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)

```bash
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:

```bash
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:

```cpp
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

```bash
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`)