Numerical Methods Collection

C++ implementations of classic numerical methods used in scientific computing and engineering, developed as part of the Numerical Methods course during my studies.

Projects Overview

Polynomial Approximation

Least squares polynomial approximation using Gaussian elimination. Fits best-fit polynomial to data points.

• Method: Normal equations + Gaussian elimination
• Features: Degree up to 5, dynamic size arrays
• Input: Data points (x, y) pairs

Numerical Differentiation

Estimates derivatives of tabulated functions using finite differences.

• Methods: Forward and backward differences
• Features: Difference tables, multiple methods
• Input: Equally spaced function values

Horner's Algorithm

Efficient polynomial evaluation using Horner's nested multiplication scheme.

• Complexity: O(n) multiplications vs O(n²) naive method
• Features: Step-by-step computation display
• Input: Polynomial coefficients and evaluation point

Numerical Integration

Approximates definite integrals using multiple quadrature methods.

• Methods:
  • Rectangle (underestimate/overestimate)
  • Trapezoid rule
  • Simpson's rule
• Features: Menu-driven, easy method comparison
• Input: Equally spaced function values

Polynomial Interpolation

Finds polynomial passing through given data points using Gaussian elimination.

• Method: Vandermonde matrix + Gaussian elimination
• Features: Exact interpolation, coefficient back substitution
• Input: Data points (exact pass-through)

General Usage

Each project has its own directory with:

• main.cpp - C++ source code
• test.txt - Sample data
• README.md - Method-specific documentation

Compilation

cd <project_directory>
g++ -o main main.cpp
./main

On Windows:

g++ -o main.exe main.cpp
main.exe

Requirements

• C++11 or later
• G++ compiler (or compatible)
• Standard library (iostream, vector, cmath, fstream, etc.)

Project Structure

num_methods/
├── approximation/      # Least squares fitting
├── differentiation/    # Finite differences
├── horner/             # Polynomial evaluation
├── integration/        # Numerical integration
├── interpolation/      # Exact polynomial fitting
└── README.md          # This file

Typical Workflow

1. Approximation: Use when you want best-fit polynomial
2. Interpolation: Use when you need exact polynomial through points
3. Differentiation: Estimate derivatives from tables
4. Integration: Approximate area under curve
5. Horner: Fast evaluation of approximated/interpolated polynomials

Notes

• All programs read from test.txt in the same directory
• Data format: space or newline separated x,y pairs
• Most use Gaussian elimination for numerical stability
• See individual README.md for method specifics