Accuracy Degradation in RealSchur

Summary

Accuracy of RealSchur have been degraded compared to release 3.4.0.

Environment

• Operating System : Linux

• Architecture : x64

  (I tested with above and verified on godbolt, but i think this is for all platforms)

• Eigen Version : 3.4.1

• Compiler Version : generic

• Compile Flags : none

• Vector Extension : none

Minimal Example

https://godbolt.org/z/z1Eq5o1Ta

Steps to reproduce

1. See link above, note the difference of two output windows (trunk and 3.4.0)

What is the current bug behavior?

Roots found by EigenSolver reside deviates from actual roots.

What is the expected correct behavior?

Roots found by EigenSolver reside within a small range of actual root.

Anything else that might help

This bug is found when running test suite of ceres solver. the code in minimal example is from its polynomial_test.

It is introduced in commit b14c5d (which have been merged to branch 3.4 as 3ee06e).