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schur.cc File Reference

#include <string>
#include "CmplxSCHUR.h"
#include "dbleSCHUR.h"
#include "fCmplxSCHUR.h"
#include "floatSCHUR.h"
#include "defun-dld.h"
#include "error.h"
#include "gripes.h"
#include "oct-obj.h"
#include "utils.h"

Include dependency graph for schur.cc:


Functions

 DEFUN_DLD (schur, args, nargout,"-*- texinfo -*-\n\ @deftypefn {Loadable Function} {@var{s} =} schur (@var{a})\n\ @deftypefnx {Loadable Function} {[@var{u}, @var{s}] =} schur (@var{a}, @var{opt})\n\ @cindex Schur decomposition\n\ The Schur 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\ @iftex\n\ @tex\n\ $S = U^T A U$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s = u' * a * u}\n\ @end ifnottex\n\ where\n\ @iftex\n\ @tex\n\ $U$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{u}\n\ @end ifnottex\n\ is a unitary matrix\n\ @iftex\n\ @tex\n\ ($U^T U$ is identity)\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ (@code{u'* u} is identity)\n\ @end ifnottex\n\ and\n\ @iftex\n\ @tex\n\ $S$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}\n\ @end ifnottex\n\ is upper triangular. The eigenvalues of\n\ @iftex\n\ @tex\n\ $A$ (and $S$)\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{a} (and @code{s})\n\ @end ifnottex\n\ are the diagonal elements of\n\ @iftex\n\ @tex\n\ $S$.\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}.\n\ @end ifnottex\n\ If the matrix\n\ @iftex\n\ @tex\n\ $A$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{a}\n\ @end ifnottex\n\ is real, then the real Schur decomposition is computed, in which the\n\ matrix\n\ @iftex\n\ @tex\n\ $U$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{u}\n\ @end ifnottex\n\ is orthogonal and\n\ @iftex\n\ @tex\n\ $S$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}\n\ @end ifnottex\n\ is block upper triangular\n\ with blocks of size at most\n\ @iftex\n\ @tex\n\ $2\\times 2$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{2 x 2}\n\ @end ifnottex\n\ along the diagonal. The diagonal elements of\n\ @iftex\n\ @tex\n\ $S$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}\n\ @end ifnottex\n\ (or the eigenvalues of the\n\ @iftex\n\ @tex\n\ $2\\times 2$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{2 x 2}\n\ @end ifnottex\n\ blocks, when\n\ appropriate) are the eigenvalues of\n\ @iftex\n\ @tex\n\ $A$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{a}\n\ @end ifnottex\n\ and\n\ @iftex\n\ @tex\n\ $S$.\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}.\n\ @end ifnottex\n\ \n\ The eigenvalues are optionally ordered along the diagonal according to\n\ the value of @code{opt}. @code{opt = \"a\"} indicates that all\n\ eigenvalues with negative real parts should be moved to the leading\n\ block of\n\ @iftex\n\ @tex\n\ $S$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}\n\ @end ifnottex\n\ (used in @code{are}), @code{opt = \"d\"} indicates that all eigenvalues\n\ with magnitude less than one should be moved to the leading block of\n\ @iftex\n\ @tex\n\ $S$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}\n\ @end ifnottex\n\ (used in @code{dare}), and @code{opt = \"u\"}, the default, indicates that\n\ no ordering of eigenvalues should occur. The leading\n\ @iftex\n\ @tex\n\ $k$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{k}\n\ @end ifnottex\n\ columns of\n\ @iftex\n\ @tex\n\ $U$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{u}\n\ @end ifnottex\n\ always span the\n\ @iftex\n\ @tex\n\ $A$-invariant\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{a}-invariant\n\ @end ifnottex\n\ subspace corresponding to the\n\ @iftex\n\ @tex\n\ $k$\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{k}\n\ @end ifnottex\n\ leading eigenvalues of\n\ @iftex\n\ @tex\n\ $S$.\n\ @end tex\n\ @end iftex\n\ @ifnottex\n\ @code{s}.\n\ @end ifnottex\n\ @end deftypefn")

Function Documentation

DEFUN_DLD ( schur  ,
args  ,
nargout   
)