This is a small C++ dense numerical linear algebra library. I mostly follow Trefethen & Bau, "Numerical Linear Algebra" and Golub & Van Loan, "Matrix Computations." The implementation uses NEON SIMD on ARM64 systems when available.

Build

The library is packaged as a C++20 named module (linalgebra):

• CMake 4.1.x
• Ninja
• LLVM Clang ≥ 18 with libc++ (Homebrew LLVM 22 is what's tested; AppleClang doesn't yet support C++20 module dependency scanning)

cmake -S . -B build -G Ninja -DCMAKE_CXX_COMPILER=/opt/homebrew/opt/llvm/bin/clang++
cmake --build build

You can explicitly disable SIMD at configure time with:

cmake -S . -B build -G Ninja \
  -DCMAKE_CXX_COMPILER=/opt/homebrew/opt/llvm/bin/clang++ \
  -DLINEAR_ALGEBRA_SIMD=NONE

Valid values are AUTO (uses available SIMD) and NONE (forces scalar fallback).

Usage

Import the module:

import linalgebra;

int main() {
    linalgebra::Matrix A{{1.0, 2.0}, {3.0, 4.0}};
    linalgebra::Vector b{5.0, 6.0};
    auto lu = linalgebra::lu_factor(A);
    auto x  = linalgebra::lu_solve(lu, b);
}

Run tests

ctest --test-dir build --output-on-failure

What's implemented

• Matrix / Vector core with SIMD matmul
• Triangular solvers (forward / backward substitution)
• LU factorization with partial pivoting (lu_factor, lu_solve)
• QR factorization — classical GS, modified GS, and Householder (qr_classical_gs, qr_modified_gs, qr_householder)
• Rank-revealing QR — Householder with column pivoting (qr_colpiv)
• Eigenvalue computation via QR iteration:
  • Unshifted QR (eigenvalues_unshifted)
  • Wilkinson-shifted QR (eigenvalues_shifted)
  • Hessenberg + Givens QR (eigenvalues_hessenberg)
  • Francis double-shift QR (eigenvalues_francis)
• Cholesky factorization (cholesky_factor, cholesky_solve)
• Symmetric tridiagonalization
• Eigenvectors via inverse iteration (eigenvectors_inverse_iteration)
• SVD — Golub-Kahan bidiagonalization + QR (svd)
• Conjugate Gradient (solve_cg)
• GMRES (solve_gmres)
• BiCGSTAB (solve_bicgstab)
• Condition number estimation
• Preconditioners (precond_jacobi, precond_ilu0)
• Least squares solver (lstsq)
• Arnoldi iteration (arnoldi)
• Matrix exponential (expm)