Python Numerical Methods Toolkit

A comprehensive collection of 20 Python programs implementing fundamental numerical methods, designed as a single, self-contained repository for students, engineers, and researchers.

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Table of Contents

• Numerical Methods Included
  • Root Finding
  • Linear Systems
  • Interpolation
  • Numerical Integration
  • Numerical Differentiation
  • Ordinary Differential Equations
  • Eigenvalue Problems
• Project Structure
• Running the Programs
• Requirements
• Customisation
• License

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Numerical Methods Included

Root Finding

#	Program	Method	Order of Convergence
01	01_bisection.py	Bisection Method	Linear
02	02_newton_raphson.py	Newton-Raphson Method	Quadratic
03	03_secant.py	Secant Method	Superlinear
04	04_false_position.py	False Position (Regula Falsi)	Linear (usually faster than bisection)

Linear Systems

#	Program	Method	Complexity
05	05_gauss_elimination.py	Gaussian Elimination (partial pivoting)	O(n^3)
06	06_lu_decomposition.py	LU Decomposition (Doolittle)	O(n^3) factorise, O(n^2) solve
07	07_jacobi.py	Jacobi Iterative Method	O(n^2) per iteration
08	08_gauss_seidel.py	Gauss-Seidel Iterative Method	O(n^2) per iteration

Interpolation

#	Program	Method	Notes
09	09_lagrange_interpolation.py	Lagrange Polynomial Interpolation	Degree n polynomial
10	10_newton_interpolation.py	Newton's Divided Difference Interpolation	Incremental construction
11	11_spline_interpolation.py	Natural Cubic Spline Interpolation	Piecewise smooth fit

Numerical Integration

#	Program	Method	Accuracy
12	12_trapezoidal.py	Composite Trapezoidal Rule	O(h^2)
13	13_simpson.py	Composite Simpson's 1/3 Rule	O(h^4)
14	14_romberg.py	Romberg Integration (Richardson extrapolation)	Exponential convergence

Numerical Differentiation

#	Program	Method	Accuracy
15	15_finite_difference.py	Forward & Central Finite Differences	O(h) / O(h^2)
16	16_higher_order_difference.py	5-Point Stencil (higher-order)	O(h^4)

Ordinary Differential Equations

#	Program	Method	Accuracy
17	17_euler_ode.py	Euler's Method	O(h)
18	18_rk4_ode.py	4th-Order Runge-Kutta	O(h^4)

Eigenvalue Problems

#	Program	Method	Notes
19	19_eigen_power.py	Power Method (dominant eigenvalue)	Iterative
20	20_eigen_qr.py	QR Algorithm (all eigenvalues)	Modified Gram-Schmidt

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Project Structure

python-numerical-methods/
├── README.md # This file
├── LICENSE # MIT License
├── requirements.txt # Optional dependencies (numpy, matplotlib)
├── .gitignore
├── Makefile # Convenience targets
├── utils.py # Shared utilities (print helpers, matrix ops)
├── src/
│ ├── 01_bisection.py
│ ├── 02_newton_raphson.py
│ ├── 03_secant.py
│ ├── 04_false_position.py
│ ├── 05_gauss_elimination.py
│ ├── 06_lu_decomposition.py
│ ├── 07_jacobi.py
│ ├── 08_gauss_seidel.py
│ ├── 09_lagrange_interpolation.py
│ ├── 10_newton_interpolation.py
│ ├── 11_spline_interpolation.py
│ ├── 12_trapezoidal.py
│ ├── 13_simpson.py
│ ├── 14_romberg.py
│ ├── 15_finite_difference.py
│ ├── 16_higher_order_difference.py
│ ├── 17_euler_ode.py
│ ├── 18_rk4_ode.py
│ ├── 19_eigen_power.py
│ └── 20_eigen_qr.py
└── tests/
└── test_all_methods.py # Automated verification of all methods

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Running the Programs

Run a single method

# From the project root directory
python src/01_bisection.py
# Or using Make
make run METHOD=01_bisection

Run all methods sequentially

make run-all

Run the test suite

make test# or
python -m pytest tests/
# or (pure stdlib)
python tests/test_all_methods.py

Example output

$ python src/01_bisection.py
╔══════════════════════════════════════════════════════════════╗
║ Bisection Method — Root Finding ║
╚══════════════════════════════════════════════════════════════╝
Enter lower bound (a): 1
Enter upper bound (b): 2
Enter tolerance: 1e-10
Enter maximum iterations: 100
────────────────────────────────────────────────────────────────
Iter a b x_mid f(x_mid)
────────────────────────────────────────────────────────────────
1 1.00000000 2.00000000 1.50000000 -0.12500000
2 1.50000000 2.00000000 1.75000000 1.10937500
...
Converged after 34 iterations.
Root = 1.5213797068
f(root) = -0.0000000003

Each program:

• Is interactive — prompts you for input values
• Displays detailed iteration tables showing convergence
• Shows error estimates comparing to exact solutions (where applicable)
• Has consistent formatted output via the shared utils.py

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Requirements

Core (zero dependencies — Python 3.7+)

All 20 programs use only the Python standard library (math, sys). No installation is needed.

Optional (for tests and plots)

pip install -r requirements.txt

Package	Purpose
numpy	Optional — used in test suite for verification
matplotlib	Optional — used in test suite for comparison plots
pytest	Optional — alternative test runner

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Customisation

To implement your own function, simply edit the f() function at the top of any source file. For example, in src/01_bisection.py:

deff(x):
# Change this to your own functionreturnmath.cos(x) -x# root near 0.7391

All root-finding, ODE, integration, and differentiation programs follow this pattern — the target function is defined at the top and clearly labelled.

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Comparison with the C++ Version

A companion C++ implementation is available with identical algorithms, method-by-method, for performance-critical applications or CUDA/OpenMP parallelisation.

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License

This project is licensed under the MIT License. See LICENSE for details. Free to use, modify, and distribute.