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## Copyright (C) 1993-2013 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);
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