A C++ numerical computing library with interactive physics simulations.

Library Modules

core — Vectors, matrices, and backend dispatch

num::Vector, num::Matrix, num::SparseMatrix, num::BandedMatrix
num::Backend — seq, blocked, simd, blas, omp, gpu
num::matmul, num::matmul_blocked, num::matmul_register_blocked, num::matmul_simd
num::matvec, num::matadd
num::dot, num::axpy, num::scale, num::norm
High-Performance Computing — cache blocking, SIMD, BLAS, backend selection
Parallel and Distributed Computing — OpenMP and CUDA backends

factorization — Direct linear solvers

Dense Matrix Factorizations
num::lu, num::lu_solve, num::lu_det, num::lu_inv
num::qr, num::qr_solve
num::thomas — O(n) tridiagonal (Thomas algorithm)

solvers — Iterative solvers

Linear Solvers
num::cg, num::cg_matfree — conjugate gradient
num::gmres — restarted GMRES (Krylov)
num::jacobi, num::gauss_seidel — stationary iterative

eigen — Eigenvalue methods

Eigenvalue Algorithms
num::power_iteration, num::inverse_iteration, num::rayleigh_iteration
num::lanczos — Krylov-Lanczos with Ritz extraction
num::eig_sym — dense symmetric eigensolver (Jacobi sweeps)

svd — Singular value decomposition

Singular Value Decomposition
num::svd — full SVD
num::svd_truncated — randomized truncated SVD

spectral — Fourier transforms

Fast Fourier Transform
num::spectral::fft, num::spectral::ifft — complex DFT / inverse DFT
num::spectral::rfft, num::spectral::irfft — real-to-complex and inverse
num::spectral::FFTPlan — precomputed plan for repeated transforms
FFTBackend::seq (Cooley-Tukey radix-2), FFTBackend::fftw (FFTW3, optional)

analysis — Quadrature and root finding

Numerical Analysis
Quadrature: num::trapz, num::simpson, num::gauss_legendre, num::adaptive_simpson, num::romberg
Root finding: num::bisection, num::newton, num::secant, num::brent

stats — Simulation observables

num::RunningStats — Welford online mean + variance, O(1) memory
num::Histogram — fixed-bin histogram; reweighting for WHAM analysis
num::autocorr_time — integrated autocorrelation time (Madras-Sokal windowing)

markov — Markov chain Monte Carlo

num::markov::metropolis_sweep — Metropolis sweep; computes exp(-beta*dE) internally
num::markov::metropolis_sweep_prob — caller-supplied acceptance probability (e.g. Boltzmann table)
num::markov::umbrella_sweep, num::markov::umbrella_sweep_prob — umbrella sampling with save/restore
num::markov::UmbrellaWindow, MetropolisStats, UmbrellaStats
num::markov::make_seeded_rng — hardware-entropy RNG seeding

sparse and banded — Structured sparse formats

num::SparseMatrix — CSR format; num::sparse_matvec
num::BandedMatrix — band-storage; num::banded_matvec, num::banded_solve

spatial — Spatial data structures and grid

num::CellList2D, num::CellList3D — linked-cell neighbour search
num::VerletList — Verlet neighbour list with skin radius
num::Grid3D — 3D scalar field with index helpers (Poisson/diffusion interop)

Physics Applications

Monte Carlo

2D Ising Model

page_app_ising

Metropolis dynamics and umbrella-sampled nucleation on a 300×300 lattice. Reproduces Brendel et al. (2005) nucleation experiments.

Library feature	Role
`num::markov::metropolis_sweep_prob`	Sweep hot path with precomputed Boltzmann table
`num::markov::umbrella_sweep_prob`	Per-sweep rejection / save-restore for window sampling
`num::SparseMatrix` + `sparse_matvec`	Neighbour-sum matrix for energy observable (SIMD path)
`num::RunningStats`, `num::Histogram`	Online mean, variance, nucleus-size distribution per window
`num::newton`	Mean-field self-consistency equation solver

Fluid Dynamics

2D SPH Fluid Simulation

page_app_fluid

Weakly compressible SPH with heat transport, rigid bodies, and particle injection. Tait EOS, cubic-spline kernel, Morris viscosity Laplacian.

Library feature	Role
`num::CellList2D`	O(1) neighbour queries; Newton-3rd-law pair traversal
`num::Backend` dispatch	`seq` (pair loop) ↔ `omp` (per-particle) selectable at runtime
`num::Vector`	Flat particle attribute arrays

3D SPH Fluid Simulation

page_app_fluid3d

3D WCSPH with opposing hose jets, heat transport, and a free-orbit camera whose orientation rotates the gravity vector.

Library feature	Role
`num::CellList3D`	3D counting-sort spatial hash; 13-stencil Newton-3rd-law pairs
`num::Backend` dispatch	`seq` (pair traversal) ↔ `omp` (per-particle)
`num::Vector`	Flat particle attribute arrays

2D Incompressible Navier-Stokes

page_app_ns

Chorin projection on a staggered MAC grid; semi-Lagrangian advection; matrix-free CG pressure solve. Kelvin–Helmholtz double shear layer initial condition.

Library feature	Role
`num::cg_matfree`	Matrix-free CG for -∇²p = r; warm-start from previous pressure field
`num::dot`	Inner products inside CG (dispatched to best available backend)
`num::Vector`	Velocity faces u, v; pressure p; intermediates u, v, rhs

Quantum Mechanics

2D Time-Dependent Schrödinger Equation

page_app_tdse

Strang operator splitting with Crank–Nicolson kinetic sweeps; Thomas algorithm for O(N) tridiagonal solves; Lanczos eigendecomposition; five interchangeable potentials.

Library feature	Role
`num::thomas`	O(N) complex tridiagonal solve per row/column (kinetic sweeps)
`num::lanczos`	Krylov subspace for lowest eigenmodes
`num::brent`	Bessel zero-finding for exact circular-well energies
`num::gauss_legendre`	Norm and energy quadrature

Electromagnetism

Electromagnetic Field Demo

page_app_em

DC current flow + magnetostatics on a 32³ voxel grid. Four Poisson problems (one electric, three magnetic) solved with matrix-free CG; interactive magnetic dipole.

Library feature	Role
`num::cg_matfree`	All four Poisson solves (variable-coefficient electric + 3× magnetic)
`num::Grid3D`	3D scalar field storage for φ, A, J, B components
`num::Vector`	Flattened field arrays passed to the solver

Status Report

Run cmake --build build --target report to generate output/REPORT.md — a full snapshot of the current build:

Build environment — compiler, flags, detected backends (BLAS, FFTW3, OpenMP, CUDA, MPI)
Test summary — pass/fail counts per suite
Benchmark tables — throughput and timing for every module (linalg, FFT, banded, analysis)
Plots — PNG throughput curves (seq vs fftw, size scaling) in output/plots/

Sections whose backend was not found at configure time are left as placeholders, so the report always reflects exactly what is installed.