Calculating the matrix condition number gives a bound on how inaccurate the solution x to the (perturbed) linear system Ax = b will be after approximation, which plays an important role in mixed-precision computing. sparse-kappa is a CPU/GPU-accelerated library for estimating condition numbers of sparse matrices using PyTorch. It supports a variety of estimation methods associated with linear solvers. sparse-kappa is designed for benchmarking condition number estimators and practical use in the science and engineering community.

Features

• GPU-Accelerated: All computations run on NVIDIA GPUs via PyTorch
• Multiple Norms: Support for 1-norm and 2-norm condition numbers
• Rich Algorithm Suite:
  • 1-norm: Hager-Higham, Power iteration, Oettli-Prager sampling, Block Hager
  • 2-norm: Power method, Lanczos, Golub-Kahan bidiagonalization
  • PyTorch integrations: SVDS, EIGSH, LOBPCG wrappers
• Flexible Solver System: LU, LSMR, CG, GMRES, Direct, Auto-selection
• GNN Prediction Module: Train reusable models that predict condition numbers directly or via inverse-norm prediction
• Smart LU Caching: Reuses factorizations for multiple solves (10-20x speedup)
• Memory Efficient: Designed for large sparse matrices

Installation

Simply via pip manager

pip install sparse-kappa

git clone https://github.com/chenxinye/sparse-kappa
pip install torch
pip install -e .

Quick Start

from sparse_kappa.backend import sparse as sp
from sparse_kappa import cond_estimate

# Create sparse matrix
A = sp.random(10000, 10000, density=0.01, format='csr')

# Estimate condition number
cond = cond_estimate(A)
print(f"κ(A) = {cond:.2e}")

# Use specific method with LU solver
cond = cond_estimate(A, norm=1, method='hager-higham', solver='lu')

Available Methods

1-Norm Methods

Method	Description	Best For	Complexity
hager	Hager algorithm (default)	High accuracy, general matrices	O(k·nnz)
power	Power iteration	Fast rough estimates	O(k·nnz)
oettli-prager	Random/adaptive sampling	Quick estimates with variants	O(m·nnz)
hager-higham	Hager-Higham (Block algorith, multiple vectors)	Improved robustness	O(k·b·nnz)

Recommended: Use solver='lu' for all 1-norm methods (10-20x faster)

2-Norm Methods

Method	Description	Best For	Complexity
svds	Partial SVD (most accurate)	Small-medium matrices (<5k)	O(k·nnz)
eigsh	Symmetric eigenvalue solver	Symmetric / Hermitian matrices	O(k·nnz)
lobpcg	Block preconditioned CG	Large matrices	O(k·nnz)
power	Power iteration	Quick estimates	O(k·nnz)
lanczos	Lanczos tridiagonalization	Medium symmetric matrices	O(k²·nnz)
lanczos_unsym	Lanczos-style condition estimation via eigsh on A^H A	Non-symmetric / rectangular matrices	O(k·nnz)
golub-kahan	Bidiagonalization	Numerically stable	O(k·nnz)
auto	Automatic selection	All cases	-

Solver Options

All 1-norm methods support flexible solver selection:

Solver	Description	Best For	Speed	Memory
auto	Automatic selection (default)	General use	Good	Low
lu	LU factorization with caching	Small matrices (<5k), multiple solves	Fastest	High
lsmr	LSMR iterative solver	Large matrices, single solve	Medium	Low
cg	Conjugate Gradient	SPD matrices	Fast	Low
bicgstab	BiCGSTAB (stabilized BiCG)	Non-symmetric matrices	Fast	Low
gmres	GMRES	Non-symmetric, when BiCGSTAB fails	Medium	Low
direct	Direct solver (no caching)	Single solve, small matrices	Fast	Medium

Legend:
k = iterations, b = block size, m = samples, nnz = non-zeros

GNN-Based Prediction

The sparse_kappa.gnn module learns a mapping from sparse matrices to condition-number related scalars. It supports two explicit strategy workflows for both 1-norm and 2-norm condition numbers:

• strategy=1: train on log10(||A^{-1}||) and compute kappa(A) = ||A|| * ||A^{-1}|| at prediction time.
• strategy=2: train on log10(kappa(A)) and predict the condition number directly.

The lower-level target API is still available: target="condition" predicts kappa(A) directly, while target="inverse_norm" predicts ||A^{-1}|| and multiplies by ||A|| at inference time.

The default graph builder turns a sparse matrix into a row/column bipartite graph. You can replace the feature extractor, model, optimizer, scheduler, loss function, and validation callback.

from sparse_kappa import make_gnn_strategy_config, train_gnn_strategy_estimator
from sparse_kappa.gnn import GNNConditionEstimator
from sparse_kappa.backend import sparse as sp

train_samples = [
    {"matrix": A0, "condition_number": 12.3, "norm_A": 4.1},
    {"matrix": A1, "condition_number": 18.9, "norm_A": 5.7},
]

# Strategy 1: inverse-norm prediction. If norm_A is not supplied, sparse-kappa
# computes ||A|| for the configured norm and derives ||A^{-1}|| = kappa(A) / ||A||.
config = make_gnn_strategy_config(norm=1, strategy=1, epochs=100, lr=1e-3)
estimator = train_gnn_strategy_estimator(
    train_samples,
    norm=1,
    strategy=1,
    val_data=None,
    config=config,
    save_path="models/gnn_strategy1_norm1.pt",
)

