original list, a copy is created using the copy() method before using reverse() . The reversed list is then compared with the original list to determine if it’s symmetrical.

The remaining code is responsible for passing each list to the is_symmetrical function and printing out the result.

Python Practice Problem 13: Sort By Number of Vowels

Given this list of strings …

… sort the list by the number of vowels in each word. Words with fewer vowels should come first.

Whenever we need to sort values in a custom order, the easiest approach is to create a helper function. In this approach, we pass the helper function to Python’s sorted() function using the key parameter. The sorting logic is defined in the helper function.

In the solution above, the custom function count_vowels uses a for loop to iterate through each character in the word, checking if it is a vowel in a case-insensitive manner. The loop increments the count variable for each vowel found and then returns it. We then simply pass the list of fruits to sorted() , along with the key=count_vowels argument.

Python Practice Problem 14: Sorting a Mixed List

Imagine you have a list with mixed data types: strings, integers, and floats:

Typically, you wouldn’t be able to sort this list, since Python cannot compare strings to numbers. However, writing a custom sorting function can help you sort this list.

Create a function that sorts the mixed list above using the following logic:

• If the element is a string, the length of the string is used for sorting.
• If the element is a number, the number itself is used.

As proposed in the exercise, a custom sorting function named custom_sort is defined to handle the sorting logic. The function checks whether each element is a string or a number using the isinstance() function. If the element is a string, it returns the length of the string for sorting; if it's a number (integer or float), it returns the number itself.

The sorted() function is then used to sort the mixed_list using the logic defined in the custom sorting function.

If you’re having a hard time wrapping your head around custom sort functions, check out this article that details how to write a custom sort function in Python .

Python Practice Problem 15: Filter and Reorder

Given another list of strings, such as the one below ..

.. create a function that does two things: filters out any words with three or fewer characters and sorts the resulting list alphabetically.

Here, we define filter_and_sort , a function that does both proposed tasks.

First, it uses a for loop to filter out words with three or fewer characters, creating a filtered_list . Then, it sorts the filtered list alphabetically using the sorted() function, producing the final sorted_list .

The function returns this sorted list, which we print out.

Want Even More Python Practice Problems?

We hope these exercises have given you a bit of a coding workout. If you’re after more Python practice content, head straight for our courses on Python Data Structures in Practice and Built-in Algorithms in Python , where you can work on exciting practice exercises similar to the ones in this article.

Additionally, you can check out our articles on Python loop practice exercises , Python list exercises , and Python dictionary exercises . Much like this article, they are all targeted towards beginners, so you should feel right at home!

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Problem Solving with Algorithms and Data Structures using Python ¶

By Brad Miller and David Ranum, Luther College

There is a wonderful collection of YouTube videos recorded by Gerry Jenkins to support all of the chapters in this text.

