Fix bwk

[JeremieGince]

Description

This pull request improves the numerical stability and correctness of the backward (gradient) computations for Pfaffian-related strategies, especially in the presence of singular or ill-conditioned matrices. The main changes ensure that the backward passes avoid using numerically unstable pseudo-inverses, compute correct gradients for singular cases, and include robust test coverage for these scenarios.

Numerical stability and correctness improvements:

• The backward methods in PfaffianBlockDet and PfaffianFDBPf now use torch.linalg.inv (LU factorization) instead of torch.linalg.pinv (SVD) to compute matrix inverses, avoiding convergence issues on ill-conditioned inputs. For singular matrices, the code replaces them with the identity matrix before inversion and computes the correct gradient using the cofactor (adjugate) matrix where necessary. [1] [2] [3]
• The docstrings and comments in these strategies have been updated to clarify the mathematical reasoning, document the new approach for singular cases, and explain the rationale for using LU-based inversion. [1] [2] [3]

Algorithmic and implementation refinements:

• The Parlett-Reid implementation has been refactored for clarity and efficiency, including improved pivoting logic and in-place matrix updates.
• A new static method _cofactor_matrix is introduced in PfaffianBlockDet to compute the cofactor matrix for singular blocks, ensuring gradients are finite and correct.

Expanded and improved test coverage:

• New tests verify that the cofactor matrix matches the classical adjugate transpose for invertible matrices, and that the backward computation for singular blocks produces finite, correct gradients.
• Additional tests ensure that the backward pass does not use the SVD-based pseudo-inverse, remains finite for odd-dimensional (always singular) matrices, and matches the closed-form gradient for both singular and ill-conditioned inputs.

These changes collectively make the Pfaffian strategies more robust and reliable for a wider range of input matrices.

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Checklist

Please complete the following checklist when submitting a PR. The PR will not be reviewed until all items are checked.

• All new features include a unit test.
  Make sure that the tests passed and the coverage is
  sufficient by running
  uv run pytest --session-timeout=600.
• All new functions and code are clearly documented.
• The code passes all pre-commit hooks.
  You can do this by running uvx pre-commit run --all-files.
• The code is type-checked using Mypy.
  You can do this by running uv run mypy src tests.

[github-actions]

☂️ Python Coverage

current status: ✅

Overall Coverage

Lines	Covered	Coverage	Threshold	Status
274	274	100%	98%	🟢

New Files

No new covered files...

Modified Files

File	Coverage	Status
src/torch_pfaffian/strategies/pfaffian_block_det.py	100%	🟢
src/torch_pfaffian/strategies/pfaffian_fdbpf.py	100%	🟢
src/torch_pfaffian/strategies/pfaffian_parlett_reid.py	100%	🟢
src/torch_pfaffian/strategies/strategy.py	100%	🟢
TOTAL	100%	🟢

updated for commit: 7606fc2 by action🐍