This repository contains a comprehensive scientific report for the "Maths for Engineering" module, detailing an independent investigation into numerical methods for solving equations of the form f(x) = 0. Such root-finding problems are crucial in engineering and technology for understanding system behaviour.
The project involved in-depth research, implementation, and critical evaluation of three key numerical methods: Interval Bisection, the Rearrangement Method (Fixed Point Iteration), and the Newton-Raphson Method. The primary focus was on applying these methods to a specific polynomial function, analysing their performance, convergence, accuracy, and potential failure modes.
The study emphasises the practical application of these methods, the importance of initial graphical analysis (using tools like Desmos), and the iterative calculation process (managed with tools like Excel).
The specific polynomial function investigated in this case study is:
f(x)= x^5 + 3.8x^4 - 46.7x^3 - 43.9x^2 + 103.5x + 48.1
(This function is known to have 5 real roots between x = -10 and +10).
- Scientific Case Study Report: This document provides a detailed theoretical background of each numerical method, the implementation process, results (including roots found to specified accuracies, iteration counts, uncertainty analysis), discussion of method failures, and a comparative conclusion. View Full Case Study Report
- Supplementary Materials: View Desmos graph of the Rearrangement Method View Desmos graph of the Newton-Raphson Method
The core objectives of this independent research project were to:
- Investigate and Understand: Gain a thorough understanding of the theory and operational mechanics of:
- The Interval Bisection Method
- The Rearrangement Method (Fixed Point Iteration)
- The Newton-Raphson Method
- Implement and Apply:
- Apply Interval Bisection to find two roots of the given function to at least 3 significant figures.
- Apply the Rearrangement Method to find the same two roots to at least 6 significant figures.
- Apply the Newton-Raphson Method to find the same two roots to at least 9 significant figures.
- Analyze Performance & Accuracy:
- Document the number of iterations required by each method for the desired accuracy.
- Determine and report the uncertainty in the final root values.
- Provide visual illustrations of each method's iterative process.
- Evaluate Robustness:
- Investigate and demonstrate cases where each method might fail or diverge.
- Discuss the conditions leading to such failures.
- Compare and Conclude:
- Critically compare the three methods based on ease of use, speed of convergence, and reliability.
- Draw conclusions supported by evidence from the results.
This project enabled the development and demonstration of several key skills valuable for research and technical roles:
- Numerical Analysis: Practical application and understanding of Interval Bisection, Fixed-Point Iteration (Rearrangement), and Newton-Raphson methods.
- Mathematical Problem-Solving: Applying mathematical theory to solve complex equations and find roots.
- Critical Thinking & Evaluation: Analysing the advantages, disadvantages, convergence properties, and failure modes of different numerical techniques.
- Scientific Reporting & Documentation: Writing a formal scientific report, including explanations of theory, methodology, results, and conclusions.
- Data Presentation: Clearly presenting numerical results, iteration counts, and uncertainties.
- Attention to Detail: Ensuring high accuracy in calculations and adherence to specified significant figures.
- Independent Research & Learning: Mastering complex topics and conducting research autonomously.
- Iterative Process Management: Understanding and executing iterative solution-finding processes.
- Numerical Calculations & Iterations: Microsoft Excel
- Graphical Analysis & Visualisation: Desmos
- Report Writing & Documentation: Microsoft Word
- Research & Analysis: Reference1, Reference2, Reference3
Copyright (c) 2025 [Sushant Jasra Kumar]
The code for all projects in this portfolio is licensed under the MIT License.
All non-code assets, including PDF documents, images, and visual designs, are dedicated to the public domain under the Creative Commons Zero v1.0 Universal (CC0 1.0).