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).

########################################################################
##
## Copyright (C) 2000-2022 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{r} =} rref (@var{A})
## @deftypefnx {} {@var{r} =} rref (@var{A}, @var{tol})
## @deftypefnx {} {[@var{r}, @var{k}] =} rref (@dots{})
## Return the reduced row echelon form of @var{A}.
##
## @var{tol} defaults to
## @code{eps * max (size (@var{A})) * norm (@var{A}, inf)}.
##
## The optional return argument @var{k} contains the vector of
## "bound variables", which are those columns on which elimination has been
## performed.
##
## @end deftypefn

function [A, k] = rref (A, tol)

  if (nargin < 1)
    print_usage ();
  endif

  if (ndims (A) > 2)
    error ("rref: A must be a 2-dimensional matrix");
  endif

  [rows, cols] = size (A);

  if (nargin < 2)
    if (isa (A, "single"))
      tol = eps ("single") * max (rows, cols) * norm (A, inf ("single"));
    else
      tol = eps * max (rows, cols) * norm (A, inf);
    endif
  endif

  used = zeros (1, cols);
  r = 1;
  for c = 1:cols
    ## Find the pivot row
    [m, pivot] = max (abs (A(r:rows,c)));
    pivot = r + pivot - 1;

    if (m <= tol)
      ## Skip column c, making sure the approximately zero terms are
      ## actually zero.
      A(r:rows, c) = zeros (rows-r+1, 1);
    else
      ## keep track of bound variables
      used(1, c) = 1;

      ## Swap current row and pivot row
      A([pivot, r], c:cols) = A([r, pivot], c:cols);

      ## Normalize pivot row
      A(r, c:cols) = A(r, c:cols) / A(r, c);

      ## Eliminate the current column
      ridx = [1:r-1, r+1:rows];
      A(ridx, c:cols) = A(ridx, c:cols) - A(ridx, c) * A(r, c:cols);

      ## Check if done
      if (r++ == rows)
        break;
      endif
    endif
  endfor
  if (nargout > 1)
    k = find (used);
  endif

endfunction


%!test
%! a = [1];
%! [r k] = rref (a);
%! assert (r, [1], 2e-8);
%! assert (k, [1], 2e-8);

%!test
%! a = [1 3; 4 5];
%! [r k] = rref (a);
%! assert (rank (a), rank (r), 2e-8);
%! assert (r, eye (2), 2e-8);
%! assert (k == [1, 2] || k == [2, 1]);

%!test
%! a = [1 3; 4 5; 7 9];
%! [r k] = rref (a);
%! assert (rank (a), rank (r), 2e-8);
%! assert (r, eye (3)(:,1:2), 2e-8);
%! assert (k, [1 2], 2e-8);

%!test
%! a = [1 2 3; 2 4 6; 7 2 0];
%! [r k] = rref (a);
%! assert (rank (a), rank (r), 2e-8);
%! assert (r, [1 0 (3-7/2); 0 1 (7/4); 0 0 0], 2e-8);
%! assert (k, [1 2], 2e-8);

%!test
%! a = [1 2 1; 2 4 2.01; 2 4 2.1];
%! tol = 0.02;
%! [r k] = rref (a, tol);
%! assert (rank (a, tol), rank (r, tol), 2e-8);
%! tol = 0.2;
%! [r k] = rref (a, tol);
%! assert (rank (a, tol), rank (r, tol), 2e-8);

## Test input validation
%!error <Invalid call> rref ()