Numerical Methods Laboratory

This repository contains Python implementations of various numerical methods covered in a Numerical Methods course. Each lab file demonstrates different computational techniques for solving mathematical problems numerically.

Overview

Numerical methods are algorithms for approximating solutions to mathematical problems that cannot be solved analytically. This collection covers fundamental techniques in error analysis, interpolation, differentiation, integration, and differential equations.

Lab Files

Lab 2 & Lab 2.1: Error Analysis

• Topics: Absolute error, relative error, significant digits
• Concepts: Error propagation, precision analysis, comparison of error types
• Key Functions: Error calculation formulas, significant digit determination

Lab 3: Differentiation and Convergence

• Topics: Symbolic derivatives, numerical differentiation, convergence order
• Concepts: Computational order of convergence, error analysis in differentiation
• Key Functions: Symbolic differentiation using SymPy, convergence rate calculation

Lab 5: Lagrange Interpolation

• Topics: Polynomial interpolation, Lagrange basis polynomials
• Concepts: Interpolation theory, basis functions, polynomial approximation
• Key Functions: Lagrange interpolation algorithm, basis term calculation

Lab 6: Interpolation and Finite Differences

• Topics: Lagrange interpolation, forward difference tables
• Concepts: Polynomial interpolation, finite difference methods
• Key Functions: Lagrange polynomial construction, difference table generation

Lab 8: Numerical Integration and Hermite Interpolation

• Topics: Trapezoidal rule, Simpson's rules (1/3 and 3/8), Hermite interpolation
• Concepts: Composite integration methods, oscillatory integral approximation, derivative-based interpolation
• Key Functions: Integration algorithms, Hermite polynomial construction

Lab 9: Gaussian Quadrature

• Topics: Gauss-Legendre, Gauss-Chebyshev, Gauss-Laguerre quadrature
• Concepts: Weighted integration, orthogonal polynomials, high-accuracy quadrature
• Key Functions: Gaussian quadrature implementations for different weight functions

Lab 10: Ordinary Differential Equations

• Topics: Euler method, Heun's method, Runge-Kutta 2nd order
• Concepts: Initial value problems, multi-step methods, predictor-corrector schemes
• Key Functions: ODE solvers for systems of equations

Dependencies

The implementations use the following Python libraries:

• numpy - Numerical computations and arrays
• sympy - Symbolic mathematics
• matplotlib - Plotting and visualization
• scipy - Scientific computing (for special functions)
• pandas - Data manipulation (for tables)

Installation

1. Ensure Python 3.7+ is installed
2. Install required packages:

   pip install numpy sympy matplotlib scipy pandas

Usage

Each lab file contains standalone functions that can be run independently. Most files include example problems with predefined data. To run a specific lab:

python lab_2.py

Or import specific functions:

from lab_5 import lagrange_interpolation
result = lagrange_interpolation([1, 2, 4], [2, 3, 1], 2.5)

Educational Purpose

These implementations are designed for learning numerical methods. They include:

• Detailed comments explaining algorithms
• Error handling for edge cases
• Step-by-step output for understanding
• References to mathematical formulas

Course Context

This repository corresponds to laboratory assignments in a Numerical Methods course, typically covering:

• Error analysis and floating-point arithmetic
• Interpolation and approximation theory
• Numerical differentiation and integration
• Solution of ordinary differential equations

Contributing

This is an educational repository. For improvements or corrections, please ensure changes maintain the pedagogical value of the implementations.