maint: Use Octave coding conventions for cuddling parentheses in scripts directory * lin2mu.m, loadaudio.m, wavread.m, accumarray.m, bicubic.m, celldisp.m, colon.m, cplxpair.m, dblquad.m, divergence.m, genvarname.m, gradient.m, int2str.m, interp1.m, interp1q.m, interp2.m, interpn.m, loadobj.m, nthargout.m, __isequal__.m, __splinen__.m, quadgk.m, quadl.m, quadv.m, rat.m, rot90.m, rotdim.m, saveobj.m, subsindex.m, triplequad.m, delaunay3.m, griddata.m, inpolygon.m, tsearchn.m, voronoi.m, get_first_help_sentence.m, which.m, gray2ind.m, pink.m, dlmwrite.m, strread.m, textread.m, textscan.m, housh.m, ishermitian.m, issymmetric.m, krylov.m, logm.m, null.m, rref.m, compare_versions.m, copyfile.m, dump_prefs.m, edit.m, fileparts.m, getappdata.m, isappdata.m, movefile.m, orderfields.m, parseparams.m, __xzip__.m, rmappdata.m, setappdata.m, swapbytes.m, unpack.m, ver.m, fminbnd.m, fminunc.m, fsolve.m, glpk.m, lsqnonneg.m, qp.m, sqp.m, configure_make.m, copy_files.m, describe.m, get_description.m, get_forge_pkg.m, install.m, installed_packages.m, is_architecture_dependent.m, load_package_dirs.m, print_package_description.m, rebuild.m, repackage.m, save_order.m, shell.m, allchild.m, ancestor.m, area.m, axes.m, axis.m, clabel.m, close.m, colorbar.m, comet.m, comet3.m, contour.m, cylinder.m, ezmesh.m, ezsurf.m, findobj.m, fplot.m, hist.m, isocolors.m, isonormals.m, isosurface.m, isprop.m, legend.m, mesh.m, meshz.m, pareto.m, pcolor.m, peaks.m, plot3.m, plotmatrix.m, plotyy.m, polar.m, print.m, __add_datasource__.m, __add_default_menu__.m, __axes_limits__.m, __bar__.m, __clabel__.m, __contour__.m, __errcomm__.m, __errplot__.m, __ezplot__.m, __file_filter__.m, __fltk_print__.m, __ghostscript__.m, __gnuplot_print__.m, __go_draw_axes__.m, __go_draw_figure__.m, __interp_cube__.m, __marching_cube__.m, __patch__.m, __pie__.m, __plt__.m, __print_parse_opts__.m, __quiver__.m, __scatter__.m, __stem__.m, __tight_eps_bbox__.m, __uigetdir_fltk__.m, __uigetfile_fltk__.m, __uiputfile_fltk__.m, quiver.m, quiver3.m, rectangle.m, refreshdata.m, ribbon.m, scatter.m, semilogy.m, shading.m, slice.m, subplot.m, surface.m, surfl.m, surfnorm.m, text.m, uigetfile.m, uiputfile.m, whitebg.m, deconv.m, mkpp.m, pchip.m, polyaffine.m, polyder.m, polygcd.m, polyout.m, polyval.m, ppint.m, ppjumps.m, ppval.m, residue.m, roots.m, spline.m, splinefit.m, addpref.m, getpref.m, setpref.m, ismember.m, setxor.m, arch_fit.m, arch_rnd.m, arch_test.m, autoreg_matrix.m, diffpara.m, fftconv.m, filter2.m, hanning.m, hurst.m, periodogram.m, triangle_sw.m, sinc.m, spectral_xdf.m, spencer.m, stft.m, synthesis.m, unwrap.m, yulewalker.m, bicgstab.m, gmres.m, pcg.m, pcr.m, __sprand_impl__.m, speye.m, spfun.m, sprandn.m, spstats.m, svds.m, treelayout.m, treeplot.m, bessel.m, factor.m, legendre.m, perms.m, primes.m, magic.m, toeplitz.m, corr.m, cov.m, mean.m, median.m, mode.m, qqplot.m, quantile.m, ranks.m, zscore.m, logistic_regression_likelihood.m, bartlett_test.m, chisquare_test_homogeneity.m, chisquare_test_independence.m, kolmogorov_smirnov_test.m, run_test.m, u_test.m, wilcoxon_test.m, z_test.m, z_test_2.m, bin2dec.m, dec2base.m, mat2str.m, strcat.m, strchr.m, strjust.m, strtok.m, substr.m, untabify.m, assert.m, demo.m, example.m, fail.m, speed.m, test.m, now.m: Use Octave coding conventions for cuddling parentheses in scripts directory.

## Copyright (C) 2008-2012 David Bateman
##
## 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} {@var{s} =} spaugment (@var{A}, @var{c})
## Create the augmented matrix of @var{A}.  This is given by
##
## @example
## @group
## [@var{c} * eye(@var{m}, @var{m}), @var{A};
##             @var{A}', zeros(@var{n}, @var{n})]
## @end group
## @end example
##
## @noindent
## This is related to the least squares solution of
## @code{@var{A} \ @var{b}}, by
##
## @example
## @group
## @var{s} * [ @var{r} / @var{c}; x] = [ @var{b}, zeros(@var{n}, columns(@var{b})) ]
## @end group
## @end example
##
## @noindent
## where @var{r} is the residual error
##
## @example
## @var{r} = @var{b} - @var{A} * @var{x}
## @end example
##
## As the matrix @var{s} is symmetric indefinite it can be factorized
## with @code{lu}, and the minimum norm solution can therefore be found
## without the need for a @code{qr} factorization.  As the residual
## error will be @code{zeros (@var{m}, @var{m})} for under determined
## problems, and example can be
##
## @example
## @group
## m = 11; n = 10; mn = max (m, n);
## A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],
##              [-1, 0, 1], m, n);
## x0 = A \ ones (m,1);
## s = spaugment (A);
## [L, U, P, Q] = lu (s);
## x1 = Q * (U \ (L \ (P  * [ones(m,1); zeros(n,1)])));
## x1 = x1(end - n + 1 : end);
## @end group
## @end example
##
## To find the solution of an overdetermined problem needs an estimate
## of the residual error @var{r} and so it is more complex to formulate
## a minimum norm solution using the @code{spaugment} function.
##
## In general the left division operator is more stable and faster than
## using the @code{spaugment} function.
## @end deftypefn

function s = spaugment (A, c)
  if (nargin < 2)
    if (issparse (A))
      c = max (max (abs (A))) / 1000;
    else
      if (ndims (A) != 2)
        error ("spaugment: expecting 2-dimenisional matrix");
      else
        c = max (abs (A(:))) / 1000;
      endif
    endif
  elseif (!isscalar (c))
    error ("spaugment: C must be a scalar");
  endif

  [m, n] = size (A);
  s = [ c * speye(m, m), A; A', sparse(n, n)];
endfunction


%!testif HAVE_UMFPACK
%! m = 11; n = 10; mn = max (m ,n);
%! A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],[-1,0,1], m, n);
%! x0 = A \ ones (m,1);
%! s = spaugment (A);
%! [L, U, P, Q] = lu (s);
%! x1 = Q * (U \ (L \ (P  * [ones(m,1); zeros(n,1)])));
%! x1 = x1(end - n + 1 : end);
%! assert (x1, x0, 1e-6);