NumericAL

A pure, header-only C++20 numerical computing library — a leaf of the SciLang ecosystem (a sibling of REAL), generic over its scalar type and depending on nothing but the C++ standard library.

NumericAL is agnostic of expressions and of any symbolic layer: it computes with concrete numeric structures — vectors, matrices, tensors — and the algorithms over them. SciLang consumes it as a numeric backend through its module seam; the symbolic↔numeric reconciliation stays in SciLang.

Capabilities

Small and complete, grown measured, honest about what is not yet here:

• numerical::vector<T> — a dense, value-semantic vector: element-wise arithmetic, scalar multiplication, dot product, tolerance comparison, shape checks.
• numerical::matrix<T> — a dense, row-major, value-semantic matrix: element-wise arithmetic, scalar / matrix–vector / matrix–matrix products, transpose, tolerance comparison, shape checks.
• numerical::lu_decomposition<T> — an LU factorisation (partial pivoting) computed once and reused: solve (any number of right-hand sides), determinant, and inverse. Free solve / determinant / inverse are one-shot conveniences.
• numerical::cholesky_decomposition<T> — A = L Lᴴ for a Hermitian positive-definite matrix (about twice as cheap as LU): the factor lower(), solve, and determinant; throws not_positive_definite_error otherwise.
• numerical::qr_decomposition<T> — A = Q R (Householder reflections, so Q stays orthonormal to rounding however ill-conditioned A is) for an m × n matrix of any shape, as NumPy's reduced qr: the factors q() (m × k) / r() (k × n), k = min(m, n), and for m ≥ n a least-squares solve of A x ≈ b, which throws singular_matrix_error when A is rank-deficient.
• numerical::hermitian_eigen — the same for a complex Hermitian matrix (real eigenvalues, a unitary V), reduced to a real tridiagonal by Householder reflections and unit phases, then diagonalised by divide and conquer, as LAPACK's zheevd does.
• numerical::symmetric_eigen — eigenvalues (ascending) and eigenvectors of a real symmetric matrix (A = V Λ Vᵀ): Householder tridiagonalisation by panels then divide and conquer, as LAPACK's dsyevd (most of the work in matrix products); implicit QL (eigen_method::tridiagonal_ql, dsyev's) or cyclic Jacobi rotations (eigen_method::jacobi) on request.
• numerical::svd_decomposition<T> — the singular value decomposition A = U Σ Vᴴ of a real or complex m × n matrix, any shape: a Householder QR, then Golub–Kahan bidiagonalisation of R (by panels, for a real one) and divide and conquer, as LAPACK's dgesdd; implicit-shift QR (svd_method::bidiagonal_qr, dgesvd's) or one-sided Jacobi on R (svd_method::jacobi, which gives every singular value to high relative accuracy) on request. singular_values() (descending), u() (m × k), v() (n × k) and vh(), with k = min(m, n); U and V stay orthonormal for a rank-deficient A.
• numerical::lstsq(a, b) — the minimum-norm least-squares solution of A x ≈ b, as NumPy's lstsq gives it, for any shape and rank (from the SVD), b a vector or a matrix of right-hand sides (as solve(a, b) takes too); pinv(a) and matrix_rank(a) from the same SVD, and lu_decomposition::log_determinant() — NumPy's slogdet, which does not underflow where the determinant does.
• numerical::tensor<T> — a dense, row-major, N-dimensional tensor (rank-1 is a vector, rank-2 a matrix, rank-0 a scalar; built directly or from a vector / matrix), with einsum — Einstein-summation index notation for contractions and general products, as in NumPy / PyTorch. einsum is the unifying primitive of the algebra: "ij,jk->ik" is the matrix product, "i,i->" the dot product, "ij->ji" the transpose, "ii->" the trace, "i,j->ij" the outer product, "ij->i" a row reduction. It takes one or more operands and sums over every label absent from the output, in a deterministic order.
• Arithmetic on tensors (tensor_ops.hpp): + - * / element by element with NumPy's broadcasting, sum, mean, max, min of every element or along an axis (sums by NumPy's pairwise blocks), reshape, and zeros, ones, full, eye, arange, linspace.

Grows in as measured: sparse storage, non-symmetric eigenvalues, FFT, numeric autodiff, iterative solvers, numerical integration, arbitrary precision.

