Numerical Analysis Project - Root Finding, System Solving, and PageRank Algorithm

This project consists of several numerical analysis exercises implemented in Python. It covers root-finding methods, solving linear systems of equations, and a PageRank-inspired algorithm for calculating the importance of pages within a network.

Table of Contents

• Project Overview
• Technologies Used
• Installation and Setup
• Features
• How to Run
• Additional Resources

Project Overview

The project is divided into four main exercises:

1. Root-Finding Methods: Implements Bisection, Newton-Raphson, and Secant methods to approximate the roots of a given function.
2. Modified Root-Finding Methods: Adjusts the standard root-finding algorithms to increase precision and solve more complex root scenarios.
3. Solving Linear Systems: Uses LU decomposition to solve systems of equations in the form (Ax = b).
4. PageRank-inspired Algorithm: Calculates page importance based on adjacency matrices and applies the power method to approximate eigenvalues and eigenvectors.

For a detailed explanation of the mathematical methods and results, refer to the full report (PDF in Greek).

Technologies Used

• Python: The project was implemented and tested in Python using basic libraries and no external dependencies.
• Markdown: Used for documentation and project presentation on GitHub.

Installation and Setup

1. Clone this repository:

   git clone https://github.com/theodougalis/Numerical-Analysis-Project---Root-Finding--System-Solving--and-PageRank-Algorithm.git

2. Make sure Python 3 is installed on your machine.
3. Navigate to the project directory and open the Python file NumericalAnalysis.py in your preferred Python environment.

Features

Exercise 1 - Root-Finding Methods

• Bisection Method: Approximates roots by iteratively narrowing down an interval.
• Newton-Raphson Method: Uses the derivative to converge quickly to a root.
• Secant Method: Similar to Newton-Raphson but does not require the derivative.

Exercise 2 - Modified Root-Finding Methods

• Customized Bisection, Newton-Raphson, and Secant methods for enhanced accuracy and performance on complex root types.

Exercise 3 - Solving Linear Systems with LU Decomposition

• Implements LU decomposition with pivoting to solve systems of linear equations.

Exercise 4 - PageRank-inspired Algorithm for Page Importance

• Uses the power method to calculate page importance based on connectivity, inspired by Google's PageRank algorithm.
• Analyzes the effect of probability (q) on the importance of randomly accessing pages.

How to Run

1. Run Root-Finding Methods:

   • To test the root-finding methods, execute:

     python NumericalAnalysis.py

   • Use functions like dixotomisif(), NRf(), and Secantf() with appropriate parameters to find roots.

2. Solve Linear Systems:

   • To solve linear systems, use the function axb() for specific matrix and vector inputs.

3. Calculate Page Importance:

   • Run the task1() through task6() functions for different setups to simulate page importance under varying conditions.

Additional Resources

• Project Report (PDF) - Contains detailed explanations of the methods, calculations, and results in Greek.

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