# numethods

A lightweight, from-scratch, object-oriented Python package implementing classic numerical methods.  
**No NumPy / SciPy solvers used**, algorithms are implemented transparently for learning and research.

## Why this might be useful

- Great for teaching/learning numerical methods step by step.
- Good reference for people writing their own solvers in C/Fortran/Julia.
- Lightweight, no dependencies.
- Consistent object-oriented API (.solve() etc).

---

## Tutorial Series

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.

### Core Tutorials

1. [Tutorial 1: Vectors and Matrices](tutorials/tutorial1_vectors.ipynb)

   - Definitions of vectors and matrices.
   - Vector operations: addition, scalar multiplication, dot product, norms.
   - Matrix operations: addition, multiplication, transpose, inverse.
   - Matrix and vector norms.
   - Examples with `numethods.linalg`.

2. [Tutorial 2: Linear Systems of Equations](tutorials/tutorial2_linear_systems.ipynb)

   - Gaussian elimination and Gauss–Jordan.
   - LU decomposition.
   - Cholesky decomposition.
   - Iterative methods: Jacobi and Gauss-Seidel.
   - Examples with `numethods.solvers`.

3. [Tutorial 3: Orthogonalization and QR Factorization](tutorials/tutorial3_orthogonalization.ipynb)

   - Inner products and orthogonality.
   - Gram–Schmidt process (classical and modified).
   - Householder reflections.
   - QR decomposition and applications.
   - Examples with `numethods.orthogonal`.

4. [Tutorial 4: Root-Finding Methods](tutorials/tutorial4_root_finding.ipynb)
   - Bisection method.
   - Fixed-point iteration.
   - Newton’s method.
   - Secant method.
   - Convergence analysis and error behavior.
   - Trace outputs for iteration history.
   - Examples with `numethods.roots`.

---

## Features

### Linear system solvers

- **LU decomposition** (with partial pivoting): `LUDecomposition`
- **Gauss-Jordan** elimination: `GaussJordan`
- **Jacobi** iterative method: `Jacobi`
- **Gauss-Seidel** iterative method: `GaussSeidel`
- **Cholesky** factorization (SPD): `Cholesky`

### Root-finding

- **Bisection**: `Bisection`
- **Fixed-Point Iteration**: `FixedPoint`
- **Secant**: `Secant`
- **Newton's method** (for roots): `NewtonRoot`

### Interpolation

- **Newton** (divided differences): `NewtonInterpolation`
- **Lagrange** polynomials: `LagrangeInterpolation`

### Orthogonalization, QR, and Least Squares

- **Classical Gram-Schmidt**: `QRGramSchmidt`
- **Modified Gram-Schmidt**: `QRModifiedGramSchmidt`
- **Householder QR** (numerically stable): `QRHouseholder`
- **QR-based linear solver** (square systems): `QRSolver`
- **Least Squares** for overdetermined systems (via QR): `LeastSquaresSolver`

### Numerical Differentiation

- **Forward difference**: `ForwardDiff`
- **Backward difference**: `BackwardDiff`
- **Central difference (2nd order)**: `CentralDiff`
- **Central difference (4th order)**: `CentralDiff4th`
- **Second derivative**: `SecondDerivative`
- **Richardson extrapolation**: `RichardsonExtrap`

### Matrix & Vector utilities

- Minimal `Matrix` / `Vector` classes
- `@` operator for **matrix multiplication**
- `*` for **scalar**-matrix multiplication
- `.T` for transpose
- Forward / backward substitution helpers
- Norms, dot products, row/column access

---

## Install (editable)

```bash
pip install -e /numethods
```

or just add `/numethods` to `PYTHONPATH`.

## Examples

```bash
python /numethods/examples/demo.py
```

## Notes

- All algorithms are implemented without relying on external linear algebra solvers.
- Uses plain Python floats and list-of-lists for matrices/vectors.
- Tolerances use a relative criterion `|Δ| ≤ tol (1 + |value|)`.