SortRule::SmallestMagn sometimes omits the last (near-zero) eigenvalues

[LTLA]

As a result of some of the issues with shift-and-invert mentioned in #126, I decided to use SortRule::SmallestMagn to get my smallest eigenvalues. However, it seems that - under certain conditions - the last eigenvalue is completely omitted from the output. In this case, the last eigenvalue is very close to zero, which probably contributes to its omission.

I have set up a reproducible example at https://github.com/LTLA/DebuggingSpectra using the debug2.* scripts. (Note the 2, not the 1 used in #126.) Again, I only use R to handle the boilerplate I/O in debug2.R, the real problem occurs in the C++ code in debug2.cpp. I compare the output of SortRule::SmallestMagn against shift-and-invert, and the output from debug2.R is:

  0.013475 # where's my last eigenvalue?
0.00652111
0.00390492

  0.00652111 # shift and invert gives me what I want
  0.00390492
-1.17311e-16

Some additional observations:

• Turning down tol = 1e-12 or lower in SymEigsSolver::compute() is enough to recover the lost eigenvalue.
• The eigenvalue is not lost when I do run_shift(test@x, test@i, test@p, order=nrow(test), 3) instead of 2.

Is this a known behavior? Incidentally, RSpectra has no problems with the same matrix in the same mode:

library(RSpectra)
eigs_sym(test, k=3, which="SM")$value
## [1] 6.521110e-03 3.904921e-03 9.011210e-16

[yixuan]

I suppose a better scheme was already proposed in #126, so let me close this one.