• 1.1. Objectives
• 1.2. Getting Started
• 1.3. What Is Computer Science?
• 1.4. What Is Programming?
• 1.5. Why Study Data Structures and Abstract Data Types?
• 1.6. Why Study Algorithms?
• 1.7. Review of Basic Python
• 1.8.1. Built-in Atomic Data Types
• 1.8.2. Built-in Collection Data Types
• 1.9.1. String Formatting
• 1.10. Control Structures
• 1.11. Exception Handling
• 1.12. Defining Functions
• 1.13.1. A Fraction Class
• 1.13.2. Inheritance: Logic Gates and Circuits
• 1.14. Summary
• 1.15. Key Terms
• 1.16. Discussion Questions
• 1.17. Programming Exercises
• 2.1.1. A Basic implementation of the MSDie class
• 2.2. Making your Class Comparable
• 3.1. Objectives
• 3.2. What Is Algorithm Analysis?
• 3.3. Big-O Notation
• 3.4.1. Solution 1: Checking Off
• 3.4.2. Solution 2: Sort and Compare
• 3.4.3. Solution 3: Brute Force
• 3.4.4. Solution 4: Count and Compare
• 3.5. Performance of Python Data Structures
• 3.7. Dictionaries
• 3.8. Summary
• 3.9. Key Terms
• 3.10. Discussion Questions
• 3.11. Programming Exercises
• 4.1. Objectives
• 4.2. What Are Linear Structures?
• 4.3. What is a Stack?
• 4.4. The Stack Abstract Data Type
• 4.5. Implementing a Stack in Python
• 4.6. Simple Balanced Parentheses
• 4.7. Balanced Symbols (A General Case)
• 4.8. Converting Decimal Numbers to Binary Numbers
• 4.9.1. Conversion of Infix Expressions to Prefix and Postfix
• 4.9.2. General Infix-to-Postfix Conversion
• 4.9.3. Postfix Evaluation
• 4.10. What Is a Queue?
• 4.11. The Queue Abstract Data Type
• 4.12. Implementing a Queue in Python
• 4.13. Simulation: Hot Potato
• 4.14.1. Main Simulation Steps
• 4.14.2. Python Implementation
• 4.14.3. Discussion
• 4.15. What Is a Deque?
• 4.16. The Deque Abstract Data Type
• 4.17. Implementing a Deque in Python
• 4.18. Palindrome-Checker
• 4.19. Lists
• 4.20. The Unordered List Abstract Data Type
• 4.21.1. The Node Class
• 4.21.2. The Unordered List Class
• 4.22. The Ordered List Abstract Data Type
• 4.23.1. Analysis of Linked Lists
• 4.24. Summary
• 4.25. Key Terms
• 4.26. Discussion Questions
• 4.27. Programming Exercises
• 5.1. Objectives
• 5.2. What Is Recursion?
• 5.3. Calculating the Sum of a List of Numbers
• 5.4. The Three Laws of Recursion
• 5.5. Converting an Integer to a String in Any Base
• 5.6. Stack Frames: Implementing Recursion
• 5.7. Introduction: Visualizing Recursion
• 5.8. Sierpinski Triangle
• 5.9. Complex Recursive Problems
• 5.10. Tower of Hanoi
• 5.11. Exploring a Maze
• 5.12. Dynamic Programming
• 5.13. Summary
• 5.14. Key Terms
• 5.15. Discussion Questions
• 5.16. Glossary
• 5.17. Programming Exercises
• 6.1. Objectives
• 6.2. Searching
• 6.3.1. Analysis of Sequential Search
• 6.4.1. Analysis of Binary Search
• 6.5.1. Hash Functions
• 6.5.2. Collision Resolution
• 6.5.3. Implementing the Map Abstract Data Type
• 6.5.4. Analysis of Hashing
• 6.6. Sorting
• 6.7. The Bubble Sort
• 6.8. The Selection Sort
• 6.9. The Insertion Sort
• 6.10. The Shell Sort
• 6.11. The Merge Sort
• 6.12. The Quick Sort
• 6.13. Summary
• 6.14. Key Terms
• 6.15. Discussion Questions
• 6.16. Programming Exercises
• 7.1. Objectives
• 7.2. Examples of Trees
• 7.3. Vocabulary and Definitions
• 7.4. List of Lists Representation
• 7.5. Nodes and References
• 7.6. Parse Tree
• 7.7. Tree Traversals
• 7.8. Priority Queues with Binary Heaps
• 7.9. Binary Heap Operations
• 7.10.1. The Structure Property
• 7.10.2. The Heap Order Property
• 7.10.3. Heap Operations
• 7.11. Binary Search Trees
• 7.12. Search Tree Operations
• 7.13. Search Tree Implementation
• 7.14. Search Tree Analysis
• 7.15. Balanced Binary Search Trees
• 7.16. AVL Tree Performance
• 7.17. AVL Tree Implementation
• 7.18. Summary of Map ADT Implementations
• 7.19. Summary
• 7.20. Key Terms
• 7.21. Discussion Questions
• 7.22. Programming Exercises
• 8.1. Objectives
• 8.2. Vocabulary and Definitions
• 8.3. The Graph Abstract Data Type
• 8.4. An Adjacency Matrix
• 8.5. An Adjacency List
• 8.6. Implementation
• 8.7. The Word Ladder Problem
• 8.8. Building the Word Ladder Graph
• 8.9. Implementing Breadth First Search
• 8.10. Breadth First Search Analysis
• 8.11. The Knight’s Tour Problem
• 8.12. Building the Knight’s Tour Graph
• 8.13. Implementing Knight’s Tour
• 8.14. Knight’s Tour Analysis
• 8.15. General Depth First Search
• 8.16. Depth First Search Analysis
• 8.17. Topological Sorting
• 8.18. Strongly Connected Components
• 8.19. Shortest Path Problems
• 8.20. Dijkstra’s Algorithm
• 8.21. Analysis of Dijkstra’s Algorithm
• 8.22. Prim’s Spanning Tree Algorithm
• 8.23. Summary
• 8.24. Key Terms
• 8.25. Discussion Questions
• 8.26. Programming Exercises

