## Copyright (C) 1996, 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/>. ## Undocumented internal function. ## -*- texinfo -*- ## @deftypefn {Function File} {} __zgpbal__ (@var{sys}) ## ## Used internally in @command{tzero}; minimal argument checking performed. ## ## Implementation of zero computation generalized eigenvalue problem ## balancing method (Hodel and Tiller, Allerton Conference, 1991) ## Based on Ward's balancing algorithm (@acronym{SIAM} J. Sci Stat. Comput., 1981). ## ## @command{__zgpbal__} computes a state/input/output weighting that attempts to ## reduced the range of the magnitudes of the nonzero elements of [@var{a}, @var{b}, ## @var{c}, @var{d}]. ## The weighting uses scalar multiplication by powers of 2, so no roundoff ## will occur. ## ## @command{__zgpbal__} should be followed by @command{zgpred}. ## @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 by R. Bruce Tenison July 3, 1994 function retsys = __zgpbal__ (Asys) if (nargin != 1 || ! isstruct (Asys)) print_usage (); endif Asys = sysupdate (Asys, "ss"); [a, b, c, d] = sys2ss (Asys); [nn, mm, pp] = abcddim (a, b, c, d); np1 = nn+1; nmp = nn+mm+pp; ## set up log vector zz, incidence matrix ff zz = zginit (a, b, c, d); ## disp("__zgpbal__: zginit returns") ## zz ## disp("/__zgpbal__") if (norm (zz)) ## generalized conjugate gradient approach xx = zgscal (a, b, c, d, zz, nn, mm, pp); for i = 1:nmp xx(i) = floor (xx(i)+0.5); xx(i) = 2.0^xx(i); endfor ## now scale a ## block 1: a = sigma a inv(sigma) for i = 1:nn a(i,1:nn) = a(i,1:nn)*xx(i); a(1:nn,i) = a(1:nn,i)/xx(i); endfor ## block 2: b= sigma a phi for j = 1:mm j1 = j+nn; b(1:nn,j) = b(1:nn,j)*xx(j1); endfor for i = 1:nn b(i,1:mm) = b(i,1:mm)*xx(i); endfor for i = 1:pp i1 = i+nn+mm; ## block 3: c = psi C inv(sigma) c(i,1:nn) = c(i,1:nn)*xx(i1); endfor for j = 1:nn c(1:pp,j) = c(1:pp,j)/xx(j); endfor ## block 4: d = psi D phi for j = 1:mm j1 = j+nn; d(1:pp,j) = d(1:pp,j)*xx(j1); endfor for i = 1:pp i1 = i + nn + mm; d(i,1:mm) = d(i,1:mm)*xx(i1); endfor endif retsys = ss (a, b, c, d); endfunction

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