eigh 0.3.3

Differentiable eigenvalue decomposition with JAX (CPU/GPU)

Differentiable Generalized Eigenvalue Decomposition

Standalone implementation of differentiable eigenvalue decomposition with CPU (LAPACK) and GPU (cuSOLVER) backends. Extracted from pyscfad.

Wheels on PyPI: https://pypi.org/project/eigh/ — Linux (manylinux_2_28, x86_64) and macOS (arm64 / Apple Silicon), Python 3.10–3.13, compatible with JAX 0.5 through 0.9+ (single forward-compatible binary). macOS x86_64 (Intel) is not shipped: JAX has no jaxlib ≥0.5 for Intel Macs. GPU path (cuSOLVER) is tested locally; CI runs CPU tests only. See Compatibility for the full support matrix.

Features

• Generalized Problems: A @ V = B @ V @ diag(W), etc.
• JAX Integrated: Full support for jit, vmap, grad, and jvp.
• High Performance: Optimized LAPACK (CPU) and cuSOLVER (GPU) kernels.
• Precision: float32/64 and complex64/128.
• Degeneracy Handling: Configurable deg_thresh for stable gradients.

Installation & Quick Start

CPU (from PyPI)

pip install eigh

This pulls a prebuilt CPU-only wheel (see Compatibility). pip install eigh always prefers the matching wheel, so it will not build a GPU extension even on a GPU node.

GPU (prebuilt eigh-cuda12 wheel)

For CUDA 12 systems, install the GPU build — a separate package name, same repository and same import eigh:

pip install eigh-cuda12

This pulls a Linux x86_64 wheel with the cuSOLVER kernel compiled in, plus the NVIDIA CUDA 12 runtime libraries (nvidia-cusolver-cu12, …) and a CUDA-12 JAX. import eigh then auto-detects the GPU backend. CUDA 12 only (see CUDA / GPU).

Why two packages? A single normal PyPI package cannot serve a small CPU wheel to CPU users and a GPU wheel to GPU users (PyPI wheel-variants are not GA). So GPU ships under the distinct name eigh-cuda12, built from this same repo. CPU users pip install eigh; GPU users pip install eigh-cuda12. This mirrors jaxlib vs jax-cuda12-plugin.

The published GPU wheel is built in CI but functionally tested only on real GPU hardware before each release (CI has no GPU). If you rely on it, sanity check on your own device.

GPU (build from source on the cluster)

If you need a CUDA version other than 12, a custom toolchain, or a platform with no eigh-cuda12 wheel, build from source on the GPU machine:

There is no prebuilt GPU wheel — you compile one on the cluster against its CUDA 12 toolkit. The key flag is --no-build-isolation: it builds the FFI handler against the jaxlib already in your environment (your cluster's CUDA jaxlib) instead of pip pulling an isolated CPU jaxlib into a sandbox.

# 0. On the cluster, load CUDA 12 and put nvcc on PATH (module load cuda/12.x ...)
nvcc --version          # confirm CUDA 12.x is visible

# 1. Install a CUDA-12 JAX matching the cluster (this provides the CUDA jaxlib)
pip install "jax[cuda12]"        # or jax[cuda12-local] to use the system CUDA

# 2. Build eigh FROM SOURCE against that jaxlib, no build isolation
pip install --no-build-isolation "scikit-build-core>=0.8" "nanobind>=1.0.0" cmake ninja
pip install --no-build-isolation --no-binary eigh eigh
#   ^ --no-binary forces a source build; CMake auto-detects nvcc and compiles
#     the cuSOLVER kernel (look for "CUDA support enabled" in the build log).

# Or from a git checkout:
#   pip install --no-build-isolation .

# 3. Verify the GPU backend loaded (no "CUDA backend not available" warning)
python -c "import eigh._core as c; print('cuda:', c._cuda_available)"

If the build prints CUDA not found - GPU support will be disabled, nvcc wasn't on PATH at build time — fix the CUDA module load and rebuild. See the CUDA / GPU compatibility notes for version constraints (CUDA 12 only).

Usage Example

import jax
import jax.numpy as jnp
# Gen. eigensolver from PySCFAD
from eigh import eigh, eigh_gen

jax.config.update("jax_enable_x64", True)
# Eigenvalue problem
A = jnp.array([[2., 1.], [1., 2.]])
B = jnp.array([[1., 1], [0.5, 1.]])
w1, v1 = eigh(A)
w2, v2 = eigh_gen(A, B)

# With gradients
grad1 = jax.grad(lambda A: eigh(A)[0].sum())(A)
grad2 = jax.grad(lambda A: eigh_gen(A, B)[0].sum())(A)
print("Eigenvalues:", w1, w2)
print("Eigenvectors:", v1, v2)
print("Gradients computed:", grad1.shape, grad2.shape)

Benchmarks

Forward/backward scaling vs. matrix size, and gradient stability as eigenvalues approach degeneracy — for the JAX eigensolvers in src/jax/. See benchmarks/suite/ for the scripts.

