approx-chol

Approximate Cholesky factorization for SDDM and graph Laplacian systems.

This package implements AC and AC(k), porting key algorithmic ideas from Laplacians.jl to make these algorithms accessible in Python.

AC(k) was introduced and analyzed by Gao, Kyng, and Spielman (2025), "AC(k): Robust Solution of Laplacian Equations by Randomized Approximate Cholesky Factorization" (SIAM Journal on Scientific Computing, November 2025, ISSN 1064-8275).

Install

pip install approx-chol

Usage

import numpy as np
from scipy.sparse import csr_array
from approx_chol import factorize

# 4-node path graph Laplacian (0-1-2-3)
row_ptrs = np.array([0, 2, 5, 8, 10], dtype=np.uint32)
col_indices = np.array([0, 1, 0, 1, 2, 1, 2, 3, 2, 3], dtype=np.uint32)
values = np.array([1.0, -1.0, -1.0, 2.0, -1.0, -1.0, 2.0, -1.0, -1.0, 1.0])

laplacian = csr_array((values, col_indices, row_ptrs), shape=(4, 4))

factor = factorize(laplacian)

# RHS must lie in the range of the Laplacian (sum to zero)
b = np.array([1.0, -1.0, 1.0, -1.0])
x = factor.solve(b)

Configuration

from approx_chol import Config, factorize

config = Config(seed=42, split=3)
factor = factorize(laplacian, config=config)

Low-level API

For direct control over CSR arrays without constructing a scipy sparse matrix:

from approx_chol import factorize_raw

factor = factorize_raw(row_ptrs, col_indices, values, n=4)
x = factor.solve(b)

API

  • factorize(matrix, config=None) — Factorize a scipy.sparse.csr_array or csr_matrix.
  • factorize_raw(row_ptrs, col_indices, values, n, config=None) — Factorize from raw CSR arrays.
  • Factor.solve(b) — Solve LDL^T x = b, returning a new array.
  • Factor.solve_into(b, out) — Solve in-place into a pre-allocated array.
  • Config(seed=0, split=None, backend=None) — Configuration for the factorization.
  • Backend.Approximate() / Backend.ExactBelow(max_dim, on_failure) — Which factorization each block gets; defaults to ExactBelow(24, ExactFailure.FallBackToApproximate).
  • Factor.fallbacks — Blocks routed to exact Cholesky and factored approximately anyway; also raises a RuntimeWarning.

Attribution

This package ports key algorithmic ideas from Laplacians.jl to make AC and AC(k) accessible in Python.

Laplacians.jl is licensed under the MIT License.

License

MIT — see LICENSE.

References

  • Gao, Yuan; Kyng, Rasmus; Spielman, Daniel (2025). AC(k): Robust Solution of Laplacian Equations by Randomized Approximate Cholesky Factorization. SIAM Journal on Scientific Computing.
  • Gao, Kyng & Spielman (2023). Robust and Practical Solution of Laplacian Equations by Approximate Elimination. https://arxiv.org/abs/2303.00709
  • Kyng & Sachdeva (2016). Approximate Gaussian Elimination for Laplacians — Fast, Sparse, and Simple. https://arxiv.org/abs/1605.02353

Project links

Data verified by PyPI on Jul 30, 2026
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  • Released:
Newer release available (0.5.0)

1 maintainer

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Avatar for appliedAI from gravatar.com appliedAI

Credits

Author: Kristof Schröder

GitHub Statistics

Data verified by PyPI on Jul 30, 2026
The GitHub source repository was provided by the project maintainers and verified by PyPI at the time of upload. Stars, forks, and open issues/PRs are derived from that repository and have not been independently verified.

License expression

MIT
View SPDX License List

Requires

Python >=3.9

Tags

laplacian sparse cholesky preconditioner sddm

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Publisher: publish.yml on aai-institute/approx-chol

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Hashes for approx_chol-0.4.0-cp39-abi3-macosx_10_12_x86_64.whl
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Provenance

The following attestation bundles were made for approx_chol-0.4.0-cp39-abi3-macosx_10_12_x86_64.whl:

Publisher: publish.yml on aai-institute/approx-chol

Attestations: Values shown here reflect the state when the release was signed and may no longer be current.

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