Numerical Methods Analysis: Precision and Errors

This repository contains a collection of numerical analysis experiments conducted as part of my Master's in Computer Engineering at Bilecik Şeyh Edebali University. The project focuses on practical implementations of mathematical concepts and their numerical stability in Python.

🔬 Project Overview

Computing is inherently limited by floating-point precision. This project explores how different algorithms behave under various conditions, specifically analyzing the trade-off between mathematical accuracy and computational limits.

Topics Covered:

• Numerical Differentiation: Implementation of Forward, Backward, and Centered difference methods. Analysis of the balance between Truncation Error and Round-off Error (the "V-curve" phenomenon).
• Taylor Series Convergence: Approximating $e^x$ and observing how the relative error decreases as the degree of the polynomial increases.
• Numerical Stability (Underflow): Handling extremely small probabilities using Log-domain summation to prevent arithmetic underflow.
• Variance Calculation Stability: Comparing naive vs. stable (two-pass) algorithms to avoid Catastrophic Cancellation when handling large datasets.

🛠 Tools & Libraries

• Python 3
• NumPy: For advanced array manipulation and mathematical stability.
• Matplotlib: For visualizing error convergence and stability analysis.

📊 Key Findings

Through these experiments, we identified that the Centered Difference method provides higher accuracy ($O(h^2)$) compared to Forward/Backward methods, but remains vulnerable to precision loss at extremely small step sizes ($h < 10^{-12}$).

---

Developed by Ahmed Azaz