/Core-Programming

Coursework solutions for the Core Programming course covering numerical methods and their implementation in Python.

Primary LanguagePython

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.