Fix tridiagonal eigensolver for x86 ubuntu machine

[smataigne]

Solve failing check on x86 ubuntu machine.

[codecov]

Codecov Report

Base: 88.09% // Head: 88.17% // Increases project coverage by +0.07% 🎉

Coverage data is based on head (68f0f2b) compared to base (839097b).
Patch coverage: 100.00% of modified lines in pull request are covered.

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #97      +/-   ##
==========================================
+ Coverage   88.09%   88.17%   +0.07%     
==========================================
  Files           9        9              
  Lines        1344     1353       +9     
==========================================
+ Hits         1184     1193       +9     
  Misses        160      160              

Impacted Files	Coverage Δ
src/skeweigen.jl	97.42% <100.00%> (+0.10%)	⬆️

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[smataigne]

@stevengj , the tests fail for the Julia 1.6 - ubuntu-latest - x86 machine because the algorithm doesn't converge and I don't find why. Do you have any idea of what could be the reason of this?

[stevengj]

Maybe your tolerance is too low for the convergence, and it's just hitting that because the roundoff errors are slightly different?

[stevengj]

You might want to look at the tolerances and convergence tests used by @andreasnoack in https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/master/src/eigenSelfAdjoint.jl

In particular, notice that it checks convergence by comparing the magnitude of the off-diagonal elements to the adjacent diagonal elements, not the norm of the whole matrix: https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/85e6867cb3b64e12e651e9e5cccc8b9385bb89cc/src/eigenSelfAdjoint.jl#L177

[smataigne]

You might want to look at the tolerances and convergence tests used by @andreasnoack in https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/master/src/eigenSelfAdjoint.jl

In particular, notice that it checks convergence by comparing the magnitude of the off-diagonal elements to the adjacent diagonal elements, not the norm of the whole matrix: https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/85e6867cb3b64e12e651e9e5cccc8b9385bb89cc/src/eigenSelfAdjoint.jl#L177

Thank you very much for the references! As I am travelling until 9th september, I will fix this issue at the end of the week.

[smataigne]

You might want to look at the tolerances and convergence tests used by @andreasnoack in https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/master/src/eigenSelfAdjoint.jl

In particular, notice that it checks convergence by comparing the magnitude of the off-diagonal elements to the adjacent diagonal elements, not the norm of the whole matrix: https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/85e6867cb3b64e12e651e9e5cccc8b9385bb89cc/src/eigenSelfAdjoint.jl#L177

@stevengj I found the origin of the error, it is not a problem of stopping criterion, however I changed it in a similar way to the references. The error is due to zero eigenvalues. I assumed that eigenvalues always came in pairs, but it is not the case for zero eigenvalues with my current shift. The assumption was however very convenient for the final complex transformation to get the eigenvectors from the matrix that transform into the real Schur form. I try the different solutions available, I think I will simply modify the shift to make appear pairs of zeroes at the end if any.