API Reference

• eigh(a, b=None, *, lower=True, eigvals_only=False, type=1, deg_thresh=1e-9) Scipy-compatible interface. type supports 1: A@v=B@v@λ, 2: A@B@v=v@λ, 3: B@A@v=v@λ.
• eigh_gen(a, b, *, lower=True, itype=1, deg_thresh=1e-9) Lower-level generalized solver.

Degenerate Eigenvalues & Gradients

Individual eigenvalue gradients are ill-defined for degenerate (repeated) eigenvalues. However, symmetric functions (like sum, var, trace) have stable gradients. The deg_thresh parameter (default 1e-9) masks divisions by near-zero gaps to maintain stability.

JAX Eigensolvers (src/jax/)

A collection of differentiable generalized eigensolvers with different strategies for handling degenerate eigenvalues in reverse-mode gradients. Useful for training pipelines where degeneracies are common.

If you just want a working solver, use stable_eigh_pyscfad / stable_eigh_gen_pyscfad from generalized_eigensolver_pyscfad.py. They wrap the fast LAPACK/cuSOLVER kernels with a Lorentzian-broadened custom VJP, so gradients stay stable when eigenvalues are (nearly) degenerate.

On Windows, or if you cannot build the C++ kernels, use stable_generalized_eigh from generalized_eigensolver_stable.py instead — same gradient treatment, pure JAX.

The remaining solvers below are kept for benchmarking and for reproducing prior work; they are not recommended as defaults.

Recommended

Solver	File	Strategy
stable_eigh_pyscfad / stable_eigh_gen_pyscfad	generalized_eigensolver_pyscfad.py	LAPACK/cuSOLVER kernels + Lorentzian-broadened VJP [2]
stable_eigh / stable_generalized_eigh (pure-JAX)	generalized_eigensolver_stable.py	Pure-JAX Cholesky + Lorentzian-broadened VJP [2]

Alternative stable solvers

Solver	File	Strategy	Gradient notes
subspace_eigh	generalized_eigensolver.py	Custom VJP: Lorentzian broadening F/(F²+ε²) [2]	Stable
subspace_generalized_eigh	generalized_eigensolver.py	Symmetry-breaking perturbation + subspace_eigh [2,4]	Stable
degen_eigh	generalized_eigensolver.py	Custom VJP: mask degenerate F_ij by threshold [1,3]	Stable only for symmetric-subspace losses
safe_generalized_eigh	generalized_eigensolver.py	Cholesky + degen_eigh	Inherits degen_eigh caveat

Baselines (not gradient-safe at degeneracies)

Solver	File	Strategy
standard_eig	generalized_eigensolver.py	scipy.linalg.eigh — non-differentiable reference
jax_eig	generalized_eigensolver.py	Plain Cholesky + jnp.linalg.eigh, default VJP
generalized_eigh	generalized_eigensolver.py	Symmetrized Cholesky with SPD shift, default VJP

Compatibility

Windows: no prebuilt wheel. The pure-JAX solvers in src/jax/ (e.g. safe_generalized_eigh, subspace_generalized_eigh, stable_generalized_eigh) work out-of-the-box — pip install jax numpy scipy and import directly from that module. The fast LAPACK/cuSOLVER-backed eigh / eigh_gen kernels (and therefore stable_eigh_pyscfad / stable_eigh_gen_pyscfad) require building from source against a local BLAS/LAPACK.

Python & JAX

The compiled CPU (LAPACK) handler registers through the XLA FFI C API, whose ABI is stable within XLA_FFI_API_MAJOR == 0. A single wheel built against the oldest supported jaxlib therefore loads on every newer one — verified working on jax/jaxlib 0.5.3, 0.6.2, 0.7.0, 0.7.2, 0.8.3, and 0.9.2 (eigh_gen correct and twice-differentiable on all of them).

Python	JAX 0.5 / 0.6	JAX 0.7+	Wheel shipped
3.10	✅	— (jax 0.7 dropped 3.10)	cp310
3.11	✅	✅	cp311
3.12	✅	✅	cp312-abi3
3.13+	—	✅	cp312-abi3 (forward-compatible)

Why three wheels, not one? nanobind's stable ABI (abi3) exists only from CPython 3.12, and abi3 is forward-compatible only (a 3.12-built wheel runs on 3.12/3.13+, never on 3.10/3.11). So 3.10 and 3.11 get version-specific wheels and 3.12 ships one abi3 wheel that also serves 3.13+.

jaxlib < 0.5 (the pre-FFI custom-call era, e.g. 0.4.x) is not supported.

CUDA / GPU

GPU support uses the classic dense cuSOLVER routines (cusolverDn{S,D}sygvd, cusolverDn{C,Z}hegvd) and is dispatched through JAX's GPU FFI under platform="gpu".

