/usr/share/octave/packages/3.2/optim-1.0.17/cpiv_bard.m is in octave-optim 1.0.17-1.
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%%
%% 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 2 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; if not, write to the Free Software
%% Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301USA
function [lb, idx, ridx, m] = cpiv_bard (v, m, incl)
%% [lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])
%%
%% v: column vector; m: matrix; incl (optional): index. length (v)
%% must equal rows (m). Finds column vectors w and l with w == v + m *
%% l, w >= 0, l >= 0, l.' * w == 0. Chooses idx, w, and l so that
%% l(~idx) == 0, l(idx) == -inv (m(idx, idx)) * v(idx), w(idx) roughly
%% == 0, and w(~idx) == v(~idx) + m(idx, ~idx).' * l(idx). idx indexes
%% at least everything indexed by incl, but l(incl) may be < 0. lb:
%% l(idx) (column vector); idx: logical index, defined above; ridx:
%% ~idx & w roughly == 0; mv: [m, v] after performing a Gauss-Jordan
%% 'sweep' (with gjp.m) on each diagonal element indexed by idx.
%% Except the handling of incl (which enables handling of equality
%% constraints in the calling code), this is called solving the
%% 'complementary pivot problem' (Cottle, R. W. and Dantzig, G. B.,
%% 'Complementary pivot theory of mathematical programming', Linear
%% Algebra and Appl. 1, 102--125. References for the current
%% algorithm: Bard, Y.: Nonlinear Parameter Estimation, p. 147--149,
%% Academic Press, New York and London 1974; Bard, Y., 'An eclectic
%% approach to nonlinear programming', Proc. ANU Sem. Optimization,
%% Canberra, Austral. Nat. Univ.).
n = length (v);
if (n > size (v, 1))
error ('first argument is no column vector'); % the most typical mistake
end
if (nargin < 3)
incl = [];
elseif (islogical (incl))
incl = find (incl);
end
nincl = 1:n;
nincl(incl) = [];
sgn = ones (n, 1);
if (length (incl) == n)
sgn = - sgn;
m = inv (m);
m = cat (2, m, m * v);
else
m = cat (2, m, v);
for id = incl(:).'
sgn(id) = -sgn(id);
m = gjp (m, id);
end
end
nz = eps; % This is arbitrary; components of w and -l are regarded as
% non-negative if >= -nz.
nl = 100 * n; % maximum number of loop repeats, after that give up
if (isempty (nincl))
ready = true;
else
ready = false;
while (~ready && nl > 0)
[vm, idm] = min (sgn(nincl) .* m(nincl, end));
if (vm >= -nz)
ready = true;
else
idm = nincl(idm);
sgn(idm) = -sgn(idm);
m = gjp (m, idm);
nl = nl - 1;
end
end
end
if (~ready)
error ('not successful');
end
idx = sgn < 0;
lb = -m(idx, end);
ridx = ~idx & abs (m(:, end)) <= nz;
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