expanded application of Spectra to complex-valued matrices: problem of inequality signs in Arnoldi.h

[dsl2501]

When I tried to amend the current best approximate codes to my complex-valued matrices, I finally came to be confronted with Arnoldi.h. Eigen/Spectra is constructed to do the job with real-valued matrices, prior consideration was minimal to predefine the inequality, absolute value (and norm), and the convergence criteria. I changed in several combinations of the following several lines and finally met the barrier. Would you please let me know the convergence of Arnoldi iteration and caveats for how to handle the inequality signs of complex valued numbers. Thanks.

• Four error message for Arnoldi.h: four lines were critical, two modified and accepted, two resisted.
  Former two if (m_beta) < (m_near_0)) ---> if (std::abs(m_beta) < std::abs(m_near_0))
  if (m_beta) < (beta_thresh)) ----> if (std::abs(m_beta) < std::abs(beta_thresh))
  Latter two if (std::norm(m_beta) > Scalar(0.717) * m_op.norm(h)) did not work
  while (count < 5 && std::abs(ortho_err) > m_eps * std::abs(m_beta)) did not work

[dsl2501]

this issue seems to be a small one of the grand problem of complex-valued matix eigendecompostion. I now move this issue to another topic.

[yixuan]

Currently, solvers for complex-valued matrices have not been implemented yet.

[dsl2501]

Thanks a lot.