Acknowledgements ¶

We are very grateful to Franklin Beedle Publishers for allowing us to make this interactive textbook freely available. This online version is dedicated to the memory of our first editor, Jim Leisy, who wanted us to “change the world.”

Python Projects

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Python Exercise with Practice Questions and Solutions

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Python Pattern printing Exercises

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Python DateTime Exercises

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Python OOPS Exercises

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• Python program to build flashcard using class in Python
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• What is a clean and Pythonic way to have multiple constructors in Python?
• How to Change a Dictionary Into a Class?
• How to create an empty class in Python?
• Student management system in Python
• How to create a list of object in Python class

Python Regex Exercises

• Validate an IP address using Python without using RegEx
• Python program to find the type of IP Address using Regex
• Converting a 10 digit phone number to US format using Regex in Python
• Python program to find Indices of Overlapping Substrings
• Python program to extract Strings between HTML Tags
• Python - Check if String Contain Only Defined Characters using Regex
• How to extract date from Excel file using Pandas?
• Python program to find files having a particular extension using RegEx
• How to check if a string starts with a substring using regex in Python?
• How to Remove repetitive characters from words of the given Pandas DataFrame using Regex?
• Extract punctuation from the specified column of Dataframe using Regex
• Extract IP address from file using Python
• Python program to Count Uppercase, Lowercase, special character and numeric values using Regex
• Categorize Password as Strong or Weak using Regex in Python
• Python - Substituting patterns in text using regex

Python LinkedList Exercises

• Python program to Search an Element in a Circular Linked List
• Implementation of XOR Linked List in Python
• Pretty print Linked List in Python
• Python Library for Linked List
• Python | Stack using Doubly Linked List
• Python | Queue using Doubly Linked List
• Program to reverse a linked list using Stack
• Python program to find middle of a linked list using one traversal
• Python Program to Reverse a linked list

Python Searching Exercises

• Binary Search (bisect) in Python
• Python Program for Linear Search
• Python Program for Anagram Substring Search (Or Search for all permutations)
• Python Program for Binary Search (Recursive and Iterative)
• Python Program for Rabin-Karp Algorithm for Pattern Searching
• Python Program for KMP Algorithm for Pattern Searching

Python Sorting Exercises

• Python Code for time Complexity plot of Heap Sort
• Python Program for Stooge Sort
• Python Program for Recursive Insertion Sort
• Python Program for Cycle Sort
• Bisect Algorithm Functions in Python
• Python Program for BogoSort or Permutation Sort
• Python Program for Odd-Even Sort / Brick Sort
• Python Program for Gnome Sort
• Python Program for Cocktail Sort
• Python Program for Bitonic Sort
• Python Program for Pigeonhole Sort
• Python Program for Comb Sort
• Python Program for Iterative Merge Sort
• Python Program for Binary Insertion Sort
• Python Program for ShellSort

Python DSA Exercises

• Saving a Networkx graph in GEXF format and visualize using Gephi
• Dumping queue into list or array in Python
• Python program to reverse a stack
• Python - Stack and StackSwitcher in GTK+ 3
• Multithreaded Priority Queue in Python
• Python Program to Reverse the Content of a File using Stack
• Priority Queue using Queue and Heapdict module in Python
• Box Blur Algorithm - With Python implementation
• Python program to reverse the content of a file and store it in another file
• Check whether the given string is Palindrome using Stack
• Take input from user and store in .txt file in Python
• Change case of all characters in a .txt file using Python
• Finding Duplicate Files with Python

