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## Copyright (C) 1994-2011 John W. Eaton
##
## 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 3 of the License, 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, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {} null (@var{A})
## @deftypefnx {Function File} {} null (@var{A}, @var{tol})
## Return an orthonormal basis of the null space of @var{A}.
##
## The dimension of the null space is taken as the number of singular
## values of @var{A} not greater than @var{tol}. If the argument @var{tol}
## is missing, it is computed as
##
## @example
## max (size (@var{A})) * max (svd (@var{A})) * eps
## @end example
## @seealso{orth}
## @end deftypefn
## Author: KH <Kurt.Hornik@wu-wien.ac.at>
## Created: 24 December 1993.
## Adapted-By: jwe
## Adapted-By: Olaf Till <olaf.till@uni-jena.de>
## This function has also been submitted to Octave (bug #33503).
function retval = __null_optim__ (A, tol)
if (isempty (A))
retval = [];
else
[U, S, V] = svd (A);
[rows, cols] = size (A);
[S_nr, S_nc] = size (S);
if (S_nr == 1 || S_nc == 1)
s = S(1);
else
s = diag (S);
endif
if (nargin == 1)
if (isa (A, "single"))
tol = max (size (A)) * (vtol = s (1) * (meps = eps ("single")));
else
tol = max (size (A)) * (vtol = s (1) * (meps = eps));
endif
elseif (nargin != 2)
print_usage ();
endif
rank = sum (s > tol);
if (rank < cols)
retval = V (:, rank+1:cols);
if (rows >= cols)
cb = columns (retval);
## Set those elements of each vector to zero whose absolute
## values are smallest and which together could be zero without
## making the angle to the originally computed vector larger
## than given by the error bound. Do this in an approximative
## but numerically feasible way.
## error bounds of basis vectors in radians, see LAPACK user
## guide, http://www.netlib.org/lapack/lug/node96.html
if (true) # test for Octave version once submitted patch is applied
# to Octave (bug #33503)
__disna__ = @ __disna_optim__;
endif
ebnd = vtol ./ (__disna__ ("R", s, rows, cols)(rank+1:cols));
## sort elements by magnitude
sb = conj (retval) .* retval;
[sb, idx] = sort (sb);
idx += repmat (0:cols:cols*(cb-1), cols, 1); # for un-sorting
## norms of vectors made by all elements up to this
sb = sqrt (cumsum (sb));
## The norm of the vectors made up by elements settable to zero
## is small enough to be approximately equal to the angle
## between the full vectors before and after setting these
## elements to zero (considering the norms of the full vectors
## being 1). Index of approximated angles not exceeding error
## bound.
zidx = sb <= repmat (ebnd, cols, 1);
## set indexed elements to zero in original basis
zidx = zidx(idx);
retval(zidx) = 0;
else
## no error bounds computable with LAPACK
retval(abs (retval) < meps) = 0;
endif
else
retval = zeros (cols, 0);
endif
endif
endfunction
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