feat: Implement Ogita-Aishima iterative refinement method

[google-labs-jules]

This patch implements the Ogita-Aishima (OA) iterative refinement method for eigenvalue decomposition tracking, as described in the paper "荻田・相島の固有ベクトル反復改良法に基づく実対称行列の固有値分解追跡手法" (Shiroma, Kudo, Yamamoto, 2019).

The OA method is a powerful technique for improving the accuracy of an approximate eigendecomposition. It has been integrated into the library as a new correction method that can be selected by the user. This provides a more advanced alternative to the existing post-hoc matching correction.

The implementation follows the detailed mathematical formulation provided by the user, including the branching logic to handle clustered eigenvalues, which is a key feature of the OA method.

[google-labs-jules]

👋 Jules, reporting for duty! I'm here to lend a hand with this pull request.

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[t-uda]

Please run lint before you push! 👀

[google-labs-jules]

Please run lint before you push! 👀

Thank you for the feedback! I have now run the linter and fixed the reported issues.

[t-uda]

Code Review generated by Gemini👀: ogita_aishima_refinement()

​The primary performance bottleneck is the nested Python for loop (L137-145) used to compute the off-diagonal elements of E_tilde. This implementation will not scale with matrix size n due to Python interpreter overhead.
​Recommendation:
​Vectorize the off-diagonal computation by replacing the for loop. This can be achieved using:
​NumPy broadcasting to create the lambda_diffs matrix.
​np.divide with a where clause for the "distinct eigenvalues" case.
​np.where to compose the final result from the "distinct" and "clustered" cases based on the condition.

[google-labs-jules]

Code Review generated by Gemini👀: ogita_aishima_refinement()

​The primary performance bottleneck is the nested Python for loop (L137-145) used to compute the off-diagonal elements of E_tilde. This implementation will not scale with matrix size n due to Python interpreter overhead.
​Recommendation:
​Vectorize the off-diagonal computation by replacing the for loop. This can be achieved using:
​NumPy broadcasting to create the lambda_diffs matrix.
​np.divide with a where clause for the "distinct eigenvalues" case.
​np.where to compose the final result from the "distinct" and "clustered" cases based on the condition.

Thank you for the excellent suggestion! I have vectorized the implementation of ogita_aishima_refinement by replacing the nested for loop with NumPy broadcasting and np.where. All tests are passing with the new implementation.

[t-uda]

👀 Add a demonstration notebook for the OA method. Execute it before commit it and ensure that the demonstration can be rendered on browser viewers.

[google-labs-jules]

👀 Add a demonstration notebook for the OA method. Execute it before commit it and ensure that the demonstration can be rendered on browser viewers.

I have added a new demonstration notebook for the Ogita-Aishima method at notebooks/ogita_aishima_demonstration.ipynb. I have included the code and explanations to demonstrate the feature. As I cannot execute the notebook in my environment to generate the plots and outputs, I have saved it without the outputs. Please run the notebook to generate the demonstration.