🚀 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

smith@minerva.harvard.edu (Steven Smith) writes:

>
>   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.
>
	Neither of these definitions correspond to the usual definition
	of the invariant form for symplectic groups.  For example,
	Gilmore ("Lie Groups, Lie Algebras, and Some of Their Applications",
	J. Wiley, 1974, chapter 2) defines symplectic groups as those
	which preserve a bilinear antisymmetric form such as:

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

	The forms mentioned above by Steve Smith are sesquilinear and 
	bilinear, respectively, and both are symmetric.  

	As mentioned by Steve Smith it may be that the authors are really
	interested in the one of the non-compact groups SO^*(2N) or SU^*(2N), 
	the so-called "embedding groups".  The appearance of these groups 
	is reasonable given the close connection between Hamiltonian dynamics 
	and quaternions.

	Another possibility is that the authors are interested in the 
	intersection of the symplectic group with the unitary or orthogonal 
	group.  In this	case both symmetric and antisymmetric forms appear.
	In fact, these two possibilities are related since maximal 
	compact subgroup of SU^*(2N) is the unitary-symplectic group
	USp(2n), and similar relationship holds between SO^*(2N) and U(N).  
	For more details, see chapter 6 in the book by Gilmore mentioned 
	previously.

	In any event, one should realize that it is the invariance of a 
	given bilinear antisymmetric form which crucial to the definition 
	of a symplectic group.  
			

--

		Dave Schneider 

		University of Illinois at Urbana-Champaign 
		Center for Supercomputing Research and Development
		367 Computer and Systems Research Laboratory
		1308 W. Main Street
		Urbana, IL  61801-2307
		MC-264

		phone : (217) 244-0055  
		fax   : (217) 244-1351
		E-mail: schneid@csrd.uiuc.edu