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
Numerical Differentiation
- Forward difference:
ForwardDiff - Backward difference:
BackwardDiff - Central difference (2nd order):
CentralDiff - Central difference (4th order):
CentralDiff4th - Second derivative:
SecondDerivative - Richardson extrapolation:
RichardsonExtrap
Eigenvalue methods
- Power Iteration (dominant eigenvalue/vector):
PowerIteration - Inverse Power Iteration (optionally shifted):
InversePowerIteration - Rayleigh Quotient Iteration:
RayleighQuotientIteration - QR eigenvalue iteration (unshifted, educational):
QREigenvalues
Matrix & Vector utilities
- Minimal
Matrix/Vectorclasses @operator for matrix multiplication*for scalar-matrix multiplication.Tfor 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|).