Python File Handling Exercises

• Python Program to Count Words in Text File
• Python Program to Delete Specific Line from File
• Python Program to Replace Specific Line in File
• Python Program to Print Lines Containing Given String in File
• Python - Loop through files of certain extensions
• Compare two Files line by line in Python
• How to keep old content when Writing to Files in Python?
• How to get size of folder using Python?
• How to read multiple text files from folder in Python?
• Read a CSV into list of lists in Python
• Python - Write dictionary of list to CSV
• Convert nested JSON to CSV in Python
• How to add timestamp to CSV file in Python

Python CSV Exercises

• How to create multiple CSV files from existing CSV file using Pandas ?
• How to read all CSV files in a folder in Pandas?
• How to Sort CSV by multiple columns in Python ?
• Working with large CSV files in Python
• How to convert CSV File to PDF File using Python?
• Visualize data from CSV file in Python
• Python - Read CSV Columns Into List
• Sorting a CSV object by dates in Python
• Python program to extract a single value from JSON response
• Convert class object to JSON in Python
• Convert multiple JSON files to CSV Python
• Convert JSON data Into a Custom Python Object
• Convert CSV to JSON using Python

Python JSON Exercises

• Flattening JSON objects in Python
• Saving Text, JSON, and CSV to a File in Python
• Convert Text file to JSON in Python
• Convert JSON to CSV in Python
• Convert JSON to dictionary in Python
• Python Program to Get the File Name From the File Path
• How to get file creation and modification date or time in Python?
• Menu driven Python program to execute Linux commands
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• Open computer drives like C, D or E using Python

Python OS Module Exercises

• Rename a folder of images using Tkinter
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• Change current working directory with Python
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• Build GUI Application for Guess Indian State using Tkinter Python
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• How to temporarily remove a Tkinter widget without using just .place?
• How to open a website in a Tkinter window?

Python Tkinter Exercises

• Create Address Book in Python - Using Tkinter
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• How to check which Button was clicked in Tkinter ?
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• Scraping Reddit with Python and BeautifulSoup
• Scraping Indeed Job Data Using Python

Python Web Scraping Exercises

• How to Scrape all PDF files in a Website?
• How to Scrape Multiple Pages of a Website Using Python?
• Quote Guessing Game using Web Scraping in Python
• How to extract youtube data in Python?
• How to Download All Images from a Web Page in Python?
• Test the given page is found or not on the server Using Python
• How to Extract Wikipedia Data in Python?
• How to extract paragraph from a website and save it as a text file?
• Automate Youtube with Python
• Controlling the Web Browser with Python
• How to Build a Simple Auto-Login Bot with Python
• Download Google Image Using Python and Selenium
• How To Automate Google Chrome Using Foxtrot and Python

Python Selenium Exercises

• How to scroll down followers popup in Instagram ?
• How to switch to new window in Selenium for Python?
• Python Selenium - Find element by text
• How to scrape multiple pages using Selenium in Python?
• Python Selenium - Find Button by text
• Web Scraping Tables with Selenium and Python
• Selenium - Search for text on page
• Python Projects - Beginner to Advanced

Python Exercise: Practice makes you perfect in everything. This proverb always proves itself correct. Just like this, if you are a Python learner, then regular practice of Python exercises makes you more confident and sharpens your skills. So, to test your skills, go through these Python exercises with solutions.

Python is a widely used general-purpose high-level language that can be used for many purposes like creating GUI, web Scraping, web development, etc. You might have seen various Python tutorials that explain the concepts in detail but that might not be enough to get hold of this language. The best way to learn is by practising it more and more.

The best thing about this Python practice exercise is that it helps you learn Python using sets of detailed programming questions from basic to advanced. It covers questions on core Python concepts as well as applications of Python in various domains. So if you are at any stage like beginner, intermediate or advanced this Python practice set will help you to boost your programming skills in Python.

List of Python Programming Exercises

In the below section, we have gathered chapter-wise Python exercises with solutions. So, scroll down to the relevant topics and try to solve the Python program practice set.

