function x = are (a, b, c, opt)

# usage: x = are (a, b, c {,opt})
#
# Solves algebraic riccati equation
#
#   a' x + x a - x b x + c = 0
#
# for identically dimensioned square matrices a, b, c.  If b (c) is not
# square, then the function attempts to use b * b' (c' * c) instead.
#
# opt is an option passed to the eigenvalue balancing routine; default
# is `B'. 
#
# see also: balance

  if (nargin == 3 || nargin == 4)
    if (nargin == 4)
      if (! (strcmp (opt, "N") || strcmp (opt, "P") ...
	     || strcmp (opt, "S") || strcmp (opt, "B") ...
	     || strcmp (opt, "n") || strcmp (opt, "p") ...
	     || strcmp (opt, "s") || strcmp (opt, "b")))
	printf ("warning: are: opt has an illegal value; setting to B");
	opt = "B";
      endif
    else
      opt = "B";
    endif
    if ((n = is_square(a)) == 0)
      error ("are: a is not square");
    endif

    if (is_controllable(a,b) == 0)
      printf("warning: are: a, b are not controllable");
    endif
    if ((m = is_square (b)) == 0)
      b = b * b';
      m = rows (b);
    endif
    if (is_observable (a, c) == 0)
      printf ("warning: are: a,c are not observable");
    endif
    if ((p = is_square (c)) == 0)
      c = c' * c;
      p = rows (c);
    endif
    if (n != m || n != p)
      error ("are: a, b, c not conformably dimensioned.");
    endif

# Should check for controllability/observability here
# use Boley-Golub (Syst. Contr. Letters, 1984) method, not the
#
#                     n-1
# rank ([ B A*B ... A^   *B]) method 

    [u, s] = schur (balance ([a, -b; -c, -a'], opt), "A");
    n1 = n + 1;
    n2 = 2 * n;
    x = u (n1:n2, 1:n) / u (1:n, 1:n);
  else
    error("usage: x = are (a, b, c)")
  endif

endfunction