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Improve schur reliability#3634

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jacksonloper wants to merge 4 commits intojosdejong:developfrom
jacksonloper:copilot/improve-schur-reliability
Open

Improve schur reliability#3634
jacksonloper wants to merge 4 commits intojosdejong:developfrom
jacksonloper:copilot/improve-schur-reliability

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@jacksonloper jacksonloper commented Jan 12, 2026

To fix #3600, shift strategy introduced to the Schur algorithm. Essentially this just takes the approach we already use to reliably get complex eigenvalues, and incorporates it here. (Well, not exactly the same approach; we use householders transforms here because need an orthogonal decomposition. But very similar!)

Copilot AI and others added 3 commits January 11, 2026 12:35
- Add Hessenberg reduction using Householder reflections
- Implement Francis double shift QR iteration for convergence
- Handle 2x2 blocks for complex conjugate eigenvalue pairs
- Add proper convergence detection and exceptional shifts
- Update tests to verify decomposition properties (A=UTU', orthogonality, quasi-triangularity)
- Add new tests for edge cases including rotation matrices with complex eigenvalues

Co-authored-by: jacksonloper <2056977+jacksonloper@users.noreply.github.com>
- Simplify Householder sigma check condition
- Clarify discriminant comment for eigenvalue classification
- Fix potential undefined access in exceptional shift calculation
- Update docstring example to describe properties rather than specific values
- Make 2x2 block test threshold relative to matrix norm

Co-authored-by: jacksonloper <2056977+jacksonloper@users.noreply.github.com>
@jacksonloper jacksonloper marked this pull request as ready for review January 27, 2026 16:37
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Schur result not quasi upper triangular

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