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) 2012-2022 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{z} =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
## @deftypefnx {} {[@var{v}, @var{z}] =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
##
## Solve the polynomial eigenvalue problem of degree @var{l}.
##
## Given an @var{n}x@var{n} matrix polynomial
##
## @code{@var{C}(@var{s}) = @var{C0} + @var{C1} @var{s} + @dots{} + @var{Cl}
## @var{s}^@var{l}}
##
## @code{polyeig} solves the eigenvalue problem
##
## @code{(@var{C0} + @var{C1} @var{z} + @dots{} + @var{Cl} @var{z}^@var{l})
## @var{v} = 0}.
##
## Note that the eigenvalues @var{z} are the zeros of the matrix polynomial.
## @var{z} is a row vector with @code{@var{n}*@var{l}} elements.  @var{v} is a
## matrix (@var{n} x @var{n}*@var{l}) with columns that correspond to the
## eigenvectors.
##
## @seealso{eig, eigs, compan}
## @end deftypefn

function [z, v] = polyeig (varargin)

  if (nargin < 1)
    print_usage ();
  endif

  n = rows (varargin{1});

  for i = 1 : nargin
    if (! issquare (varargin{i}))
      error ("polyeig: coefficients must be square matrices");
    endif
    if (rows (varargin{i}) != n)
      error ("polyeig: coefficients must have the same dimensions");
    endif
  endfor

  ## matrix polynomial degree
  l = nargin - 1;

  ## form needed matrices
  C = [ zeros(n * (l - 1), n), eye(n * (l - 1));
       -cell2mat(varargin(1:end-1)) ];

  D = [ eye(n * (l - 1)), zeros(n * (l - 1), n);
        zeros(n, n * (l - 1)), varargin{end} ];

  ## solve generalized eigenvalue problem
  if (nargout < 2)
    z = eig (C, D);
  else
    [z, v] = eig (C, D);
    v = diag (v);
    ## return n-element eigenvectors normalized so that the infinity-norm = 1
    z = z(1:n,:);
    t = max (z);    # max() takes the abs if complex.
    z ./= t;
  endif

endfunction


%!shared C0, C1
%! C0 = [8, 0; 0, 4];
%! C1 = [1, 0; 0, 1];

%!test
%! z = polyeig (C0, C1);
%! assert (z, [-8; -4]);

%!test
%! [v,z] = polyeig (C0, C1);
%! assert (z, [-8; -4]);
%! z = diag (z);
%! d = C0*v + C1*v*z;
%! assert (norm (d), 0.0);

## Test input validation
%!error <Invalid call> polyeig ()
%!error <coefficients must be square matrices> polyeig (ones (3,2))
%!error <coefficients must have the same dimensions>
%! polyeig (ones (3,3), ones (2,2))