Python List Exercises

• Python program to interchange first and last elements in a list
• Python program to swap two elements in a list
• Python | Ways to find length of list
• Maximum of two numbers in Python
• Minimum of two numbers in Python

>> More Programs on List

Python String Exercises

• Python program to check whether the string is Symmetrical or Palindrome
• Reverse words in a given String in Python
• Ways to remove i’th character from string in Python
• Find length of a string in python (4 ways)
• Python program to print even length words in a string

>> More Programs on String

Python Tuple Exercises

• Python program to Find the size of a Tuple
• Python – Maximum and Minimum K elements in Tuple
• Python – Sum of tuple elements
• Python – Row-wise element Addition in Tuple Matrix
• Create a list of tuples from given list having number and its cube in each tuple

>> More Programs on Tuple

Python Dictionary Exercises

• Python | Sort Python Dictionaries by Key or Value
• Handling missing keys in Python dictionaries
• Python dictionary with keys having multiple inputs
• Python program to find the sum of all items in a dictionary
• Python program to find the size of a Dictionary

>> More Programs on Dictionary

Python Set Exercises

• Find the size of a Set in Python
• Iterate over a set in Python
• Python – Maximum and Minimum in a Set
• Python – Remove items from Set
• Python – Check if two lists have atleast one element common

>> More Programs on Sets

• Python – Assigning Subsequent Rows to Matrix first row elements
• Python – Group similar elements into Matrix

>> More Programs on Matrices

>> More Programs on Functions

• Python | Find the Number Occurring Odd Number of Times using Lambda expression and reduce function

>> More Programs on Lambda

• Programs for printing pyramid patterns in Python

>> More Programs on Python Pattern Printing

• Python program to get Current Time
• Get Yesterday’s date using Python
• Python program to print current year, month and day
• Python – Convert day number to date in particular year
• Get Current Time in different Timezone using Python

>> More Programs on DateTime

>> More Programs on Python OOPS

• Python – Check if String Contain Only Defined Characters using Regex

>> More Programs on Python Regex

>> More Programs on Linked Lists

>> More Programs on Python Searching

• Python Program for Bubble Sort
• Python Program for QuickSort
• Python Program for Insertion Sort
• Python Program for Selection Sort
• Python Program for Heap Sort

>> More Programs on Python Sorting

• Program to Calculate the Edge Cover of a Graph
• Python Program for N Queen Problem

>> More Programs on Python DSA

• Read content from one file and write it into another file
• Write a dictionary to a file in Python
• How to check file size in Python?
• Find the most repeated word in a text file
• How to read specific lines from a File in Python?

>> More Programs on Python File Handling

• Update column value of CSV in Python
• How to add a header to a CSV file in Python?
• Get column names from CSV using Python
• Writing data from a Python List to CSV row-wise

>> More Programs on Python CSV

>> More Programs on Python JSON

• Python Script to change name of a file to its timestamp

>> More Programs on OS Module

• Python | Create a GUI Marksheet using Tkinter
• Python | ToDo GUI Application using Tkinter
• Python | GUI Calendar using Tkinter
• File Explorer in Python using Tkinter
• Visiting Card Scanner GUI Application using Python

>> More Programs on Python Tkinter

NumPy Exercises

• How to create an empty and a full NumPy array?
• Create a Numpy array filled with all zeros
• Create a Numpy array filled with all ones
• Replace NumPy array elements that doesn’t satisfy the given condition
• Get the maximum value from given matrix

>> More Programs on NumPy

Pandas Exercises

• Make a Pandas DataFrame with two-dimensional list | Python
• How to iterate over rows in Pandas Dataframe
• Create a pandas column using for loop
• Create a Pandas Series from array
• Pandas | Basic of Time Series Manipulation

>> More Programs on Python Pandas

>> More Programs on Web Scraping

• Download File in Selenium Using Python
• Bulk Posting on Facebook Pages using Selenium
• Google Maps Selenium automation using Python
• Count total number of Links In Webpage Using Selenium In Python
• Extract Data From JustDial using Selenium

>> More Programs on Python Selenium

• Number guessing game in Python
• 2048 Game in Python
• Get Live Weather Desktop Notifications Using Python
• 8-bit game using pygame
• Tic Tac Toe GUI In Python using PyGame

>> More Projects in Python

In closing, we just want to say that the practice or solving Python problems always helps to clear your core concepts and programming logic. Hence, we have designed this Python exercises after deep research so that one can easily enhance their skills and logic abilities.

