update packages

[CreaPhase.git] / octave_packages / m / linear-algebra / null.m

diff --git a/octave_packages/m/linear-algebra/null.m b/octave_packages/m/linear-algebra/null.m
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+## Copyright (C) 1994-2012 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
+
+function retval = null (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)) * s (1) * eps ("single");
+      else
+        tol = max (size (A)) * s (1) * eps;
+      endif
+    elseif (nargin != 2)
+      print_usage ();
+    endif
+
+    rank = sum (s > tol);
+
+    if (rank < cols)
+      retval = V (:, rank+1:cols);
+      if (isa (A, "single"))
+        retval(abs (retval) < eps ("single")) = 0;
+      else
+        retval(abs (retval) < eps) = 0;
+      endif
+    else
+      retval = zeros (cols, 0);
+    endif
+  endif
+
+endfunction
+
+%!test
+%! A = 0;
+%! assert(null(A), 1);
+
+%!test
+%! A = 1;
+%! assert(null(A), zeros(1,0))
+
+%!test
+%! A = [1 0; 0 1];
+%! assert(null(A), zeros(2,0));
+
+%!test
+%! A = [1 0; 1 0];
+%! assert(null(A), [0 1]')
+
+%!test
+%! A = [1 1; 0 0];
+%! assert(null(A), [-1/sqrt(2) 1/sqrt(2)]', eps)
+
+%!test
+%! tol = 1e-4;
+%! A = [1 0; 0 tol-eps];
+%! assert(null(A,tol), [0 1]')
+
+%!test
+%! tol = 1e-4;
+%! A = [1 0; 0 tol+eps];
+%! assert(null(A,tol), zeros(2,0));
+
+%!error null()