## Copyright (C) 1993, 1994, 1995 Auburn University.  All Rights Reserved.
## 
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## later version.
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## FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
## for more details.
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## along with Octave; see the file COPYING.  If not, write to the Free
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## -*- texinfo -*-
## @deftypefn {Function File} {@var{x} = } dlyap (@var{a}, @var{b})
## Solve the discrete-time Lyapunov equation
## 
##   @strong{Inputs}
##   @table @var
##     @item a
##     @var{n} by @var{n} matrix
##     @item b
##     Matrix: @var{n} by @var{n}, @var{n} by @var{m}, or @var{p} by @var{n}.
##   @end table
## 
##   @strong{Outputs}
##   @var{x}: matrix satisfying appropriate discrete time Lyapunov equation.
##   Options:
##   @itemize @bullet
##     @item @var{b} is square: solve @code{a x a' - x + b = 0}
##     @item @var{b} is not square: @var{x} satisfies either
##       @example
##       a x a' - x + b b' = 0
##       @end example
##       @noindent
##       or
##       @example
##  a' x a - x + b' b = 0,
## 	@end example
## 	@noindent
##     whichever is appropriate.
##   @end itemize
##   
## @strong{Method}
##   Uses Schur decomposition method as in Kitagawa,
##     @cite{An Algorithm for Solving the Matrix Equation @var{X} =
##     @var{F}@var{X}@var{F}' + @var{S}},
##   International Journal of Control, Volume 25, Number 5, pages 745--753
##   (1977). 
## 
## Column-by-column solution method as suggested in
##   Hammarling, @cite{Numerical Solution of the Stable, Non-Negative
##   Definite Lyapunov Equation}, IMA Journal of Numerical Analysis, Volume
##   2, pages 303--323 (1982).
## 
## @end deftypefn
 
function x = dlyap (a, b)

  ## Written by A. S. Hodel (scotte@eng.auburn.edu) August 1993.

  if ((n = is_square (a)) == 0)
    warning ("dlyap: a must be square");
  endif

  if ((m = is_square (b)) == 0)
    [n1, m] = size (b);
    if (n1 == n)
      b = b*b';
      m = n1;
    else
      b = b'*b;
      a = a';
    endif
  endif

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

  ## Solve the equation column by column.

  [u, s] = schur (a);
  b = u'*b*u;

  j = n;
  while (j > 0)
    j1 = j;

    ## Check for Schur block.

    if (j == 1)
      blksiz = 1;
    elseif (s (j, j-1) != 0)
      blksiz = 2;
      j = j - 1;
    else
      blksiz = 1;
    endif

    Ajj = kron (s (j:j1, j:j1), s) - eye (blksiz*n);

    rhs = reshape (b (:, j:j1), blksiz*n, 1);

    if (j1 < n)
      rhs2 = s*(x (:, (j1+1):n) * s (j:j1, (j1+1):n)');
      rhs = rhs + reshape (rhs2, blksiz*n, 1);
    endif

    v = - Ajj\rhs;
    x (:, j) = v (1:n);

    if(blksiz == 2)
      x (:, j1) = v ((n+1):blksiz*n);
    endif

    j = j - 1;

  endwhile

  ## Back-transform to original coordinates.

  x = u*x*u';

endfunction