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

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This release

0.5.0

Aug 11, 2026

0.4.0

Jul 30, 2026

0.3.1

Jul 10, 2026

0.3.0

Jun 15, 2026

0.2.0

May 26, 2026

0.1.0

Mar 11, 2026

approx-chol 0.5.0

approx-chol

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