## Copyright (C) 2012 Fotios Kasolis
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
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## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn  {Function File} address@hidden =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
## @deftypefnx {Function File} {[ @var{v}, @var{z} ] =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
## Solve the polynomial eigenvalue problem of degree @var{l}.
##
## Given a polynomial matrix @var{C0 + C1 s + @dots{} + Cl s^l} polyeig 
## solves the generalized eigenvalue problem C v = z D v with C the
## first companion matrix and D = [I, 0; 0, @var{Cl}]. Note that the
## eigenvalues are the roots of the matrix polynomial.
## @end deftypefn

## Author: Fotios Kasolis

function [ z, varargout ] = polyeig (varargin)
  
  if ( nargout > 2 )
    print_usage ();
  endif
  
  inn = numel (varargin);
  
  if ! ( issquare (varargin{1}) )
    error ("polyeig: coefficients must be square matrices");
  endif

  nin = numel (varargin);
  n = size (varargin{1}, 1);

  for cnt = 2 : nin
    T = varargin{cnt};
    nr = size (T, 1);
    if ! ( issquare (T) && isequal(n, nr) )
       error ("polyeig: coefficients must be square matrices with the same dimensions");
    endif
  endfor

  # matrix polynomial degree
  l = inn - 1;

  # form needed matrices
  C = [ zeros(n * (l - 1), n), eye(n * (l - 1)); -cell2mat(varargin(1 : end - 1)) ];
  D = [ eye(n * (l - 1)), zeros(n * (l - 1), n); zeros(n, n * (l - 1)), varargin{end} ];

  % solve generalized eigenvalue problem C v = z D v
  if ( isequal (nargout, 1) )
    z = eig (C, D);
  else
    [ z, v ] = eig (C, D);
    varargout{1} = diag (v);
  endif

endfunction

# TO DO!
# When condeig is implemented allow third output 
# argument with condition number with respect to
# each eigenvalue.