🚀 go-pugleaf

goh@nyquist.usc.edu (K.C.Goh) writes:

> Does anybody out there know of any numerical routines
> for the computation of the eigenvalues of a Hamiltonian
> matrix (or the zeros of a Hamiltonian Pencil) which exploit
> the Hamiltonian eigenvalue (zero) structure?

See C. F. Van Loan (1984), A symplectic method for approximating all
the eigenvalues of a Hamiltonian matrix, Lin. Alg. Appl., 61, 233-252,
and the references therein.  There is also more recent work by
Bunse-Gerstner et al.  I have never encountered any public domain
software for these type of problems.

By the way, these papers define the symplectic group to be the group
of transformations leaving invariant the form
       _      _              _
     x y  + x y  + . . . + x y
      1 1    2 2            n n
 _                                                                  *
(y denotes the complex conjugate of y).  This is really the group SO(n);
the symplectic group leaves invariant the form

     x y  + x y  + . . . + x y .
      1 1    2 2            n n

I don't know if this matters to you; they do this because the
Hermitian pops up in the (complex) Riccati equation.


Steven Smith