Can SYEVX with lower accuracy (ABSTOL > 0) be faster than SYEV?

[EduardFedorenkov]

Hello everyone!
I have like theoretical question connecting with SYEVX algorithm. I ask community to help me with understanding, please.

If we are talking about matrices (N ~ 3000) with all eigenvalues are close enough in term of STEIN condition (w[j] - w[j - 1] < ORTOL). I have about 95-98% of STEIN time in reorthogonalization part. When I change the code and always skip reorthogonalization I figure out that my tests pass the validation in term of orthogonality with the similar accuracy as SYEV function. But unfortunately, some of Netlib Lapack internal tests failed.

According to this situation I have some questions:

1. Is ORTOL parameter enough to determine the case where the reorthogonalization is needed? Can we really omit reorthogonalization part in some cases? What matrix parameters influence on it?
2. Can SYEVX(ABSTOL > 0, V case) theoretically be faster than SYEV(V case) in single thread case and in term of parallel computations?
3. Does anyone have any suggestions on optimizations of SYEVX(ABSTOL > 0, V case)? Because it seems that solving eigen problem with lower accuracy can be faster than solving with maximum available accuracy. But I don't see it in my experiments.