Before revision:  The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow , and optimize over the perturbation size in the outer iteration  .
After revision:  The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow , and optimize over the perturbation size in the outer iteration , which is shown to converge quadratically in the typical case of a defective coalescence of simple eigenvalues approaching the imaginary axis .