mathkit

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A unified toolkit for computational mathematics, spanning the field end to end: Newton's method chasing roots and Poincare sections of the driven Duffing oscillator, Mandelbrot sets and the Feigenbaum route to chaos, Dijkstra's shortest paths and finite-field arithmetic, the Central Limit Theorem in action and Bessel functions in closed form -- with each domain's docs tracing the field's own foundational breakthroughs in chronological, pedagogical order, every historical milestone linked directly to the runnable code that reproduces it. 14 domain subpackages, one consistent NumPy-based API -- built directly on numpy/scipy for anything they already implement (decompositions, eigensolvers, quadrature, statistical distributions, optimization routines, and more), hand-rolling an algorithm from scratch only where no numpy/scipy equivalent exists (e.g. graph algorithms, the simplex method, modular arithmetic) or where the algorithm's own iterate behavior is the pedagogical subject (e.g. root-finder convergence history, autodiff). No hard dependency on networkx, cvxpy, or SageMath. Sharing common ODE integrators throughout. Conventionally imported as mk.

All 14 domains from mathkit-spec.md's build plan are implemented -- see Subpackages for the full list, or browse the docs at https://mathkit.readthedocs.io.

Install

python -m venv .venv && source .venv/bin/activate
pip install -e ".[dev]"

Published on PyPI as mathematicskit (the name mathkit was already taken by an unrelated package) -- pip install mathematicskit, then import mathkit as mk as usual.

Quick start

import mathkit as mk

spline = mk.numerical_analysis.CubicSpline(x=[0, 1, 2, 3], y=[0, 1, 0, 1], boundary="natural")
print(spline.evaluate(1.5))

Subpackages

Domain subpackages, each with runnable examples linked below:

• mathkit.numerical_analysis -- root finding (bisection, Newton-Raphson, secant, fixed-point, hand-rolled for their convergence history), Lagrange/Newton (hand-rolled) plus scipy-backed cubic-spline interpolation, numpy-backed Chebyshev nodes, numpy.linalg.lstsq-based polynomial regression, and numpy.linalg.cond-based error/condition-number analysis.
• mathkit.linalg -- LU/QR/Cholesky decompositions and eigenvalue computation via scipy.linalg/numpy.linalg, power/inverse iteration (hand-rolled), SVD, conjugate gradient and GMRES via scipy.sparse.linalg, and least-squares stability (normal equations vs. QR vs. numpy.linalg.lstsq).
• mathkit.calculus -- finite-difference derivatives with Richardson extrapolation (hand-rolled), scipy.integrate-backed trapezoidal/Simpson/Gauss-Legendre/adaptive quadrature, forward-mode (dual numbers) and reverse-mode (backpropagation) automatic differentiation (hand-rolled), and Taylor/Maclaurin series.
• mathkit.ode_dynamics -- Jacobian-linearization fixed-point stability (node/saddle/spiral/center), phase portraits, the logistic map's Feigenbaum route to chaos, saddle-node/pitchfork/Hopf bifurcation normal forms, the Van der Pol limit cycle, and Poincare sections of the driven Duffing oscillator.
• mathkit.fractals_chaos -- Lyapunov exponent estimation, box-counting fractal dimension, Numba-accelerated Mandelbrot/Julia set generation, iterated function systems (Barnsley fern, Sierpinski triangle/carpet), and cellular automata (Wolfram rules, Conway's Game of Life).
• mathkit.optimization -- gradient descent and nonlinear conjugate gradient (hand-rolled, iterate-path-exposing), Newton's method and BFGS via scipy.optimize.minimize, Lagrange multipliers and KKT-condition verification, a quadratic-penalty method, and linear programming via scipy.optimize.linprog.
• mathkit.probability -- binomial/Poisson/geometric/uniform/exponential/normal/gamma distributions via scipy.stats (plus mathkit's own MGFs), Monte Carlo integration with variance reduction (importance sampling, control variates), Law of Large Numbers/Central Limit Theorem simulation, and discrete-time Markov chains.
• mathkit.statistics -- descriptive statistics, z/t/chi-square hypothesis tests and one-way ANOVA, confidence intervals for means/proportions/variances, OLS linear regression with residual diagnostics, and bootstrap resampling via scipy.stats.bootstrap.
• mathkit.number_theory -- the extended Euclidean algorithm and modular inverses, fast modular exponentiation, primality testing (trial division, Miller-Rabin) and prime generation, the Chinese Remainder Theorem, continued fractions and best rational approximations, Euler's totient/Mobius/divisor-sum functions, and linear/Pell Diophantine equation solvers (hand-rolled -- exact-integer number theory has no numpy/scipy equivalent).
• mathkit.combinatorics -- permutation/combination counting via scipy.special.perm/comb (sequence generation via itertools), a hand-rolled Pascal's triangle, integer partitions and Young/Ferrers diagrams, the inclusion-exclusion principle and derangements, and Stirling/Catalan/Bell numbers (hand-rolled -- no scipy/numpy equivalent).
• mathkit.graph_theory -- a lightweight own graph container (no networkx); shortest paths (Dijkstra/Bellman-Ford/Floyd-Warshall) and minimum spanning tree (Kruskal) via scipy.sparse.csgraph, with a hand-rolled Prim's kept for comparison; maximum flow/minimum cut via scipy.sparse.csgraph.maximum_flow; hand-rolled graph coloring (greedy, exact backtracking); and spectral graph theory (Laplacian via scipy, eigendecomposition via numpy.linalg.eigh).
• mathkit.abstract_algebra -- cyclic and permutation groups with Cayley tables and group-property checks, subgroup/coset enumeration, finite field arithmetic (GF(p)/GF(p^n) via irreducible polynomials), and polynomial ring arithmetic over Z/Q/finite fields (hand-rolled throughout -- no scipy/numpy equivalent).
• mathkit.geometry -- convex hull via scipy.spatial.ConvexHull (with a hand-rolled Graham scan comparison), Delaunay triangulation/Voronoi diagrams via scipy.spatial, hand-rolled segment intersection and point-in-polygon tests, polygon area/centroid via the shoelace formula, and curvature/arc-length/the Frenet-Serret frame for parametric curves.
• mathkit.special_functions -- the gamma/beta functions and Bessel functions via scipy.special, orthogonal polynomial families (Legendre/Chebyshev/Hermite/Laguerre) via numpy.polynomial with numerically-verified orthogonality, and the discrete Fourier transform via numpy.fft alongside a hand-rolled naive-DFT-vs-radix-2-FFT pedagogical speed comparison.

