130 lines
4.0 KiB
Markdown
130 lines
4.0 KiB
Markdown
# numethods
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A lightweight, from-scratch, object-oriented Python package implementing classic numerical methods.
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**No NumPy / SciPy solvers used**, algorithms are implemented transparently for learning and research.
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## Why this might be useful
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- Great for teaching/learning numerical methods step by step.
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- Good reference for people writing their own solvers in C/Fortran/Julia.
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- Lightweight, no dependencies.
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- Consistent object-oriented API (.solve() etc).
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---
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## Tutorial Series
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This package comes with a set of Jupyter notebooks designed as a structured tutorial series in **numerical methods**, both mathematically rigorous and hands-on with code.
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### Core Tutorials
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1. [Tutorial 1: Vectors and Matrices](tutorials/tutorial1_vectors.ipynb)
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- Definitions of vectors and matrices.
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- Vector operations: addition, scalar multiplication, dot product, norms.
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- Matrix operations: addition, multiplication, transpose, inverse.
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- Matrix and vector norms.
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- Examples with `numethods.linalg`.
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2. [Tutorial 2: Linear Systems of Equations](tutorials/tutorial2_linear_systems.ipynb)
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- Gaussian elimination and Gauss–Jordan.
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- LU decomposition.
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- Cholesky decomposition.
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- Iterative methods: Jacobi and Gauss-Seidel.
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- Examples with `numethods.solvers`.
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3. [Tutorial 3: Orthogonalization and QR Factorization](tutorials/tutorial3_orthogonalization.ipynb)
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- Inner products and orthogonality.
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- Gram–Schmidt process (classical and modified).
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- Householder reflections.
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- QR decomposition and applications.
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- Examples with `numethods.orthogonal`.
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4. [Tutorial 4: Root-Finding Methods](tutorials/tutorial4_root_finding.ipynb)
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- Bisection method.
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- Fixed-point iteration.
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- Newton’s method.
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- Secant method.
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- Convergence analysis and error behavior.
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- Trace outputs for iteration history.
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- Examples with `numethods.roots`.
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- [Polynomial Regression Demo](tutorials/polynomial_regression.ipynb)
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- Step-by-step example of polynomial regression.
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- Shows how to fit polynomials of different degrees to data.
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- Visualizes fitted curves against the original data.
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---
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## Features
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### Linear system solvers
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- **LU decomposition** (with partial pivoting): `LUDecomposition`
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- **Gauss-Jordan** elimination: `GaussJordan`
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- **Jacobi** iterative method: `Jacobi`
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- **Gauss-Seidel** iterative method: `GaussSeidel`
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- **Cholesky** factorization (SPD): `Cholesky`
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### Root-finding
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- **Bisection**: `Bisection`
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- **Fixed-Point Iteration**: `FixedPoint`
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- **Secant**: `Secant`
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- **Newton's method** (for roots): `NewtonRoot`
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### Interpolation
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- **Newton** (divided differences): `NewtonInterpolation`
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- **Lagrange** polynomials: `LagrangeInterpolation`
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### Orthogonalization, QR, and Least Squares
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- **Classical Gram-Schmidt**: `QRGramSchmidt`
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- **Modified Gram-Schmidt**: `QRModifiedGramSchmidt`
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- **Householder QR** (numerically stable): `QRHouseholder`
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- **QR-based linear solver** (square systems): `QRSolver`
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- **Least Squares** for overdetermined systems (via QR): `LeastSquaresSolver`
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### Numerical Differentiation
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- **Forward difference**: `ForwardDiff`
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- **Backward difference**: `BackwardDiff`
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- **Central difference (2nd order)**: `CentralDiff`
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- **Central difference (4th order)**: `CentralDiff4th`
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- **Second derivative**: `SecondDerivative`
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- **Richardson extrapolation**: `RichardsonExtrap`
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### Matrix & Vector utilities
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- Minimal `Matrix` / `Vector` classes
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- `@` operator for **matrix multiplication**
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- `*` for **scalar**-matrix multiplication
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- `.T` for transpose
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- Forward / backward substitution helpers
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- Norms, dot products, row/column access
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---
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## Install (editable)
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```bash
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pip install -e /numethods
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```
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or just add `/numethods` to `PYTHONPATH`.
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## Examples
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```bash
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python /numethods/examples/demo.py
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```
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## Notes
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- All algorithms are implemented without relying on external linear algebra solvers.
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- Uses plain Python floats and list-of-lists for matrices/vectors.
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- Tolerances use a relative criterion `|Δ| ≤ tol (1 + |value|)`.
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