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+## Copyright (C) 1996, 1998 Auburn University.  All rights reserved.
+##
+## This file is part of Octave.
+##
+## Octave 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 2, or (at your option) any
+## later version.
+##
+## Octave 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 Octave; see the file COPYING.  If not, write to the Free
+## Software Foundation, 59 Temple Place, Suite 330, Boston, MA 02111 USA.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{x} =} zgscal (@var{f}, @var{z}, @var{n}, @var{m}, @var{p})
+## Generalized conjugate gradient iteration to
+## solve zero-computation generalized eigenvalue problem balancing equation
+## @math{fx=z};
+## called by @code{zgepbal}
+## @end deftypefn
+
+## References:
+## ZGEP: Hodel, "Computation of Zeros with Balancing," 1992, submitted to  LAA
+## Generalized CG: Golub and Van Loan, "Matrix Computations, 2nd ed" 1989
+
+## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
+## Created: July 24, 1992
+## Conversion to Octave R. Bruce Tenison July 3, 1994
+
+function x = zgscal (a, b, c, d, z, n, m, p)
+
+  ## initialize parameters:
+  ## Givens rotations, diagonalized 2x2 block of F, gcg vector initialization
+
+  nmp = n+m+p;
+
+  ## x_0 = x_{-1} = 0, r_0 = z
+  x = zeros(nmp,1);
+  xk1 = x;
+  xk2 = x;
+  rk1 = z;
+  k = 0;
+
+  ## construct balancing least squares problem
+  F = eye(nmp);
+  for kk=1:nmp
+    F(1:nmp,kk) = zgfmul(a,b,c,d,F(:,kk));
+  endfor
+
+  [U,H,k1] = krylov(F,z,nmp,1e-12,1);
+  if(!is_square(H))
+    if(columns(H) != k1)
+      error("zgscal(tzero): k1=%d, columns(H)=%d",k1,columns(H));
+    elseif(rows(H) != k1+1)
+      error("zgscal: k1=%d, rows(H) = %d",k1,rows(H));
+    elseif ( norm(H(k1+1,:)) > 1e-12*norm(H,"inf") )
+      zgscal_last_row_of_H = H(k1+1,:)
+      error("zgscal: last row of H nonzero (norm(H)=%e)",norm(H,"inf"))
+    endif
+    H = H(1:k1,1:k1);
+    U = U(:,1:k1);
+  endif
+
+  ## tridiagonal H can still be rank deficient, so do permuted qr
+  ## factorization
+  [qq,rr,pp] = qr(H);   # H = qq*rr*pp'
+  nn = rank(rr);
+  qq = qq(:,1:nn);
+  rr = rr(1:nn,:);            # rr may not be square, but "\" does least
+  xx = U*pp*(rr\qq'*(U'*z));  # squares solution, so this works
+  ## xx1 = pinv(F)*z;
+  ## zgscal_x_xx1_err = [xx,xx1,xx-xx1]
+  return;
+
+  ## the rest of this is left from the original zgscal;
+  ## I've had some numerical problems with the GCG algorithm,
+  ## so for now I'm solving it with the krylov routine.
+
+  ## initialize residual error norm
+  rnorm = norm(rk1,1);
+
+  xnorm = 0;
+  fnorm = 1e-12 * norm([a,b;c,d],1);
+
+  ## dummy defines for MATHTOOLS compiler
+  gamk2 = 0;      omega1 = 0;      ztmz2 = 0;
+
+  ## do until small changes to x
+  len_x = length(x);
+  while ((k < 2*len_x) & (xnorm> 0.5) & (rnorm>fnorm))|(k == 0)
+    k = k+1;
+
+    ## solve F_d z_{k-1} = r_{k-1}
+    zk1= zgfslv(n,m,p,rk1);
+
+    ## Generalized CG iteration
+    ## gamk1 = (zk1'*F_d*zk1)/(zk1'*F*zk1);
+    ztMz1 = zk1'*rk1;
+    gamk1 = ztMz1/(zk1'*zgfmul(a,b,c,d,zk1));
+
+    if(rem(k,len_x) == 1) omega = 1;
+    else                  omega = 1/(1-gamk1*ztMz1/(gamk2*omega1*ztmz2));
+    endif
+
+    ## store x in xk2 to save space
+    xk2 = xk2 + omega*(gamk1*zk1 + xk1 - xk2);
+
+    ## compute new residual error: rk = z - F xk, check end conditions
+    rk1 = z - zgfmul(a,b,c,d,xk2);
+    rnorm = norm(rk1);
+    xnorm = max(abs(xk1 - xk2));
+
+    ## printf("zgscal: k=%d, gamk1=%e, gamk2=%e, \nztMz1=%e ztmz2=%e\n", ...
+    ##   k,gamk1, gamk2, ztMz1, ztmz2);
+    ## xk2_1_zk1 = [xk2 xk1 zk1]
+    ## ABCD = [a,b;c,d]
+    ## prompt
+
+    ## get ready for next iteration
+    gamk2 = gamk1;
+    omega1 = omega;
+    ztmz2 = ztMz1;
+    [xk1,xk2] = swap(xk1,xk2);
+  endwhile
+  x = xk2;
+
+  ## check convergence
+  if (xnorm> 0.5 & rnorm>fnorm)
+    warning("zgscal(tzero): GCG iteration failed; solving with pinv");
+
+    ## perform brute force least squares; construct F
+    Am = eye(nmp);
+    for ii=1:nmp
+      Am(:,ii) = zgfmul(a,b,c,d,Am(:,ii));
+    endfor
+
+    ## now solve with qr factorization
+    x = pinv(Am)*z;
+  endif
+endfunction