Custom SortRule

[toonijn]

This pull request provides the ability to use user-defined selection criteria, without breaking API usage.

Upon using Spectra, I was very impressed by its usability but was missing a way to add a custom eigenvalue selection rule. In my case I wanted to select the smallest non-zero eigenvalues. Adding this functionality to Spectra was a bit more involved than expected. As such, this pull request is quite large.

To make user defined selection criteria work the EigenvalueSorter has now a std::function member which it uses as a sort target. To avoid fundamentally breaking API compatibility. Values of the enum SortRule are implicitly cast to an appropriate EigenvalueSorter.

One of the drawbacks of this approach is that we now rely upon the compiler to optimize many calls to this std::function. In my opinion, this is not a big issue because the number of eigenvalues always will be relatively small. Also, the overhead of O(n log(n)) function calls is negligible with respect to the O(n³) QR-decomposition in each step. Alternatively, if you find this overhead unacceptable, I may be able to rewrite it with some more templates. Such that argsort becomes a virtual function where the target-function is a template parameter and can be inlined. This would reduce the virtual function calls from n log(n) to 1, with the cost of much more complex code base, and a less readable implementation.

How to use a user-defined selection criterium:

GenEigsSolver<...> eigsSolver(...);
eigsSolver.init();
EigenvalueSorter<std::complex<double>> closestToTarget{[](std::complex<double> eigenvalue) {
    return std::abs(eigenvalue - 10.);
}};
eigsSolver.compute(closestToTarget);

[shivupa]

Hi I'm not a maintainer, and I do think this flexibility is nice. However for your case couldn't you use Spectra::GenEigsRealShiftSolver with $\sigma=0.0$ combined with the Spectra::SortRule::SmallestMagn?

[toonijn]

Maybe I am mistaken, but I think SmallestMagn also selects the zero eigenvalues. My custom sortrule is something like:

[](const std::complex<Scalar> &r) {
    if (r.real() < 1e-5 || abs(r.imag()) > 1e-5)
        return Eigen::NumTraits<Scalar>::infinity();
    return r.real();
}

which selects the smallest real eigenvalues which are larger than 1e-5.

[shivupa]

Oh I understand now. You are right.

[yixuan]

Hi @toonijn, I apologize that at this moment I don't have time to look at the PR carefully, but I expect this would be a nice feature. I'll handle it once I get some free time. Thanks.