engmath

Engineering mathematics with Python - numerical methods you can read, run and trust.

engmath teaches the numerical and symbolic methods engineers use every day - root finding, linear and nonlinear systems, interpolation, integration, differentiation, smoothing, curve fitting, ODEs, boundary-value problems, PDEs, Fourier analysis, optimisation and linear programming, Monte Carlo simulation, sparse matrices and symbolic mathematics with SymPy - in two complementary ways:

• A small library of readable, from-scratch implementations. Each method is a few dozen lines of plain NumPy that you can read like a textbook: Newton's method, natural cubic splines via the Thomas algorithm, Savitzky-Golay filters, Levenberg-Marquardt, adaptive Runge-Kutta, Crank-Nicolson.
• A course of 21 Jupyter notebooks (about 21 hours of study), with worked solutions to all 83 exercises in solutions/. Each notebook states its learning objectives, prerequisites and time; opens up the algorithm in a few lines of plain Python checked against the library; and ends with exercises, including an "implement it yourself" task. The problems are original engineering ones: pipe friction and the Moody diagram, particle settling, heat conduction with generation, vibration modes, steam-table interpolation, energy from logged power data, identifying a cooling law, peak detection, thermocouple calibration, Antoine-equation fitting, pipe roughness from pressure drops, batch-reactor optimisation, heat penetration, cantilever deflection, a three-reservoir network, fin heat transfer, machine-vibration diagnosis, production planning, tank design, structural failure probability and a heated circuit board.

Every algorithm is validated against SciPy/NumPy, exact analytic solutions, NIST certified values and its theoretical order of convergence - 106 checks in docs/VALIDATION.md, rerun on every push.

Start learning

#	Notebook	Engineering problem	Numerical ideas
00	Python & NumPy primer	laminar pipe flow rate	arrays, vectorisation, functions, plots
01	Errors & uncertainty	pump efficiency	round-off, truncation, propagation, Monte Carlo, significant figures
02	Root finding	Colebrook friction factor, sand settling	bisection, Newton, secant, Brent, convergence order
03	Linear systems	heated wall, 2D plate, vibrating chain	Gauss, Thomas, Gauss-Seidel, eigenvalues
04	Interpolation	vapour-pressure table	linear, splines, transforming data, Runge's phenomenon
05	Integration	machine energy use, tank filling	trapezoid, Simpson, Romberg, Gauss-Legendre
06	Differentiation	cooling law of a forging	uneven-grid derivatives, noise amplification
07	Smoothing	overlapping detector peaks	moving average, Savitzky-Golay, LOWESS
08	Linear least squares	thermocouple calibration	QR vs normal equations, residuals, robust fitting
09	Nonlinear regression	Antoine equation	Levenberg-Marquardt, parameter correlation, starting values
10	Implicit models	pipe roughness from pressure drops	root finding inside fitting, golden section, Nelder-Mead
11	ODEs	batch reactor A → B → C	Euler, RK4, adaptive steps, stiffness
12	PDEs	heat penetration into steel	FTCS stability, Crank-Nicolson, inverse problems
13	Symbolic maths (SymPy)	cantilever deflection	derivations, error terms of numerical methods, lambdify, Jacobians
14	Nonlinear systems	three-reservoir pipe network	Newton for systems, line search, basins of attraction
15	Boundary-value problems	pin fin, radiating fin	shooting, finite differences, Robin conditions, nonlinear BVPs
16	Fourier analysis	vibration diagnosis of a pump	DFT, FFT, windowing, resolution, aliasing, filtering
17	Constrained optimisation & LP	production planning, tank design	simplex method, shadow prices, penalty method, formulation pitfalls
18	Monte Carlo	high-dimensional integrals, diffusion, failure probability	1/√N law, random walks, rare events
19	Sparse matrices & 2D PDEs	chip heating a circuit board	5-point Laplacian, Kronecker products, CG, transient implicit solves
20	Working with data files	real NIST and Mauna Loa CO₂ data, a messy logger file	reading, inspecting and cleaning files; trend and seasonal analysis

Real data included

engmath.datasets ships data files to practise on - read them with NumPy or pandas exactly as you would read your own:

File	Content
Chwirut2.dat, Hahn1.dat, ENSO.dat	real NIST reference measurements (ultrasonic calibration, thermal expansion of copper, Pacific pressure record) with certified results - public domain
co2_mauna_loa_monthly.csv	real monthly CO₂ at Mauna Loa, 1958-2026 (NOAA, public-domain dedication), kept unchanged including its header quirk
thermocouple_calibration.csv, pipe_pressure_drop.csv, thermocouple_10mm.csv, pump_vibration.csv	course measurement data (synthetic, generated by tools/make_course_data.py)
process_log_messy.csv	a deliberately messy data-logger file (unit change, gap, duplicates, error codes, spikes)

from engmath import datasets
import numpy as np
data = np.loadtxt(datasets.path("Chwirut2.dat"), skiprows=60)
print(datasets.info("co2_mauna_loa_monthly.csv"))

See the notebook guide for times, prerequisites and suggested paths. Open any notebook on GitHub to read it with all results and figures, or run it:

git clone https://github.com/ktwyw/engmath && cd engmath
pip install -e ".[notebooks]"
jupyter lab notebooks/

or open it in Google Colab (https://colab.research.google.com/github/ktwyw/engmath/blob/main/notebooks/<name>.ipynb)

• the first cell installs engmath automatically.