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The Python REPL

Introduction.

In this chapter, you will learn how to write and run your first lines of Python code at the Python REPL also called the Python prompt. You will learn how to use Python as a calculator and be introduced to Python variables and Python's print() function. By the end of this chapter, you will be able to:

Open and close the Python REPL

Compute mathematical calculations using the Python REPL

Use the output from the Python REPL as input in another problem

Import the math and statistics modules from the Python Standard Library and use their functions

Assign values to variables

Use variables in calculations

Create strings

Combine and compare strings

Python-bloggers

Data science news and tutorials - contributed by python bloggers, solving recurrence relations.

Posted on May 9, 2024 by John Mount in Data science | 0 Comments

Introduction

A neat bit of “engineering mathematics” is solving recurrence relations . The solution method falls out of the notation itself, and harkens back to a time where formal sums were often used in place of vector subscript notation.

Unfortunately the variety of such solutions is small enough to allow teaching by memorization. In this note I try to avoid memorization, and motivate the methodology. I feel this is facilitated by separating a number of often smeared together representations (formulas, sequences, vectors, linear checks, characteristic polynomial, and polynomial check families) into distinct realizations. We are also going to emphasize calculating and confirming claims in Python.

The problem

A simple form of the recurrence problem is to write down a general solution for a subscripted family of linear equations such as the following

Fn+2 = Fn+1 + Fn

where n is a subscript varying over all positive integers.

Such a relation or equation can arise in number of situations or applications:

• Time series analysis .
• Estimating run times of algorithms.
• Combinatorics.

The above example is in fact the Fibonacci sequence .

The question is: if we are given initial conditions F1 = 1 and F2 = 1 , what is Fn (for general non-negative integer n )?

In this case the Wikipedia solution is Fn = (r1n - r2n)/sqrt(5) where:

• r1 = (1 + sqrt(5))/2
• r2 = (1 - sqrt(5))/2

Natural questions include:

• Why is the solution in this form?
• How do you find r1 and r2 ?

We will set up some notation and then solve a few examples.

The Solution

Vector space notation.

Let’s formalize our notation a bit.

First let’s settle on working over the vector space of all infinite sequences of real numbers with non-negative subscripts. This is just saying we consider the infinite sequence F = (F1, F2, F3, ...) we are solving for as one of many possible sequences. We can use “ R[Z+] ” as the group-ring style naming of this vector space. Now consider any such vector v that obeys the recurrence equations:

vn+2 = vn+1 + vn for n = 1, 2, ...

Let “ S ” denote the subset of R[Z+] that obey all of the above linear recurrence checks. We claim a few things about S (the set of solutions to our current example system):

• The all zeroes vector is in S . So S is non-empty.
• If v obeys all of the above constraints then so does c v for any constant c.
• If u and v obey all of the above constraints then so does u + v .
• By induction, all vn are completely determined by the values v1 and v2 .

The first three observations tell us S is a vector subspace of R[Z+] . The fourth observation tells us this vector subspace is 2 dimensional, so any solution can be written as the linear combination of two basis solutions.

Inspecting the claimed Fibonacci solution

The neat trick.

The core of the solution follows from a neat trick: replace the subscripts with superscripts. This is very powerful trick. Let’s see that in action.

We gamble and hope some of the solutions are of the following simple form: [rn | n = 1, 2, ...] = (r, r2, r3, ...) , where r is a to be determined (possibly complex ) number.

Our claim is: if the number r is a solution to the polynomial equation (in x )

then v = [rn | n = 1, 2, ...] satisfies

. The polynomial is called “the characteristic polynomial” of the linear recurrence checks.

The “trick” to this is: if x = r is a root of, or satisfies, x2 = x + 1 , then it also satisfies xn+2 = xn+1 + xn (which is equivalent to xn x2 = xn x + xn 1 ) for all n ≥ 0 . So rn+2 = rn+1 + rn for all n ≥ 0 which is exactly the claim v = [rn | n = 1, 2, ...] is a solution to all of the subscripted v -checks. It pays to think of the characteristic polynomial p(x) as shorthand for the family of “check polynomials” xn p(x) for n = 0, 1, 2, ... . However, for some problems we will need to appeal directly to the family of check polynomials.