Aspect	Support	Notes
CUDA major version	CUDA 12 only	JAX ≥0.5 ships only cuda12 plugins (jax[cuda12]). CUDA 11 is not supported — it would require jax[cuda11], dropped in modern JAX.
Built/tested toolkit	CUDA 12.8.1	The local GPU env (setup_gpu_env_clean.sh) targets 12.8.1; any CUDA 12.x toolkit should build.
cuSOLVER API	CUDA 8+	The *sygvd/*hegvd dense API is long-stable, so there is no upper CUDA-12 bound from the API surface.
Compute capability	nvcc default for the toolkit	No explicit -arch is set; PTX JITs forward to newer GPUs. Set CMAKE_CUDA_ARCHITECTURES to target a specific SM.
FFI ABI across JAX	Same MAJOR == 0 stability as CPU	The GPU handler is forward-compatible across jax 0.5→0.9 just like the CPU one.

Where the GPU path breaks / what to know:

• Prebuilt GPU wheel: pip install eigh-cuda12 (Linux x86_64, CUDA 12). The default pip install eigh is CPU-only — pairing it with jax[cuda12] does not enable GPU, because the CUDA kernel must be compiled into the wheel and the CPU wheel does not contain it.
• GPU wheels are built in CI but not GPU-tested there (no GPU runner); CI only checks the extension loads. Verify on real hardware before relying on a release.
• CUDA 11 clusters are unsupported (see table) — use a CUDA 12 module/env.
• For other CUDA versions or platforms, build from source (below). The build auto-detects CUDA via check_language(CUDA); set EIGH_REQUIRE_CUDA=ON to make a missing nvcc a hard error instead of silently skipping the GPU module.
• A GPU build is not portable the way the CPU wheel is: it must match the cluster's CUDA 12 runtime and driver.

Clusters / HPC

• glibc: Linux wheels target manylinux_2_28 (glibc 2.28) — runs on RHEL 8+, Ubuntu 18.04+, and most current HPC. CentOS 7 / RHEL 7 (glibc 2.17, EOL) is not supported; build from source there.
• BLAS/LAPACK: auditwheel bundles OpenBLAS + libgfortran into the Linux wheel, so cluster nodes do not need a system BLAS installed.
• GPU nodes: build from source as above; the published wheel is CPU-only.

References

• [1] Kasim, M. F., & Vinko, S. M. Learning the exchange–correlation functional from nature with fully differentiable density functional theory. Phys. Rev. Lett. 127, 126403 (2021). https://doi.org/10.1103/PhysRevLett.127.126403
• [2] Colburn, S., & Majumdar, A. Inverse design and flexible parameterization of meta-optics using algorithmic differentiation. Communications Physics 4, 54 (2021). https://doi.org/10.1038/s42005-021-00568-6
• [3] JAX Issue #2748 — Differentiable eigh with degeneracies. https://github.com/jax-ml/jax/issues/2748
• [4] JAX Issue #5461 — Stable generalized eigh. https://github.com/jax-ml/jax/issues/5461

Development & Testing

• Requirements: CMake 3.18+, C++17, JAX, NumPy, LAPACK/CUDA.
• Tests:

  pytest tests/test_eigh.py     # Core functionality
  pytest tests/test_eigh_gen.py # Generalized itypes
  pytest tests/test_eigh_jit.py # JIT & vmap
  

• GPU Setup:

  source setup_gpu_env_clean.sh
  ./run_gpu.sh python example_simple.py
  

License & Citation

Apache License 2.0. If used in research, please cite:

@software{sokolov2026eigh,
  author={Sokolov, Igor},
  title={Eigh: Differentiable eigenvalue decomposition with jax (cpu/gpu)},
  url={https://github.com/Brogis1/eigh},
  year={2026}
}

@software{pyscfad,
  author = {Zhang, Xing},
  title = {PySCFad: Automatic Differentiation for PySCF},
  url = {https://github.com/fishjojo/pyscfad},
  year = {2021-2025}
}

@article{10.1063/5.0118200,
    author = {Zhang, Xing and Chan, Garnet Kin-Lic},
    title = {Differentiable quantum chemistry with PySCF for molecules and materials at the mean-field level and beyond},
    journal = {The Journal of Chemical Physics},
    volume = {157},
    number = {20},
    pages = {204801},
    year = {2022},
    month = {11},
    issn = {0021-9606},
    doi = {10.1063/5.0118200},
    url = {https://doi.org/10.1063/5.0118200},
}

@article{sokolov2026xc,
  title = {Quantum-enhanced neural exchange-correlation functionals},
  author = {Sokolov, Igor O. and Both, Gert-Jan and Bochevarov, Art D. and Dub, Pavel A. and Levine, Daniel S. and Brown, Christopher T. and Acheche, Shaheen and Barkoutsos, Panagiotis Kl. and Elfving, Vincent E.},
  journal = {Phys. Rev. A},
  volume = {113},
  issue = {1},
  pages = {012427},
  numpages = {24},
  year = {2026},
  month = {Jan},
  publisher = {American Physical Society},
  doi = {10.1103/m51l-fys2},
  url = {https://link.aps.org/doi/10.1103/m51l-fys2}
}