Use the library

import numpy as np
from engmath import roots, interpolate, integrate, fitting, odes, reporting

roots.newton(lambda x: x**3 - 2*x - 5, 2.0)            # RootResult with iteration history
s = interpolate.CubicSpline(T_table, p_table); s(35.4)  # natural cubic spline
integrate.simpson(power, time)                          # works for unevenly spaced data
fit = fitting.levenberg_marquardt(model, x, y, p0)      # coefficients, SEs, correlation, history
t, y = odes.rk23(rhs, y0, 0, 10, rtol=1e-6)             # adaptive Runge-Kutta
reporting.format_uncertainty(2.29987, 0.08907)          # '2.30 ± 0.09'

Module	Methods
roots	bisection, Newton-Raphson, secant, Brent; observed convergence order
optimize	golden-section search, Nelder-Mead simplex
linalg	Gaussian elimination with pivoting, Thomas algorithm, Jacobi/Gauss-Seidel, power iteration
interpolate	linear (no silent extrapolation), natural cubic spline, Lagrange polynomial
integrate	trapezoid, cumulative trapezoid, Simpson (uneven spacing), Romberg, Gauss-Legendre
differentiate	forward/central differences, 3-point gradient for uneven data, second derivative, Richardson
smoothing	moving average, Savitzky-Golay (with derivatives), LOWESS (robust option)
fitting	QR least squares with standard errors, Huber robust regression, Levenberg-Marquardt
odes	Euler, RK4, adaptive Bogacki-Shampine RK23, backward Euler for stiff systems
pdes	1D diffusion: explicit FTCS (with stability check), Crank-Nicolson
reporting	value ± uncertainty with correct significant figures; first-order error propagation
nonlinear	Newton's method for systems with numerical or supplied Jacobian and backtracking line search
bvp	shooting method; finite differences for linear BVPs with Dirichlet and Robin (ghost-node) conditions
fourier	DFT from the definition, radix-2 FFT from scratch, amplitude spectra with Hann window, FFT filtering
optimize (also)	simplex linear programming with shadow prices; quadratic penalty method
montecarlo	Monte Carlo integration with standard errors; lattice random walks
sparse	sparse 1D/2D Laplacians via Kronecker products, 2D Poisson solver, conjugate gradient
datasets	paths, sources and licences of the bundled data files

SymPy is used in the notebooks (optional dependency: pip install "engmath[notebooks]").

The library is for learning and for small, transparent calculations. For production work use SciPy - the notebooks always show the equivalent SciPy call and compare the two.

Gallery

Moody diagram from a root finder	Two correlated parameters: a long valley
Why explicit schemes need small steps	Runge's phenomenon: more points, worse fit
A bearing defect found in a vibration spectrum	53,521 unknowns: a chip heating a circuit board

Validation

python docs/validate.py checks, among others: roots against scipy.optimize.brentq and observed orders (Newton 2, secant 1.618); Gauss elimination, Thomas and iterative solvers against numpy.linalg.solve; the natural spline, its derivative and integral against scipy.interpolate; observed orders of the trapezoid (2), Simpson (4), Euler (1) and RK4 (4); Savitzky-Golay against scipy.signal.savgol_filter; Huber regression against statsmodels; Levenberg-Marquardt against the NIST StRD Misra1a certified values; RK4 against solve_ivp; FTCS and Crank-Nicolson against the exact erfc solution; Newton for systems against an independent one-equation reduction and scipy.optimize.root; shooting and finite differences against the exact fin solution (observed order 2); the FFT against numpy.fft; the simplex method (optimum and shadow prices) against scipy.optimize.linprog; the penalty method against SLSQP and the analytic optimum; Monte Carlo against exact n-ball volumes and random-walk theory; the sparse Poisson solver against manufactured solutions; Levenberg-Marquardt against NIST's certified results for the real Chwirut2 and ENSO datasets, read from the bundled files; and the reporting rules.

Related

For engineering statistics - confidence and tolerance intervals, hypothesis tests, design of experiments, control charts, reliability, gauge R&R - see the companion library engstat. For fluid mechanics, see fluidmech.

Author

Yanwei Wang - personal open-source project. GitHub @ktwyw · ORCID 0000-0002-8488-9833 · wangyanwei@gmail.com

License

MIT - see LICENSE. Contributions welcome: see CONTRIBUTING.md.