In our case, the roots, or solutions, to this polynomial equation are the roots x2 - x - 1 = 0 . By the quadratic formula :

A harder example

Suppose we want to solve the recurrence:

Wn+2 = 6 Wn+1 - 9 Wn for n = 1, 2, ...

The above are the “ W checks” we now want to satisfy. A solution to these is a specific vector of values we can substitute in for the W ‘s, such that none of the equations are false.

We will try the earlier solution strategy. We are then interested in roots of the corresponding characteristic polynomial:

x2 - 6 x + 9 = 0

The above polynomial factors into (x - 3)2 . So r1 = r2 = 3 . So we know [3n | n = 1, 2, ...] is a solution to the W checks.

The space of solutions is again 2 dimensional. So to parameterize over all the possible solutions, we need a second linear independent solution. The question then is: how do we find a second linear independent solution?

Dealing with repeated roots

When our characteristic polynomial p(x) has repeated roots, the characteristic polynomial is not expressive enough to represent some solutions. However, the corresponding family of check polynomials xn p(x) for n = 0, 1, ... are rich enough to find the extra solutions. We will use that when a polynomial p(x) has a repeated root (that is: for some number r , the fact that (x-r)2 divides into p(x) with no remainder), then p(x) shares that root with p'(x) (where p'(x) is the derivative of p(x) with respect to x ).

Take the earlier “ W check” polynomials p(x) = xn+2 - 6 xn+1 + 9 xn . Define the polynomial y(x) = x p'(x) = (n+2) xn+2 - 6 (n+1) xn+1 + 9 n xn . y(x) itself is (by inspection) the check polynomials corresponding to the following (new) linear recurrence checks:

(n+2) Yn+2 = 6 (n+1) Yn+1 - 9 (n) Yn for n = 1, 2, ...

As y(3) = 0 (true because 3 is a double root of p(x) ) we know [3n | n = 1, 2, ...] is a solution to the new Y linear recurrence checks. Substitute this valid Y solution [3n | n = 1, 2, ...] into the Y checks to derive the following family of (true or valid) equations.

((n+2) 3n+2) = 6 ((n+1) 3n+1) - 9 ((n) 3n) for n = 1, 2, ...

Confirming the last claim

Frankly, this sort of argument doesn’t fully sink in until one confirms some examples and calculations. The derivation is a bit of “proof by change of notation”, which is never very satisfying.

So: let’s confirm some calculations claimed in the example to try to build some familiarity with the items discussed.

Back to the example

As we now have the required number of linearly independent solutions (2 solutions: [3n | n = 1, 2, ...] and [n 3n | n = 1, 2, ...] ), we can solve for any specified initial conditions (as demonstrated earlier).

Believe it or not, we are done.

The general solution strategy is as follows.

For a general homogeneous linear recurrence of the form:

vn+k = ck-1 vn+k-1 + ... + c0 vn

Move to the characteristic polynomial equation:

xk = ck-1 xk-1 + ... + c0

We can generate a basis for all solutions as v = [ nk rn | n = 1, 2, ...] where r is a root of the characteristic polynomial, and k is a non-negative integer smaller than the degree of multiplicity of the root r . Once we have enough linear independent solutions, we can write any other solution as a linear combination of what we have.

We call all of the above the “swap subscripts (general and powerful) to powers (specific and weak)” strategy (though there are obviously a few more steps and details to the method).

Our solution strategy was exchanging powerful subscripts (which can implement any integer keyed lookup table) with less powerful superscripts (that can only express powers). We can lift any solution found in the weaker world (power series) to the more general one (subscripted sequences). We don’t always find enough power series solutions, and in that case we transform the problem to find more solutions to modified polynomials. The trick is to track the details of how the transforms operate on both our vector solutions and check polynomials.

The above system is general, it can solve a lot of problems. We concentrated on calculating over vector values. Related methods include the umbral calculus , the study of shift operators , and the Laplace transform .

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