view scripts/linear-algebra/rank.m @ 30875:5d3faba0342e

doc: Ensure documentation lists output argument when it exists for all m-files. For new users of Octave it is best to show explicit calling forms in the documentation and to show a return argument when it exists. * bp-table.cc, shift.m, accumarray.m, accumdim.m, bincoeff.m, bitcmp.m, bitget.m, bitset.m, blkdiag.m, celldisp.m, cplxpair.m, dblquad.m, flip.m, fliplr.m, flipud.m, idivide.m, int2str.m, interpft.m, logspace.m, num2str.m, polyarea.m, postpad.m, prepad.m, randi.m, repmat.m, rng.m, rot90.m, rotdim.m, structfun.m, triplequad.m, uibuttongroup.m, uicontrol.m, uipanel.m, uipushtool.m, uitoggletool.m, uitoolbar.m, waitforbuttonpress.m, help.m, __additional_help_message__.m, hsv.m, im2double.m, im2frame.m, javachk.m, usejava.m, argnames.m, char.m, formula.m, inline.m, __vectorize__.m, findstr.m, flipdim.m, strmatch.m, vectorize.m, commutation_matrix.m, cond.m, cross.m, duplication_matrix.m, expm.m, orth.m, rank.m, rref.m, trace.m, vech.m, cast.m, compare_versions.m, delete.m, dir.m, fileattrib.m, grabcode.m, gunzip.m, inputname.m, license.m, list_primes.m, ls.m, mexext.m, movefile.m, namelengthmax.m, nargoutchk.m, nthargout.m, substruct.m, swapbytes.m, ver.m, verLessThan.m, what.m, fminunc.m, fsolve.m, fzero.m, optimget.m, __fdjac__.m, matlabroot.m, savepath.m, campos.m, camroll.m, camtarget.m, camup.m, camva.m, camzoom.m, clabel.m, diffuse.m, legend.m, orient.m, rticks.m, specular.m, thetaticks.m, xlim.m, xtickangle.m, xticklabels.m, xticks.m, ylim.m, ytickangle.m, yticklabels.m, yticks.m, zlim.m, ztickangle.m, zticklabels.m, zticks.m, ellipsoid.m, isocolors.m, isonormals.m, stairs.m, surfnorm.m, __actual_axis_position__.m, __pltopt__.m, close.m, graphics_toolkit.m, pan.m, print.m, printd.m, __ghostscript__.m, __gnuplot_print__.m, __opengl_print__.m, rotate3d.m, subplot.m, zoom.m, compan.m, conv.m, poly.m, polyaffine.m, polyder.m, polyint.m, polyout.m, polyreduce.m, polyvalm.m, roots.m, prefdir.m, prefsfile.m, profexplore.m, profexport.m, profshow.m, powerset.m, unique.m, arch_rnd.m, arma_rnd.m, autoreg_matrix.m, bartlett.m, blackman.m, detrend.m, durbinlevinson.m, fftconv.m, fftfilt.m, fftshift.m, fractdiff.m, hamming.m, hanning.m, hurst.m, ifftshift.m, rectangle_lw.m, rectangle_sw.m, triangle_lw.m, sinc.m, sinetone.m, sinewave.m, spectral_adf.m, spectral_xdf.m, spencer.m, ilu.m, __sprand__.m, sprand.m, sprandn.m, sprandsym.m, treelayout.m, beta.m, betainc.m, betaincinv.m, betaln.m, cosint.m, expint.m, factorial.m, gammainc.m, gammaincinv.m, lcm.m, nthroot.m, perms.m, reallog.m, realpow.m, realsqrt.m, sinint.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, pascal.m, rosser.m, toeplitz.m, vander.m, wilkinson.m, center.m, corr.m, cov.m, discrete_cdf.m, discrete_inv.m, discrete_pdf.m, discrete_rnd.m, empirical_cdf.m, empirical_inv.m, empirical_pdf.m, empirical_rnd.m, kendall.m, kurtosis.m, mad.m, mean.m, meansq.m, median.m, mode.m, moment.m, range.m, ranks.m, run_count.m, skewness.m, spearman.m, statistics.m, std.m, base2dec.m, bin2dec.m, blanks.m, cstrcat.m, deblank.m, dec2base.m, dec2bin.m, dec2hex.m, hex2dec.m, index.m, regexptranslate.m, rindex.m, strcat.m, strjust.m, strtrim.m, strtrunc.m, substr.m, untabify.m, __have_feature__.m, __prog_output_assert__.m, __run_test_suite__.m, example.m, fail.m, asctime.m, calendar.m, ctime.m, date.m, etime.m: Add return arguments to @deftypefn macros where they were missing. Rename variables in functions (particularly generic "retval") to match documentation. Rename some return variables for (hopefully) better clarity (e.g., 'ax' to 'hax' to indicate it is a graphics handle to an axes object).
author Rik <rik@octave.org>
date Wed, 30 Mar 2022 20:40:27 -0700
parents 796f54d4ddbf
children 597f3ee61a48
line wrap: on
line source

########################################################################
##
## Copyright (C) 1993-2022 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################

## -*- texinfo -*-
## @deftypefn  {} {@var{k} =} rank (@var{A})
## @deftypefnx {} {@var{k} =} 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 equals the dimension of the row and column space.  The function
## @code{orth} may be used to compute an orthonormal basis of the column space.
##
## For testing if a system @code{@var{A}*@var{x} = @var{b}} of linear equations
## is solvable, one can use
##
## @example
## rank (@var{A}) == rank ([@var{A} @var{b}])
## @end example
##
## In this case, @code{@var{x} = @var{A} \ @var{b}} finds a particular solution
## @var{x}.  The general solution is @var{x} plus the null space of matrix
## @var{A}.  The function @code{null} may be used to compute a basis of the
## null space.
##
## Example:
##
## @example
## @group
## A = [1 2 3
##      4 5 6
##      7 8 9];
## rank (A)
##   @result{} 2
## @end group
## @end example
##
## @noindent
## In this example, the number of linearly independent rows is only 2 because
## the final row is a linear combination of the first two rows:
##
## @example
## A(3,:) == -A(1,:) + 2 * A(2,:)
## @end example
##
## @seealso{null, orth, sprank, svd, eps}
## @end deftypefn

function k = rank (A, tol)

  if (nargin < 1)
    print_usage ();
  endif

  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
  else
    sigma = svd (A);
    tolerance = tol;
  endif

  k = 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);