| | DEFUN_DLD (schur, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{S} =} schur (@var{A})\n\
@deftypefnx {Loadable Function} {@var{S} =} schur (@var{A}, \"complex\")\n\
@deftypefnx {Loadable Function} {[@var{U}, @var{S}] =} schur (@var{A}, @var{opt})\n\
@cindex Schur decomposition\n\
The Schur@tie{}decomposition is used to compute eigenvalues of a\n\
square matrix, and has applications in the solution of algebraic\n\
Riccati equations in control (see @code{are} and @code{dare}).\n\
@code{schur} always returns\n\
@tex\n\
$S = U^T A U$\n\
@end tex\n\
@ifnottex\n\
@code{@var{S} = @var{U}' * @var{A} * @var{U}}\n\
@end ifnottex\n\
where\n\
@tex\n\
$U$\n\
@end tex\n\
@ifnottex\n\
@var{U}\n\
@end ifnottex\n\
is a unitary matrix\n\
@tex\n\
($U^T U$ is identity)\n\
@end tex\n\
@ifnottex\n\
(@code{@var{U}'* @var{U}} is identity)\n\
@end ifnottex\n\
and\n\
@tex\n\
$S$\n\
@end tex\n\
@ifnottex\n\
@var{S}\n\
@end ifnottex\n\
is upper triangular. The eigenvalues of\n\
@tex\n\
$A$ (and $S$)\n\
@end tex\n\
@ifnottex\n\
@var{A} (and @var{S})\n\
@end ifnottex\n\
are the diagonal elements of\n\
@tex\n\
$S$.\n\
@end tex\n\
@ifnottex\n\
@var{S}.\n\
@end ifnottex\n\
If the matrix\n\
@tex\n\
$A$\n\
@end tex\n\
@ifnottex\n\
@var{A}\n\
@end ifnottex\n\
is real, then the real Schur@tie{}decomposition is computed, in which the\n\
matrix\n\
@tex\n\
$U$\n\
@end tex\n\
@ifnottex\n\
@var{U}\n\
@end ifnottex\n\
is orthogonal and\n\
@tex\n\
$S$\n\
@end tex\n\
@ifnottex\n\
@var{S}\n\
@end ifnottex\n\
is block upper triangular\n\
with blocks of size at most\n\
@tex\n\
$2\\times 2$\n\
@end tex\n\
@ifnottex\n\
@code{2 x 2}\n\
@end ifnottex\n\
along the diagonal. The diagonal elements of\n\
@tex\n\
$S$\n\
@end tex\n\
@ifnottex\n\
@var{S}\n\
@end ifnottex\n\
(or the eigenvalues of the\n\
@tex\n\
$2\\times 2$\n\
@end tex\n\
@ifnottex\n\
@code{2 x 2}\n\
@end ifnottex\n\
blocks, when\n\
appropriate) are the eigenvalues of\n\
@tex\n\
$A$\n\
@end tex\n\
@ifnottex\n\
@var{A}\n\
@end ifnottex\n\
and\n\
@tex\n\
$S$.\n\
@end tex\n\
@ifnottex\n\
@var{S}.\n\
@end ifnottex\n\
\n\
A complex decomposition may be forced by passing \"complex\" as @var{opt}.\n\
\n\
The eigenvalues are optionally ordered along the diagonal according to\n\
the value of @var{opt}. @code{@var{opt} = \"a\"} indicates that all\n\
eigenvalues with negative real parts should be moved to the leading\n\
block of\n\
@tex\n\
$S$\n\
@end tex\n\
@ifnottex\n\
@var{S}\n\
@end ifnottex\n\
(used in @code{are}), @code{@var{opt} = \"d\"} indicates that all eigenvalues\n\
with magnitude less than one should be moved to the leading block of\n\
@tex\n\
$S$\n\
@end tex\n\
@ifnottex\n\
@var{S}\n\
@end ifnottex\n\
(used in @code{dare}), and @code{@var{opt} = \"u\"}, the default, indicates\n\
that no ordering of eigenvalues should occur. The leading\n\
@tex\n\
$k$\n\
@end tex\n\
@ifnottex\n\
@var{k}\n\
@end ifnottex\n\
columns of\n\
@tex\n\
$U$\n\
@end tex\n\
@ifnottex\n\
@var{U}\n\
@end ifnottex\n\
always span the\n\
@tex\n\
$A$-invariant\n\
@end tex\n\
@ifnottex\n\
@var{A}-invariant\n\
@end ifnottex\n\
subspace corresponding to the\n\
@tex\n\
$k$\n\
@end tex\n\
@ifnottex\n\
@var{k}\n\
@end ifnottex\n\
leading eigenvalues of\n\
@tex\n\
$S$.\n\
@end tex\n\
@ifnottex\n\
@var{S}.\n\
@end ifnottex\n\
@end deftypefn") |
| | DEFUN_DLD (rsf2csf, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Function File} {[@var{U}, @var{T}] =} rsf2csf (@var{UR}, @var{TR})\n\
Convert a real, upper quasi-triangular Schur@tie{}form @var{TR} to a complex,\n\
upper triangular Schur@tie{}form @var{T}.\n\
\n\
Note that the following relations hold: \n\
\n\
@code{@var{UR} * @var{TR} * @var{UR}' = @var{U} * @var{T} * @var{U}'} and\n\
@code{@var{U}' * @var{U}} is the identity matrix I.\n\
\n\
Note also that @var{U} and @var{T} are not unique.\n\
@seealso{schur}\n\
@end deftypefn") |
