## Copyright (C) 1993-2015 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} {} rank (@var{A})
## @deftypefnx {Function File} {} rank (@var{A}, @var{tol})
## Compute the rank of matrix @var{A}, using the singular value decomposition.
##
## The rank is taken to be the number of singular values of @var{A} that
## are greater than the specified tolerance @var{tol}.  If the second
## argument is omitted, it is taken to be
##
## @example
## tol = max (size (@var{A})) * sigma(1) * eps;
## @end example
##
## @noindent
## where @code{eps} is machine precision and @code{sigma(1)} is the largest
## singular value of @var{A}.
##
## The rank of a matrix is the number of linearly independent rows or
## columns and determines how many particular solutions exist to a system
## of equations.  Use @code{null} for finding the remaining homogenous
## solutions.
##
## Example:
##
## @example
## @group
## x = [1 2 3
##      4 5 6
##      7 8 9];
## rank (x)
##   @result{} 2
## @end group
## @end example
##
## @noindent
## The number of linearly independent rows is only 2 because the final row
## is a linear combination of -1*row1 + 2*row2.
##
## @seealso{null, sprank, svd}
## @end deftypefn

## Author: jwe

function retval = rank (A, tol)

  if (nargin == 1)
    sigma = svd (A);
    if (isempty (sigma))
      tolerance = 0;
    else
      if (isa (A, "single"))
        tolerance = max (size (A)) * sigma (1) * eps ("single");
      else
        tolerance = max (size (A)) * sigma (1) * eps;
      endif
    endif
  elseif (nargin == 2)
    sigma = svd (A);
    tolerance = tol;
  else
    print_usage ();
  endif

  retval = sum (sigma > tolerance);

endfunction


%!test
%! A = [1 2 3 4 5 6 7;
%!      4 5 6 7 8 9 12;
%!      1 2 3.1 4 5 6 7;
%!      2 3 4 5 6 7 8;
%!      3 4 5 6 7 8 9;
%!      4 5 6 7 8 9 10;
%!      5 6 7 8 9 10 11];
%! assert (rank (A), 4);

%!test
%! A = [1 2 3 4 5 6 7;
%!      4 5 6 7 8 9 12;
%!      1 2 3.0000001 4 5 6 7;
%!      4 5 6 7 8 9 12.00001;
%!      3 4 5 6 7 8 9;
%!      4 5 6 7 8 9 10;
%!      5 6 7 8 9 10 11];
%! assert (rank (A), 4);

%!test
%! A = [1 2 3 4 5 6 7;
%!      4 5 6 7 8 9 12;
%!      1 2 3 4 5 6 7;
%!      4 5 6 7 8 9 12.00001;
%!      3 4 5 6 7 8 9;
%!      4 5 6 7 8 9 10;
%!      5 6 7 8 9 10 11];
%! assert (rank (A), 3);

%!test
%! A = [1 2 3 4 5 6 7;
%!      4 5 6 7 8 9 12;
%!      1 2 3 4 5 6 7;
%!      4 5 6 7 8 9 12;
%!      3 4 5 6 7 8 9;
%!      4 5 6 7 8 9 10;
%!      5 6 7 8 9 10 11];
%! assert (rank (A), 3);

%!test
%! A = eye (100);
%! assert (rank (A), 100);

%!assert (rank ([]), 0)
%!assert (rank ([1:9]), 1)
%!assert (rank ([1:9]'), 1)

%!test
%! A = [1, 2, 3; 1, 2.001, 3; 1, 2, 3.0000001];
%! assert (rank (A), 3);
%! assert (rank (A,0.0009), 1);
%! assert (rank (A,0.0006), 2);
%! assert (rank (A,0.00000002), 3);

