Python Numerical Computing Utils

A comprehensive collection of Python implementations for fundamental numerical methods used in scientific computing, engineering, and applied mathematics. This repository serves as a learning resource and practical toolbox for solving problems involving linear systems, nonlinear equations, numerical differentiation, integration, and interpolation.

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Features

• Linear Algorithms: Methods for solving linear systems, including direct and iterative techniques.
• Non-Linear Algorithms: Root-finding methods for nonlinear equations.
• Numerical Differentiation: Techniques for approximating derivatives.
• Numerical Integration: Methods for evaluating definite integrals numerically.
• Numerical Interpolation: Algorithms for constructing new data points within the range of a discrete set.

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Directory Structure

python-numerical-computing-utils/
│
├── linear-algorithms/
│   ├── gauss_elimination_method.py         # Classical Gauss Elimination for linear systems
│   ├── gauss_jordan_method.py              # Gauss-Jordan elimination (row reduction)
│   ├── gauss_seidel_iterative_method.py    # Gauss-Seidel iterative solver
│   ├── jacobi_iterative_method.py          # Jacobi iterative solver
│   ├── lu_decomposition_method.py          # LU decomposition for matrix factorization
│   ├── test.py                            # Tests for linear algorithms
│   └── .gitignore
│
├── non-linear-algorithms/
│   ├── bisection_method.py                 # Bisection root-finding method
│   ├── fixed_point_iteration_method.py     # Fixed-point iteration for roots
│   ├── newton_raphson_method.py            # Newton-Raphson method for nonlinear equations
│   ├── regula_falsi_method.py              # Regula Falsi (False Position) method
│   ├── regula_secant_method.py             # Secant method for root finding
│   ├── test.py                            # Tests for non-linear algorithms
│   └── .gitignore
│
├── numerical-differentiation/
│   ├── backward_differentiation.py         # Backward finite difference for derivatives
│   ├── forward_differentiation.py          # Forward finite difference for derivatives
│   ├── test.py                            # Tests for numerical differentiation
│   └── .gitignore
│
├── numerical-integration/
│   ├── simpsons_one_by_three_rule.py       # Simpson's 1/3 rule for integration
│   ├── simpsons_three_by_eight_rule.py     # Simpson's 3/8 rule for integration
│   ├── trapzoidals_rule.py                 # Trapezoidal rule for integration
│   ├── test.py                            # Tests for numerical integration
│   └── .gitignore
│
├── numerical-interpolation/
│   ├── gauss_forward_central_difference.py # Gauss forward central difference interpolation
│   ├── lagrange_interpolation.py           # Lagrange polynomial interpolation
│   ├── newtons_backward_difference.py      # Newton's backward difference interpolation
│   ├── newtons_divided_difference.py       # Newton's divided difference interpolation
│   ├── newtons_forward_difference.py       # Newton's forward difference interpolation
│   ├── test.py                            # Tests for numerical interpolation
│   └── .gitignore
│
└── README.md                               # Project documentation (this file)

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Module Overview

Linear Algorithms

• Direct Methods: gauss_elimination_method.py, gauss_jordan_method.py, lu_decomposition_method.py
• Iterative Methods: jacobi_iterative_method.py, gauss_seidel_iterative_method.py
• Testing: test.py for unit tests and usage examples

Non-Linear Algorithms

• Root-finding: Bisection, Fixed-point, Newton-Raphson, Regula Falsi, Secant methods
• All implementations include test scripts for validation

Numerical Differentiation

• Forward and backward finite difference methods for approximating derivatives of functions
• Accompanied by test scripts

Numerical Integration

• Trapezoidal rule, Simpson’s 1/3 and 3/8 rules for numerical integration of functions
• Includes tests for demonstration

Numerical Interpolation

• Lagrange and Newton methods for interpolation of discrete data
• Gauss forward central difference for interpolation
• Example and test scripts included

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Getting Started

1. Clone the Repository

   git clone https://github.com/PythoneerSamurai/python-numerical-computing-utils.git
   cd python-numerical-computing-utils

2. Run Tests or Examples

   • Each submodule contains a test.py file to demonstrate usage.

3. Dependencies

   • Only standard Python libraries used (unless noted in the script headers).

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Contribution

Issues, improvements, and new numerical algorithms are welcome! Please open an issue or submit a pull request.

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License

CC BY License.

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Author

PythoneerSamurai