Add a useful package (from Source forge) for octave

[CreaPhase.git] / octave_packages / optim-1.2.0 / private / __null_optim__.m

diff --git a/octave_packages/optim-1.2.0/private/__null_optim__.m b/octave_packages/optim-1.2.0/private/__null_optim__.m
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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