1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
|
########################################################################
##
## Copyright (C) 2012-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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.
##
## Octave is distributed in the hope that it will be useful, but
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{z} =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
## @deftypefnx {} {[@var{v}, @var{z}] =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
##
## Solve the polynomial eigenvalue problem of degree @var{l}.
##
## Given an @var{n}x@var{n} matrix polynomial
##
## @code{@var{C}(@var{s}) = @var{C0} + @var{C1} @var{s} + @dots{} + @var{Cl}
## @var{s}^@var{l}}
##
## @code{polyeig} solves the eigenvalue problem
##
## @code{(@var{C0} + @var{C1} @var{z} + @dots{} + @var{Cl} @var{z}^@var{l})
## @var{v} = 0}.
##
## Note that the eigenvalues @var{z} are the zeros of the matrix polynomial.
## @var{z} is a row vector with @code{@var{n}*@var{l}} elements. @var{v} is a
## matrix (@var{n} x @var{n}*@var{l}) with columns that correspond to the
## eigenvectors.
##
## @seealso{eig, eigs, compan}
## @end deftypefn
function [z, v] = polyeig (varargin)
if (nargin < 1)
print_usage ();
endif
n = rows (varargin{1});
for i = 1 : nargin
if (! issquare (varargin{i}))
error ("polyeig: coefficients must be square matrices");
endif
if (rows (varargin{i}) != n)
error ("polyeig: coefficients must have the same dimensions");
endif
endfor
## matrix polynomial degree
l = nargin - 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 (nargout < 2)
z = eig (C, D);
else
[z, v] = eig (C, D);
v = diag (v);
## return n-element eigenvectors normalized so that the infinity-norm = 1
z = z(1:n,:);
t = max (z); # max() takes the abs if complex.
z ./= t;
endif
endfunction
%!shared C0, C1
%! C0 = [8, 0; 0, 4];
%! C1 = [1, 0; 0, 1];
%!test
%! z = polyeig (C0, C1);
%! assert (z, [-8; -4]);
%!test
%! [v,z] = polyeig (C0, C1);
%! assert (z, [-8; -4]);
%! z = diag (z);
%! d = C0*v + C1*v*z;
%! assert (norm (d), 0.0);
## Test input validation
%!error <Invalid call> polyeig ()
%!error <coefficients must be square matrices> polyeig (ones (3,2))
%!error <coefficients must have the same dimensions>
%! polyeig (ones (3,3), ones (2,2))
|
