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).

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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

   • 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

   • 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

   • 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

   • Bisection method.
   • Fixed-point iteration.
   • Newton’s method.
   • Secant method.
   • Convergence analysis and error behavior.
   • Trace outputs for iteration history.
   • Examples with numethods.roots.

• Polynomial Regression Demo
  • Step-by-step example of polynomial regression.
  • Shows how to fit polynomials of different degrees to data.
  • Visualizes fitted curves against the original data.

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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

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Install (editable)

pip install -e /numethods

or just add /numethods to PYTHONPATH.

Examples

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|).