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. See Tutorials.

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

Eigenvalue methods

• Power Iteration (dominant eigenvalue/vector): PowerIteration
• Inverse Power Iteration (optionally shifted): InversePowerIteration
• Rayleigh Quotient Iteration: RayleighQuotientIteration
• QR eigenvalue iteration (unshifted, educational): QREigenvalues

Singular Value Decomposition

• SVD via eigen-decomposition of (A^T A): SVD

ODE solvers

Initial value problem solvers for ( y'(t) = f(t,y), ; y(t_0)=y_0 )

• Euler's method (explicit, first order): Euler
• Heun's method / Improved Euler (2nd order): Heun
• Runge-Kutta 2 (midpoint, 2nd order): RK2
• Runge-Kutta 4 (classic, 4th order): RK4
• Backward Euler (implicit, requires Newton iteration): BackwardEuler
• Trapezoidal rule (implicit, 2nd order): ODETrapezoidal
• Adams-Bashforth (multistep explicit): AdamsBashforth
• Adams-Moulton (multistep implicit): AdamsMoulton
• Predictor-Corrector (AB predictor + AM corrector): PredictorCorrector
• Adaptive Runge–Kutta (RK45) (Fehlberg/Dormand–Prince, step control): RK45

Quadrature (Numerical Integration)

• Trapezoidal rule (composite): Trapezoidal
• Simpson's rule (composite, even n): Simpson
• Gauss-Legendre quadrature (2 and 3 point): GaussLegendre

Numerical Differentiation

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

Curve Fitting

• Polynomial least squares fit: PolyFit
• Linear regression with custom basis functions: LinearFit
• Exponential fit (via log transform): ExpFit
• Nonlinear least squares (Gauss-Newton / Levenberg-Marquardt): NonlinearFit

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)

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|).
• ODE implicit solvers use Newton’s method with finite-difference Jacobian approximation.
• Curve fitting supports polynomial, linear basis, exponential, and general nonlinear regression.
• Visualization requires matplotlib.