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Numerical-Methods

Exercises for the graduate course in Numerical Methods for Differential Equations

This repository contains my implementation of numerical methods covered in the course.

Course Contents

1. Numerical Basics

1.1 Interpolation

Polynomial interpolation
Vandermonde systems
Lagrange interpolation
Spline interpolation
B-splines

1.2 Approximation and Regression

Least squares approximation
Linear regression
Polynomial regression
Weighted least squares
QR decomposition

2. Numerical Integration (Quadrature)

Basic Quadrature Rules

Left rectangle rule
Right rectangle rule
Midpoint rule
Trapezoidal rule
Simpson's rule (Kepler’s rule)

Newton–Cotes Formulas

Polynomial-based quadrature
Degree of accuracy

Gaussian Quadrature

Optimal quadrature points
Maximum degree of accuracy

3. Error Analysis

Interpolation Error

Error polynomial
Error bounds
Conditioning effects

Quadrature Error

Degree of accuracy
Error estimates
Convergence behaviour

4. Composite Quadrature Methods

Composite rectangle rule
Composite midpoint rule
Composite trapezoidal rule
Composite Simpson's rule
Convergence order

5. Extrapolation Methods

Romberg extrapolation
Accuracy improvement strategies

6. Numerical Methods for Differential Equations

Initial value problems
Discretisation methods
Stability considerations
Convergence analysis