# Copyright (C) 1996,1998 A. Scottedward Hodel 
#
# This file is part of Octave. 
#
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# later version. 
# 
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# for more details.
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function x = zgscal(a,b,c,d,z,n,m,p)
  # x = zgscal(f,z,n,m,p) generalized conjugate gradient iteration to 
  # solve zero-computation generalized eigenvalue problem balancing equation 
  # fx=z
  # called by zgepbal
  #
  # References:
  # ZGEP: Hodel, "Computation of Zeros with Balancing," 1992, submitted to  LAA
  # Generalized CG: Golub and Van Loan, "Matrix Computations, 2nd ed" 1989
  
  # A. S. Hodel July 24 1992
  # Conversion to Octave R. Bruce Tenison July 3, 1994


  #**************************************************************************
  #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