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) 2004-2022 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/>.
##
########################################################################
##
## Original version by Paul Kienzle distributed as free software in the
## public domain.

## -*- texinfo -*-
## @deftypefn {} {@var{y} =} nthroot (@var{x}, @var{n})
##
## Compute the real (non-complex) @var{n}-th root of @var{x}.
##
## @var{x} must have all real entries and @var{n} must be a scalar.
## If @var{n} is an even integer and @var{x} has negative entries then
## @code{nthroot} aborts and issues an error.
##
## Example:
##
## @example
## @group
## nthroot (-1, 3)
## @result{} -1
## (-1) ^ (1 / 3)
## @result{} 0.50000 - 0.86603i
## @end group
## @end example
## @seealso{realsqrt, sqrt, cbrt}
## @end deftypefn

function y = nthroot (x, n)

  if (nargin != 2)
    print_usage ();
  endif

  if (iscomplex (x))
    error ("nthroot: X must not contain complex values");
  endif

  if (! isreal (n) || ! isscalar (n) || n == 0)
    error ("nthroot: N must be a real nonzero scalar");
  endif

  if (n == 3)
    y = cbrt (x);
  elseif (n == -3)
    y = 1 ./ cbrt (x);
  elseif (n < 0)
    y = 1 ./ nthroot (x, -n);
  else
    ## Compute using power.
    integer_n = n == fix (n);
    if (integer_n && mod (n, 2) == 1)
      y = abs (x) .^ (1/n) .* sign (x);
    elseif (any (x(:) < 0))
      error ("nthroot: N must be an odd integer if X contains negative values");
    else
      y = x .^ (1/n);
    endif

    if (integer_n && n > 0 && isfinite (n))
      if (isscalar (y) && y == 0)
        ## Don't apply correction which leads to division by zero (bug #43492)
      else
        ## FIXME: What is this correction for?
        y = ((n-1)*y + x ./ (y.^(n-1))) / n;
        y = merge (isfinite (y), y, x);
      endif
    endif
  endif

endfunction


%!assert (nthroot (-32, 5), -2)
%!assert (nthroot (81, 4), 3)
%!assert (nthroot (Inf, 4), Inf)
%!assert (nthroot (-Inf, 7), -Inf)
%!assert (nthroot (-Inf, -7), 0)

## This should not generate a division by zero warning
%!test <*43492>
%! warnmsg = lastwarn ();
%! assert (nthroot (0, 2), 0);
%! assert (lastwarn (), warnmsg);

## Test input validation
%!error <Invalid call> nthroot ()
%!error <Invalid call> nthroot (1)
%!error <X must not contain complex values> nthroot (1+j, 2)
%!error <N must be a real nonzero scalar> nthroot (1, i)
%!error <N must be a real nonzero scalar> nthroot (1, [1 2])
%!error <N must be a real nonzero scalar> nthroot (1, 0)
%!error <N must be an odd integer> nthroot (-1, 2)