Shared infrastructure, used across the subpackages above rather than standalone toolkits:

• mathkit.constants -- mathematical constants and default numerical tolerances shared across subpackages.
• mathkit.integrators -- shared numerical ODE integrators (RK4, leapfrog, Yoshida4, adaptive Dormand-Prince), used by mathkit.ode_dynamics.

Test

Tests live alongside each subpackage, at mathkit/<name>/tests/.

pytest                                              # everything
pytest mathkit/numerical_analysis/tests             # a single subpackage

# docstring examples, across every subpackage:
MPLBACKEND=Agg pytest --doctest-modules mathkit --ignore-glob="*/tests/*"

Both commands, plus ruff check/ruff format --check, run in CI on every PR (.github/workflows/ci.yml) across Python 3.9-3.12 on Linux and macOS. See CONTRIBUTING.md before opening a PR.

Coverage

MPLBACKEND=Agg pytest -q --cov=mathkit --cov-report=term

620 tests, 96% line coverage overall. Per-subpackage coverage:

visualizers/ modules are smoke-tested only (correct return type/shape, or that anim.save() succeeds) rather than covered line-by-line, per the testing convention in CLAUDE.md. integrators sits lower because several of its fixed-step/adaptive methods aren't exercised directly by its own tests, only indirectly through the two subpackages (ode_dynamics, fractals_chaos) that call into it.

Docs

Built docs are hosted at https://cpoli.github.io/mathkit/, served from the gh-pages branch. To build locally:

pip install -e ".[docs]"
cd docs && make html

See docs/source/history/ for a chronology of each domain's foundational results, linked to the corresponding implementation at each step.

Citation

If you use mathkit in your research, please cite it — see CITATION.cff.

Contributing

See CONTRIBUTING.md. Please note that this project follows the Contributor Covenant.

License

MIT -- see LICENSE.