## 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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## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
## General Public License for more details.
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## 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} {@var{z} =} 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 @var{n*n} matrix polynomial @var{C(s)} = @var{C0 + C1 s + @dots{} + Cl s^l} polyeig 
## solves the eigenvalue problem (@var{C0} + @var{C1} + @dots{} + @var{Cl})v = 0.
## Note that the eigenvalues @var{z} are the zeros of the matrix polynomial. @var{z} is a
## @var{lxn} vector and @var{v} is a @var{(n x n)l} matrix with columns that correspond to
## the eigenvectors.
## @seealso{eig, eigs, compan}
## @end deftypefn

## Author: Fotios Kasolis

function [ z, varargout ] = polyeig (varargin)
  
  if ( nargout > 2 )
    print_usage ();
  endif

  nin = numel (varargin);

  n = zeros (1, nin);

  for cnt = 1 : nin
    if ! ( issquare (varargin{cnt}) )
       error ("polyeig: coefficients must be square matrices");
    endif
    n(cnt) = size (varargin{cnt}, 1);
  endfor

  if numel (unique (n)) > 1
       error ("polyeig: coefficients must have the same dimensions");
  endif
  n = unique (n);

  # matrix polynomial degree
  l = nin - 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
  if ( isequal (nargout, 1) )
    z = eig (C, D);
  else
    [ z, v ] = eig (C, D);
    varargout{1} = v;
    % return n-element eigenvectors normalized so
    % that the infinity-norm = 1
    z = z(1:n,:);
    % max() takes the abs if complex:
    t = max(z);
    z /= diag(t);
  endif

endfunction

%!test
%! C0 = [8, 0; 0, 4]; C1 = [1, 0; 0, 1];
%! [v,z] = polyeig (C0, C1);
%! assert (isequal (z(1), -8), true);
%! d = C0*v + C1*v*z
%! assert (isequal (norm(d), 0.0), true);