numerics

numerics is a C++20 library of numerical methods. It covers dense, banded and sparse linear algebra, Krylov solvers, eigenvalues and the SVD, ODE integrators, PDE operators, spectral transforms, quadrature, root finding, graphs and sampling. The arithmetic lives in num::kernel, a set of allocation-free loops over raw pointers with no dependencies. BLAS, LAPACK, OpenMP, CUDA, FFTW and SuiteSparse are used when they are present.

I started numerics to pull the code from my research and my numerical methods courses (fluid simulation, stochastic modeling, graph algorithms, linear algebra) into one library. I maintain it myself, and it keeps growing alongside my research, so use it with appropriate caution!!

Every feature below is explained by an example, a complete program in examples/ shown with the output it prints.

Setup

include(FetchContent)
FetchContent_Declare(numerics
    GIT_REPOSITORY https://github.com/AdityaDendukuri/numerics.git
    GIT_TAG v1.0.0)
FetchContent_MakeAvailable(numerics)
target_link_libraries(my_program PRIVATE numerics::numerics)

Target	Contents
numerics::numerics	Everything, with each backend the configuration found
numerics::core	Containers, concepts and the portable kernels, no external dependency
numerics::kernel	num::kernel alone
numerics::blas, lapack, openmp, fftw, suitesparse, simd, mpi, cuda	One backend each

#include <numerics.hpp> includes the whole library.

Standout features from my PhD research

Feature	Example
Refactoring after a local change, with Woodbury-corrected LU while few states differ and block LU that keeps every block before the first change	16_factor_reuse
Solving with \(A + PQ^T\) from a factorization of \(A\)	17_woodbury_low_rank_update
\(\operatorname{diag}(A^{-1})\) of a sparse M-matrix from a block of random probes	19_probed_inverse_diagonal
Shifted systems \((sI - A)x = b\) at many shifts from one Hessenberg reduction, and Talbot inversion	03_resolvent_and_expv, 13_talbot_spectral_validation

Feature index

Area	Features	Example
Containers	vec<T>, mat<T>, spmat, num::ops expressions, in-place matvec, axpy, selection helpers	00
Direct solvers	lu, lu(A, no_pivot), cholesky, qr least squares, thomas, det, inverse, rcond, one solve(F, b, x) for all	01
Iterative solvers	cg, pcg with Jacobi, minres, gmres, sparse and matrix-free operators	02
Shifted systems	hessenberg_resolvent, shift, solve_batch, expv	03, 12, 13
Eigenvalues	eig_sym, lanczos, power_iteration, svd	04
ODEs	ode_rk4, ode_rk45, ode_verlet, ode_yoshida4	05
PDEs	3D fields, field_solver::solve_poisson, SPD operators reaching CG	06, 15
Spectral	fft, ifft, rfft, irfft, backends	07
Roots and quadrature	bisection, newton, secant, brent, trapz, simpson, gauss_legendre, adaptive_simpson, romberg	08
Statistics	running_stats, histogram	09
Banded matrices	band_mat, banded LU with pivoting, many right-hand sides	10
Plotting	gnuplot figures, plots in the terminal	11
Laws and diagnostics	assume_spd, make_spd, compile-time requirements, num::unsafe, diagnostic presets	14, 15
Factor reuse	lu(R, no_pivot, previous, changed), lu(R, blocks(levels), previous, changed)	16
Low-rank updates	low_rank_difference, woodbury_solver, transpose(F)	17
Precision and conditioning	rcond, inverse_norm1_estimate, lu(A, mixed_precision)	18
Inverse diagonal	inverse_diagonal(F, A)	19
Backends and kernel	num::seq, blas, omp, lapack, num::kernel on raw pointers	20
Graphs	graph, dijkstra, minimum_spanning_tree, disjoint_set, Laplacian PCG with ApproxChol	21

Reference

Page	Contents
Containers	Every container and the operations on it
Concepts	Every concept and law
Kernels	The BLAS layer, num::kernel
Linear algebra	Factorizations, solvers, eigenvalues, the SVD
ODEs	Integrators and their step ranges
PDEs and fields	Grids, fields, stencils, diffusion, Poisson
Spectral	FFT and sine transforms
Quadrature and roots	Integration rules and root finding
Statistics and sampling	Running statistics, random engines, MCMC
Data structures	Graphs, queues, union-find, cell lists
Algorithms	BFS, DFS, Dijkstra, spanning trees, generators
All names	One page per public name, by directory
Benchmark report	Measured performance