numethods

A small numerical methods library implementing the classical algorithms from MATH 240 (Computational Mathematics) at Lake Forest College, with full test coverage, convergence benchmarks, and worked examples.

The package covers Taylor expansion, root-finding, finite differences, quadrature, ODE solvers, direct and iterative linear solvers, unconstrained optimization, and linear regression—every routine implemented from scratch on top of NumPy, validated against SciPy / scikit-learn, and benchmarked to confirm the textbook convergence rates.

Quick start

git clone https://github.com/it-malek/compmath.git
cd compmath
pip install -e ".[dev]"

from numethods import newton, rk4, simpson

# Solve x^2 - 2 = 0 with Newton's method
root = newton(lambda x: x**2 - 2, x0=1.0, fprime=lambda x: 2*x)
print(root.root)        # 1.4142135623730951
print(root.iterations)  # 5
print(root.converged)   # True

# Integrate sin(x) from 0 to pi via composite Simpson
import numpy as np
val = simpson(np.sin, 0.0, np.pi, n=20).value
print(val)              # ≈ 2.0000

# Solve dy/dt = -y, y(0)=1 with RK4
sol = rk4(lambda t, y: -y, t_span=(0.0, 1.0), y0=1.0, n_steps=50)
print(sol.y[-1])        # ≈ exp(-1) = 0.3679

What's included

Module	Methods
numethods.taylor	taylor_polynomial, taylor_remainder_bound
numethods.rootfinding	bisection, newton, secant, fixed_point
numethods.differentiation	forward_diff, backward_diff, central_diff, richardson_extrapolation, numerical_jacobian
numethods.integration	trapezoid, simpson, romberg, monte_carlo, adaptive_simpson
numethods.ode	euler, midpoint, rk4, rk45_adaptive
numethods.linsys	lu_decomposition, lu_solve, gauss_elimination, jacobi, gauss_seidel, sor, conjugate_gradient
numethods.optimization	gradient_descent, gradient_descent_backtracking, newton_minimize
numethods.regression	OLSRegression, RidgeRegression, polynomial_features

Every iterative method returns a structured result (RootResult, OptimizationResult, IterativeSolveResult, ...) that includes the full convergence history, so you can plot it, inspect failed runs, or use it in tests.

Examples

The examples/ folder contains the original lab notebooks, cleaned up and rewritten on top of the library. Each runs top-to-bottom with no errors and ends with a polished plot.

Notebook	Topic
examples/00_recap_numpy_sympy.ipynb	NumPy / SymPy / Matplotlib refresher
examples/01_taylor_polynomials.ipynb	Taylor expansion and Lagrange remainder
examples/02_rootfinding.ipynb	Bisection, Newton, secant on $\sqrt{2}$ and $\cos x = x$
examples/03_integration_and_differentiation.ipynb	Quadrature, finite differences, $\pi$ via half-disc area
examples/04_odes_and_dynamical_systems.ipynb	Damped oscillator, Lotka–Volterra, SIR sweep on $R_0$
examples/05_linear_systems.ipynb	LU vs Jacobi vs Gauss–Seidel vs CG
examples/06_optimization.ipynb	Gradient descent and Newton on Rosenbrock
examples/07_regression.ipynb	OLS and ridge with polynomial features

Benchmarks

Two notebooks under benchmarks/:

• convergence_rates.ipynb – log-log plots showing each method's empirical convergence order matches theory ($\mathcal{O}(h)$, $\mathcal{O}(h^2)$, $\mathcal{O}(h^4)$, quadratic for Newton).
• runtime_vs_scipy.ipynb – head-to-head timing against SciPy on the same inputs. SciPy wins on speed (it's compiled), but the gap on small problems is modest.

Sample plot from the convergence notebook (ODE solvers, exponential decay):

The slopes of 1, 2, and 4 visible above match the theoretical orders of the Euler, midpoint, and RK4 schemes exactly.

Testing

pytest                           # 118 tests, ≈2 seconds
pytest --cov=numethods           # 92% line coverage
ruff check numethods tests       # style / lints

Method reference

See docs/method_reference.md for a one-page "I want to do X, which method should I use?" cheat sheet.

Course attribution

The original lab assignments come from MATH 240 (Introduction to Computational Mathematics) at Lake Forest College. The library and infrastructure here are an expanded, productionized version of that work.

License

MIT – see LICENSE.