Decomposition precision [not urgent]

[yixuan]

Decomposition precision [not urgent]

[yixuan]

Hi @jdbancal,

For your first question, tol is indeed the tolerance threshold for convergence, and m_prec can be regarded as a number that is "approximately zero". m_prec is used internally in the following scenarios:

• To judge whether an eigenvalue is complex or real. This is because the general eigen solver will return all eigenvalues as complex numbers, but usually there exist some real ones, and I use m_prec to filter them out.
• To test orthogonality of vectors. When constructing an orthogonal basis of some linear space, the numerical orthogonality gradually breaks down as iterations go on. The algorithm will correct the orthogonality if its value goes beyond m_prec.
• In convergence test, tol is the relative tolerance, and tol * m_prec is the absolute tolerance, so actually the criterion is pretty stringent.

As for the value of m_prec, it was taken directly from ARPACK, and I don't have a very strong reason why it is this specific number. I guess the rationale is that m_prec should be of larger magnitude than eps, for example the complex eigenvalues roughly have precision of similar order as eps, and hence we need a looser threshold to filter out the real ones.

[jdbancal]

Thanks for your update on the source code regarding epsilons!
The precision of the eigenvalue decompositions is better now

[yixuan]

Thanks for the testing. Let me first close this issue, and you can reopen it once you encounter any more in the future.