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lyap.m

## Copyright (C) 1996, 1997, 2000, 2002, 2004, 2005, 2006, 2007
##               Auburn University.  All rights reserved.
##
##
## This program 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.
##
## This program 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 this program; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {} lyap (@var{a}, @var{b}, @var{c})
## @deftypefnx {Function File} {} lyap (@var{a}, @var{b})
## Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart
## algorithm (Communications of the @acronym{ACM}, 1972).
##
## If @var{a}, @var{b}, and @var{c} are specified, then @code{lyap} returns
## the solution of the  Sylvester equation
## @iftex
## @tex
##   $$ A X + X B + C = 0 $$
## @end tex
## @end iftex
## @ifinfo
## @example
##     a x + x b + c = 0
## @end example
## @end ifinfo
## If only @code{(a, b)} are specified, then @command{lyap} returns the
## solution of the Lyapunov equation
## @iftex
## @tex
##   $$ A^T X + X A + B = 0 $$
## @end tex
## @end iftex
## @ifinfo
## @example
##     a' x + x a + b = 0
## @end example
## @end ifinfo
## If @var{b} is not square, then @code{lyap} returns the solution of either
## @iftex
## @tex
##   $$ A^T X + X A + B^T B = 0 $$
## @end tex
## @end iftex
## @ifinfo
## @example
##     a' x + x a + b' b = 0
## @end example
## @end ifinfo
## @noindent
## or
## @iftex
## @tex
##   $$ A X + X A^T + B B^T = 0 $$
## @end tex
## @end iftex
## @ifinfo
## @example
##     a x + x a' + b b' = 0
## @end example
## @end ifinfo
## @noindent
## whichever is appropriate.
##
## Solves by using the Bartels-Stewart algorithm (1972).
## @end deftypefn

## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
## Created: August 1993
## Adapted-By: jwe

function x = lyap (a, b, c)

  if (nargin != 3 && nargin != 2)
    print_usage ();
  endif

  if ((n = issquare(a)) == 0)
    error ("lyap: a is not square");
  endif

  if (nargin == 2)

    ## Transform Lyapunov equation to Sylvester equation form.

    if ((m = issquare (b)) == 0)
      if ((m = rows (b)) == n)

        ## solve a x + x a' + b b' = 0

        b = b * b';
        a = a';
      else

        ## Try to solve a'x + x a + b' b = 0.

        m = columns (b);
        b = b' * b;
      endif

      if (m != n)
        error ("lyap: a, b not conformably dimensioned");
      endif
    endif

    ## Set up Sylvester equation.

    c = b;
    b = a;
    a = b';

  else

    ## Check dimensions.

    if ((m = issquare (b)) == 0)
      error ("lyap: b must be square in a sylvester equation");
    endif

    [n1, m1] = size(c);

    if (n != n1 || m != m1)
      error("lyap: a,b,c not conformably dimensioned");
    endif
  endif

  ## Call octave built-in function.

  x = syl (a, b, c);

endfunction

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