add exceptional shift#1603
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Summary of Changes
This PR fixes some cases where the search for eigenvalue could hang. This was solved by using the "Exceptional Shift" method in Wilkinson, J. H., & Reinsch, C. (1971).
Handbook for automatic computation (Vol. 2: Linear algebra).
Details
I first checked why the Lapack algorithm would converge and not hang. I looked at dlahqr procedure (https://netlib.org/lapack/explore-html/d8/df4/group__lahqr_ga100049ab080cef756846702a33e8e28f.html) saw the mention of the exceptional shift and the Ahues & Tisseur stopping criterion.
The last one was a red herring, I implemented it but the test matrix from #611 would still hang. I then implemented the exceptional shift and it worked. I tried tracing back the magical value (0.75 & -0.4375), and discovered https://www.netlib.org/eispack/hqr.f that led me to the original source:
Wilkinson, J. H., & Reinsch, C. (1971).
Handbook for automatic computation (Vol. 2: Linear algebra).
You can view the orignal magical number and algorithm implementation here : https://www.scribd.com/document/568239788/Wilkinson-Reinsch1971-Book-HandbookForAutomaticComputatio#page=373
AI was used to find the original paper
Testing
I implemented a test with the matrix from #611 that now pass successfully
Fixes #611