Introduction to Core Programming

Course Overview

This repository contains my coursework solutions for the Core Programming course. The course covers numerical methods and their implementation in Python, including root-finding, solving linear systems, polynomial interpolation, numerical calculus, and solving ODEs.

Directory Structure

Calculation (CAL)

1. CAL1.py:

   • Implements Bernoulli numbers and their applications in mathematical functions.
   • Includes the bernoulli function to compute Bernoulli numbers.
   • Includes the pn function to compute values using Bernoulli numbers.

2. CAL2.py:

   • Implements numerical differentiation using forward, backward, and central difference formulas.
   • Includes visualisation of the numerical differentiation results.

3. CAL3.py:

   • Implements numerical integration using trapezoidal, Simpson's, and midpoint rules.
   • Includes visualisation of the numerical integration results.

Linear Systems (LIN)

1. LIN1.py:

   • Implements Gaussian elimination for solving linear systems.
   • Includes operation count and pivoting strategies.

2. LIN2.py:

   • Implements Jacobi and Gauss-Seidel methods for solving linear systems iteratively.
   • Includes matrix norms and convergence analysis.

3. LIN3.py:

   • Implements matrix factorisation techniques.
   • Includes special matrices like diagonally dominant and symmetric positive definite matrices.

Analysis of Functions (ACF)

1. ACF1.py:

   • Implements root-finding methods: Bisection method, fixed-point iteration, and Newton’s method.
   • Includes convergence analysis of these methods.

2. ACF2.py:

   • Implements polynomial interpolation using Lagrange polynomials.
   • Includes error analysis of the interpolation.

3. ACF3.py:

   • Implements numerical solutions for ordinary differential equations (ODEs) using Euler’s method and higher-order Runge-Kutta methods.
   • Includes local truncation error analysis and visualisation.

Learning Outcomes

Upon successful completion of this course, students will be able to:

• Solve nonlinear equations using root-finding methods, and analyse their convergence.
• Solve linear systems of equations using direct methods, and analyse their computational complexity.
• Solve linear systems of equations using iterative techniques, and analyse their convergence.
• Approximate functions by polynomial interpolants, and analyse their accuracy.
• Approximate derivatives and definite integrals using numerical differentiation and integration, and analyse their convergence.
• Approximate ODEs using numerical methods, and analyse their convergence.
• Implement reusable codes in Python.