# Load and predict one matrix or a list of matrices.
estimator = GNNConditionEstimator.load("models/gnn_strategy1_norm1.pt")
result = estimator.predict(sp.random(100, 100, density=0.02, format="csr"), return_dict=True)
print(result["condition_number"], result["norm_A"], result["norm_Ainv"])

# Strategy 2: direct condition-number prediction.
direct_config = make_gnn_strategy_config(norm=2, strategy=2, epochs=100)
direct_estimator = train_gnn_strategy_estimator(train_samples, norm=2, strategy=2, config=direct_config)
pred = direct_estimator.predict(A_test)

Customization hooks follow the same shape:

import torch

estimator.fit(
    train_samples,
    val_data=val_samples,
    optimizer_factory=lambda params: torch.optim.Adam(params, lr=5e-4),
    scheduler_factory=lambda opt: torch.optim.lr_scheduler.CosineAnnealingLR(opt, T_max=50),
    loss_fn=torch.nn.SmoothL1Loss(),
    validator=my_validation_callback,
)

Examples

Example 1: Compare Methods

from sparse_kappa.backend import sparse as sp
from sparse_kappa import cond_estimate

A = sp.random(2000, 2000, density=0.005, format='csr')

# Compare 1-norm methods
methods_1 = ['hager-higham', 'power', 'oettli-prager', 'block-hager']
for method in methods_1:
    result = cond_estimate(A, norm=1, method=method, solver='lu', 
                          return_dict=True)
    print(f"{method:15s}: κ={result['condition_number']:.4e}, "
          f"iters={result['iterations']}")

# Compare 2-norm methods
methods_2 = ['svds', 'lanczos', 'golub-kahan']
for method in methods_2:
    cond = cond_estimate(A, norm=2, method=method)
    print(f"{method:12s}: κ={cond:.4e}")

Example 2: LU Solver (for 1-norm)

# Highly recommended: use LU solver for Hager-Higham
result = cond_estimate(A, norm=1, method='hager-higham',
                      solver='lu', return_dict=True)

print(f"Condition number: {result['condition_number']:.4e}")
print(f"Solver info:")
print(f"  Type: {result['solver_info']['solver_A']['method']}")
print(f"  Factorized: {result['solver_info']['solver_A']['factorized']}")
print(f"  Solves: {result['solver_info']['solver_A']['solve_count']}")

Example 3: Oettli-Prager Variants

# Adaptive (most accurate)
result = cond_estimate(A, norm=1, method='oettli-prager',
                      solver='lu', variant='adaptive', max_iter=15)

# Random sampling (fastest)
result = cond_estimate(A, norm=1, method='oettli-prager',
                      solver='lu', variant='random', max_iter=20)

# Hybrid (balanced)
result = cond_estimate(A, norm=1, method='oettli-prager',
                      solver='lu', variant='hybrid', max_iter=15)

Example 4: Custom Solver Parameters

# LSMR with relaxed tolerance for large matrices
result = cond_estimate(A, norm=1, method='hager-higham',
                      solver='lsmr',
                      solver_kwargs={'atol': 1e-3, 'maxiter': 20})

# CG for symmetric matrices
A_spd = A @ A.T + sp.eye(A.shape[0]) * 10
result = cond_estimate(A_spd, norm=1, method='hager-higham',
                      solver='cg',
                      solver_kwargs={'atol': 1e-3, 'maxiter': 30})

Performance Tips

1. Auto mode is recommended for first-time usage
2. For symmetric matrices, use eigsh or lanczos
3. For large sparse matrices (>10k), use golub-kahan or lobpcg
4. For highest accuracy on small matrices, use svds
5. Increase max_iter if convergence fails

Testing

# Run all tests
pytest tests/ -v

# Run specific test file
pytest tests/test_norm2.py -v

# Run with coverage
pytest tests/ --cov=sparse_kappa

License

MIT License

Contributing

Contributions welcome! Please submit issues and pull requests on GitHub.

References

• Hager, W. W. (1984). "Condition estimates." SIAM J. Sci. Stat. Comput., 5(2), 311-316.
• Higham, N. J., & Tisseur, F. (2000). "A block algorithm for matrix 1-norm estimation." SIAM J. Matrix Anal. Appl., 21(4), 1185-1201.
• Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations (4th ed.). Johns Hopkins University Press.
• Saad, Y. (2011). Numerical Methods for Large Eigenvalue Problems (2nd ed.). SIAM.
• Oettli, W., & Prager, W. (1964). "Compatibility of approximate solution of linear equations." Numerische Mathematik, 6(1), 405-409.
• Van der Vorst, H. A. (1992). "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems." SIAM J. Sci. Stat. Comput., 13(2), 631-644.

Citation

If you use this library in your research, please cite:

@misc{carson2026estimatingconditionnumbergraph,
      title={Estimating condition number with Graph Neural Networks}, 
      author={Erin Carson and Xinye Chen},
      year={2026},
      eprint={2603.10277},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2603.10277}, 
}