syevd! is reporting more errors respect to syevr!

[VinceNeede]

syevd! chosen as default algorithm for juliav v1.12.0+beta is reporting more errors respect to syevr!, creating
problems when a large number of eigen decompositions must be performed.

The same matrix can give LAPACKError (with info>0) on a device and pass on another and viceversa, furthermore the same matrix can give error or not based on the number of threads used, meaning that some instability is present (at least in OpenBLAS, haven't tested MKL yet).

test code with matrix saved in HDF5

The matrix is of medium size $658\times 658$ and resulted from a long computation, so it is stored in a `HDF5` file, if you tell me how can I attach the file for the debug I will send it to you. The code is just the following:

using HDF5
using LinearAlgebra

mat = nothing
h5open("mat_error.h5", "r") do file
	global mat = Hermitian(file["mat"][])
end

@info "exectuing with 2 threads"
BLAS.set_num_threads(2)
eigen(mat)

@info "exectuing with 1 thread"
BLAS.set_num_threads(1)
eigen(mat)

Error obtained

The output is

[ Info: exectuing with 2 threads
[ Info: exectuing with 1 thread
ERROR: LoadError: LAPACKException(326121)
Stacktrace:
  [1] chklapackerror_positive(::Int64)
    @ LinearAlgebra.LAPACK ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/lapack.jl:44
  [2] chklapackerror
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/lapack.jl:40 [inlined]
  [3] _syevd!(jobz::Char, uplo::Char, A::Matrix{Float64})
    @ LinearAlgebra.LAPACK ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/lapack.jl:5463
  [4] syevd!
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/lapack.jl:5324 [inlined]
  [5] #eigen!#251
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:14 [inlined]
  [6] eigen!
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:12 [inlined]
  [7] #_eigen#253
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:55 [inlined]
  [8] _eigen
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:53 [inlined]
  [9] #eigen#252
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:49 [inlined]
 [10] eigen
    @ ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:48 [inlined]
 [11] eigen(A::Hermitian{Float64, Matrix{Float64}})
    @ LinearAlgebra ~/.julia/juliaup/julia-1.12.0-beta1+0.x64.linux.gnu/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetriceigen.jl:48
 [12] top-level scope
    @ ~/Tesi/lapack_error/study_from_mat.jl:15
 [13] include(mapexpr::Function, mod::Module, _path::String)
    @ Base ./Base.jl:304
 [14] top-level scope
    @ REPL[2]:1
in expression starting at /home/bisogno/Tesi/lapack_error/study_from_mat.jl:15