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polyeig: return eigenvectors of the _correct_ size
author Ed Meyer <eem2314@gmail.com>
date Thu, 19 Jul 2012 04:53:34 +0100
parents 504fec921af5
children 2ad5e6212cd7
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1 ## Copyright (C) 2012 Fotios Kasolis
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {@var{z} =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
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21 ## @deftypefnx {Function File} {[ @var{v}, @var{z} ] =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
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22 ## Solve the polynomial eigenvalue problem of degree @var{l}.
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23 ##
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24 ## Given a @var{n*n} matrix polynomial @var{C(s)} = @var{C0 + C1 s + @dots{} + Cl s^l} polyeig
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25 ## solves the eigenvalue problem (@var{C0} + @var{C1} + @dots{} + @var{Cl})v = 0.
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26 ## Note that the eigenvalues @var{z} are the zeros of the matrix polynomial. @var{z} is a
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27 ## @var{lxn} vector and @var{v} is a @var{(n x n)l} matrix with columns that correspond to
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28 ## the eigenvectors.
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29 ## @seealso{eig, eigs, compan}
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30 ## @end deftypefn
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31
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32 ## Author: Fotios Kasolis
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33
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34 function [ z, varargout ] = polyeig (varargin)
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35
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36 if ( nargout > 2 )
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37 print_usage ();
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38 endif
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39
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40 nin = numel (varargin);
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41
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42 n = zeros (1, nin);
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43
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44 for cnt = 1 : nin
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45 if ! ( issquare (varargin{cnt}) )
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46 error ("polyeig: coefficients must be square matrices");
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47 endif
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48 n(cnt) = size (varargin{cnt}, 1);
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49 endfor
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50
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51 if numel (unique (n)) > 1
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52 error ("polyeig: coefficients must have the same dimensions");
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53 endif
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54 n = unique (n);
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55
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56 # matrix polynomial degree
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57 l = nin - 1;
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58
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59 # form needed matrices
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60 C = [ zeros(n * (l - 1), n), eye(n * (l - 1)); -cell2mat(varargin(1 : end - 1)) ];
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61 D = [ eye(n * (l - 1)), zeros(n * (l - 1), n); zeros(n, n * (l - 1)), varargin{end} ];
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62
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63 % solve generalized eigenvalue problem
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64 if ( isequal (nargout, 1) )
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65 z = eig (C, D);
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66 else
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67 [ z, v ] = eig (C, D);
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68 varargout{1} = v;
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69 % return n-element eigenvectors normalized so
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70 % that the infinity-norm = 1
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71 z = z(1:n,:);
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72 % max() takes the abs if complex:
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73 t = max(z);
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74 z /= diag(t);
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75 endif
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76
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77 endfunction
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78
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79 %!test
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80 %! C0 = [8, 0; 0, 4]; C1 = [1, 0; 0, 1];
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81 %! [v,z] = polyeig (C0, C1);
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82 %! assert (isequal (z(1), -8), true);
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83 %! d = C0*v + C1*v*z
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84 %! assert (isequal (norm(d), 0.0), true);