Build

make test        # build and run the test suite
make coverage    # line-coverage summary + HTML report
make sanitize    # tests under AddressSanitizer + UndefinedBehaviorSanitizer
make lint        # clang-tidy
make format      # uncrustify, in place
make doc         # API reference (Doxygen) with embedded coverage

Override the compiler with make test CXX=g++-14.

numerical::numerical is the CMake target — add_subdirectory, FetchContent, or an installed config package:

# After `cmake --install <build> --prefix <prefix>`:
find_package(numerical CONFIG REQUIRED)
target_link_libraries(app PRIVATE numerical::numerical)

Releasing

make release computes the next calendar version YYYY.M.PATCH — the patch resets each month, the first release of a month is .0 (PEP 440 drops leading zeros, so 2026.6.1, never 2026.06.001) — bumps it in pyproject.toml and python/numerical/__init__.py, then commits, tags and pushes from a clean main. Pushing the tag drives release.yml, which checks the tag matches the version, builds the abi3 wheels (Linux x86-64/aarch64, macOS universal, Windows) and the sdist, and publishes to PyPI via Trusted Publishing (OIDC, no stored secret).

One-time PyPI setup (before the first release): create a Trusted Publisher for the project numeric-al — owner RECHE23, repository NumericAL, workflow release.yml, environment pypi — and create the matching GitHub Environment named pypi. Until that exists the build/sdist jobs still run, but the publish step fails; the pushed tag remains a valid versioned snapshot regardless.

Python binding

pip install numeric-al        # the distribution is numeric-al; the module is numerical

A CPython binding (stable ABI, one cp310 abi3 wheel serves 3.10+) exposes the unifying object numerical.Tensor (rank-1 a vector, rank-2 a matrix, rank-0 a scalar; float64 or complex128) and the algebra over it. Every function takes a Tensor or anything asarray reads — a NumPy array, a nested list, a number:

import numerical
a = [[1, 2], [3, 4]]
numerical.einsum("ij,jk->ik", a, [[5, 6], [7, 8]])   # matrix product
numerical.einsum("i,i->", [1, 2, 3], [1, 2, 3])       # dot
numerical.solve(a, [1.0, 1.0])                         # a @ x = b

A failure is a numerical.LinAlgError: a ValueError, as numpy.linalg.LinAlgError is, and a numerical.error. A Tensor indexes (t[i, j], t[1:5:2, ::-1]), iterates, transposes (t.T) and multiplies (a @ b) as a NumPy array of rank 1 or 2 does, and computes as one:

t = numerical.arange(6).reshape(2, 3)
(t + 1) * [10, 0, 1] / 2      # element-wise, broadcast as NumPy's
t.sum(), t.mean(axis=0), t.max(axis=-1)
numerical.zeros((2, 2)), numerical.eye(3), numerical.linspace(0, 1, 5)
numerical.exp(t), numerical.sqrt(t), abs(t), t ** 2

Shapes that do not broadcast raise a ValueError worded as NumPy's. max and min take float64 only (NumPy orders complex numbers lexicographically; NumericAL does not order them).

The decomposition layer is exposed too (a numpy.linalg-style surface): determinant(a), slogdet(a) → (sign, logabsdet), inverse(a), pinv(a), matrix_rank(a), lstsq(a, b), cholesky(a), qr(a) → (Q, R), eigh(a) → (values, vectors), svd(a) → (U, S, Vh) — NumPy's svd(a, full_matrices=False), so (U * S) @ Vh is a. Every one accepts float64 and complex128; eigh of a complex Hermitian matrix gives real values and complex vectors. eigh(a, method) and svd(a, method) choose the algorithm: divide and conquer by default (LAPACK's dsyevd/zheevd, dgesdd), "tridiagonal_ql" or "bidiagonal_qr" (dsyev's, dgesvd's), or "jacobi".

