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########################################################################
##
## Copyright (C) 2018-2024 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{lambda} =} ordeig (@var{A})
## @deftypefnx {} {@var{lambda} =} ordeig (@var{A}, @var{B})
## Return the eigenvalues of quasi-triangular matrices in their order of
## appearance in the matrix @var{A}.
##
## The quasi-triangular matrix @var{A} is usually the result of a Schur
## factorization. If called with a second input @var{B} then the generalized
## eigenvalues of the pair @var{A}, @var{B} are returned in the order of
## appearance of the matrix @code{@var{A}-@var{lambda}*@var{B}}. The pair
## @var{A}, @var{B} is usually the result of a QZ decomposition.
##
## @seealso{ordschur, ordqz, eig, schur, qz}
## @end deftypefn
function lambda = ordeig (A, B)
if (nargin < 1)
print_usage ();
endif
if (! isnumeric (A) || ! issquare (A))
error ("ordeig: A must be a square matrix");
endif
n = length (A);
if (nargin == 1)
B = eye (n);
if (isreal (A))
if (! is_quasitri (A))
error ("ordeig: A must be quasi-triangular (i.e., upper block triangular with 1x1 or 2x2 blocks on the diagonal)");
endif
else
if (! istriu (A))
error ("ordeig: A must be upper-triangular when it is complex");
endif
endif
else
if (! isnumeric (B) || ! issquare (B))
error ("ordeig: B must be a square matrix");
elseif (length (B) != n)
error ("ordeig: A and B must be the same size");
endif
if (isreal (A) && isreal (B))
if (! is_quasitri (A) || ! is_quasitri (B))
error ("ordeig: A and B must be quasi-triangular (i.e., upper block triangular with 1x1 or 2x2 blocks on the diagonal)");
endif
else
if (! istriu (A) || ! istriu (B))
error ("ordeig: A and B must both be upper-triangular if either is complex");
endif
endif
endif
## Start of algorithm
lambda = zeros (n, 1);
i = 1;
while (i <= n)
if (i == n || (A(i+1,i) == 0 && B(i+1,i) == 0))
lambda(i) = A(i,i) / B(i,i);
else
a = B(i,i) * B(i+1,i+1);
b = - (A(i,i) * B(i+1,i+1) + A(i+1,i+1) * B(i,i));
c = A(i,i) * A(i+1,i+1) - ...
(A(i,i+1) - B(i,i+1)) * (A(i+1,i) - B(i+1,i));
if (b > 0)
lambda(i) = 2*c / (-b - sqrt (b^2 - 4*a*c));
i += 1;
lambda(i) = (-b - sqrt (b^2 - 4*a*c)) / 2 / a;
else
lambda(i) = (-b + sqrt (b^2 - 4*a*c)) / 2 / a;
i += 1;
lambda(i) = 2*c / (-b + sqrt (b^2 - 4*a*c));
endif
endif
i += 1;
endwhile
endfunction
## Check whether a matrix is quasi-triangular
function retval = is_quasitri (A)
if (length (A) <= 2)
retval = true;
else
v = diag (A, -1) != 0;
retval = (all (tril (A, -2)(:) == 0) && all (v(1:end-1) + v(2:end) < 2));
endif
endfunction
%!test
%! A = toeplitz ([0, 1, 0, 0], [0, -1, 0, 0]);
%! T = schur (A);
%! lambda = ordeig (T);
%! assert (lambda, [1.61803i; -1.61803i; 0.61803i; -0.61803i], 1e-4);
%!test
%! A = toeplitz ([0, 1, 0, 0], [0, -1, 0, 0]);
%! B = toeplitz ([0, 0, 0, 1], [0, -1, 0, 2]);
%! [AA, BB] = qz (A, B, 'real');
%! assert (isreal (AA) && isreal (BB));
%! lambda = ordeig (AA, BB);
%! assert (lambda, [0.5+0.86603i; 0.5-0.86603i; i; -i], 1e-4);
%! [AA, BB] = qz (A, B, 'complex');
%! assert (iscomplex (AA) && iscomplex (BB));
%! lambda = ordeig (AA, BB);
%! assert (lambda, diag (AA) ./ diag (BB));
## Check trivial 1x1 case
%!test <*55779>
%! lambda = ordeig ([6], [2]);
%! assert (lambda, 3);
## Test input validation
%!error <Invalid call> ordeig ()
%!error <A must be a square matrix> ordeig ('a')
%!error <A must be a square matrix> ordeig ([1, 2, 3])
%!error <A must be quasi-triangular> ordeig (magic (3))
%!error <A must be upper-triangular> ordeig ([1, 0; i, 1])
%!error <B must be a square matrix> ordeig (1, 'a')
%!error <B must be a square matrix> ordeig (1, [1, 2])
%!error <A and B must be the same size> ordeig (1, ones (2,2))
%!error <A and B must be quasi-triangular>
%! ordeig (triu (magic (3)), magic (3))
%!error <A and B must both be upper-triangular>
%! ordeig ([1, 1; 0, 1], [1, 0; i, 1])
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