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.

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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

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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

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3. Error Analysis

Interpolation Error

• Error polynomial
• Error bounds
• Conditioning effects

Quadrature Error

• Degree of accuracy
• Error estimates
• Convergence behaviour

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4. Composite Quadrature Methods

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

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5. Extrapolation Methods

• Romberg extrapolation
• Accuracy improvement strategies

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6. Numerical Methods for Differential Equations

• Initial value problems
• Discretisation methods
• Stability considerations
• Convergence analysis