A Tensor is a citizen of the scientific Python stack, zero-copy:

import numpy as np, torch
np.asarray(t)          # via __array_interface__ (a read-only view)
np.from_dlpack(t)      # via __dlpack__
torch.from_dlpack(t)   # the same DLPack capsule — JAX / CuPy consume it too
numerical.asarray(np.eye(3))   # the other direction (a copy)

asarray copies any NumPy array — bool, integer, float32/64 or complex64/128 elements, in either byte order, at any strides (a transposed, sliced or reversed view) — as well as a nested list or tuple of numbers and a plain number (rank 0), into float64, or complex128 for complex input.

numerical.get_include() returns the header directory, so the C++ library can be located through its Python install. make python builds the extension in place; make python-test runs the binding suite — including a seeded differential fuzzer that checks hundreds of random einsum specifications and linear-algebra problems against NumPy within tolerance (when NumPy is importable, as in CI). PETSc (petsc4py) has no such exchange protocol — bridging it would require an explicit copy into a PETSc object.

Threads and kernels

The kernels share a pool of threads, started the first time a kernel's work pays for more than one: one per physical core the process may run on, where the system tells them (Linux, macOS; elsewhere one per hardware thread), since two threads on one core's hyperthreads share its multiply-add units. numerical.set_num_threads(n) (C++: numerical::set_max_threads), or NUMERICAL_NUM_THREADS read at import, caps the threads a kernel uses, the caller's included, as OMP_NUM_THREADS caps BLAS; at 1 no thread is started. A process forked while the pool exists runs its kernels alone. Each library that embeds NumericAL's headers has a pool and a cap of its own.

float64 and complex128 matrix products, and the decompositions through them, run on a kernel chosen at run time, which numerical.gemm_path() (C++: gemm_path_in_use) reports:

• "portable" — the compiler's own vectors, on any processor;
• "avx2", "avx512" — on x86-64, where cpuid reports the instructions and the system saves their registers (the wheels are built for the baseline x86-64 and carry these tiles beside it);
• "amx" — on Apple M1 to M4, the matrix coprocessor Apple's Accelerate uses, which Apple does not document. It is used only on a known chip family, outside a virtual machine, and once a product computed on it equals std::fma in order to the bit; NUMERICAL_AMX=0, read by the Python package at import, keeps it off (its self-test included). C++ reads no environment: limit_gemm_path below amx before the first product does the same, and compiling with NUMERICAL_NO_AMX removes it.

numerical.limit_gemm_path(name) (C++: limit_gemm_path), or NUMERICAL_GEMM at import, caps the kernel at name or a more portable one. Every element of a product is summed in ascending order of the depth on any number of threads, so a result is the same, to the bit, whatever the thread count; across kernels it differs in the last bits, since the AVX and AMX kernels fuse each multiply-add and the portable one rounds as the compiler does (GCC and Clang may fuse c + a·b there, as they do by default on arm64: the wheels keep the compilers' defaults, which a separate rounding of each product would cost half the speed of a product of order 512). Past the portable kernel, a complex product runs as one real product of twice the order, [Re A | −Im A] times [[Re B, Im B], [Im B, −Re B]] (by blocks), on that kernel and its threads: each element then sums the products of A's real parts, then those of its imaginary parts, where the portable kernel sums whole complex products in order.

Benchmarks

make bench-linalg times NumericAL against the BLAS/LAPACK NumPy ships, each library alone in its process, every answer checked first; make fuzz-linalg checks every decomposition against NumPy on adversarial matrices, on every kernel. On an M1 Pro against Accelerate, at order 1024, matrix products, solve and inverse are as fast or faster (0.94×), Cholesky, QR and eigh within 1.0–1.2×, svd and lstsq 1.3–1.4×; small problems (order 64–256) take 1.5–3× Accelerate's time. Against OpenBLAS on x86-64 (i9-10900X, i9-9900K) large problems (order 512 and 1024) take 1.1–2.5×, eigh 2.6–2.9×. The tables, the method and where the gaps are: BENCHMARKS.md.

Numerical discipline

Floating-point accuracy is documented, never claimed exact where it cannot be: every decomposition is backward stable, checked against its defining property and against NumPy within bounds of the order times the machine epsilon. A result is the same, to the bit, on any number of threads; between kernels and processors it may differ in the last bits (see Threads and kernels). Same gate bar as the rest of the ecosystem: 100% coverage, clang and g++-14, lint, format, MISRA-oriented analysis and sanitizers green.

License

MIT — see LICENSE.

Author

René Chenard