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########################################################################
##
## Copyright (C) 2008-2024 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{angle} =} subspace (@var{A}, @var{B})
## Determine the largest principal angle between two subspaces
## spanned by the columns of matrices @var{A} and @var{B}.
## @end deftypefn
## Reference:
## Andrew V. Knyazev, Merico E. Argentati:
## Principal Angles between Subspaces in an A-Based Scalar Product:
## Algorithms and Perturbation Estimates.
## SIAM Journal on Scientific Computing, Vol. 23 no. 6, pp. 2008-2040
##
## other texts are also around...
function ang = subspace (A, B)
if (nargin != 2)
print_usage ();
elseif (ndims (A) != 2 || ndims (B) != 2)
error ("subspace: A and B must be 2-dimensional arrays");
elseif (rows (A) != rows (B))
error ("subspace: column dimensions of A and B must match");
endif
A = orth (A);
B = orth (B);
c = A'*B;
scos = min (svd (c));
if (scos^2 > 1/2)
if (columns (A) >= columns (B))
c = B - A*c;
else
c = A - B*c';
endif
ssin = max (svd (c));
ang = asin (min (ssin, 1));
else
ang = acos (scos);
endif
endfunction
%!assert (subspace (1, 1), 0)
%!assert (subspace ([1, 0]', [1, 1; 0, 1]'), 0, 3*eps)
%!assert (subspace ([1, 0, 1]', [1, 1, 0; 1, -1, 0]'), pi/4, 3*eps)
%!assert (subspace ([1 5 0 0; -3 2 0 0]', [0 0 4 2; 0 0 4 3]'), pi/2)
%!assert (subspace ([1 1 1 1; 1 2 3 4]', [1 -1 -1 1]'), pi/2)
%!test
%! ## For small angle between subspaces
%! theta = pi/200;
%! Ry = [cos(theta), 0, sin(theta);0, 1, 0;-sin(theta), 0, cos(theta)];
%! a = Ry*[3*e, 0, 0]';
%! b = [1, 1, 0; 1, -1, 0]';
%! assert (theta, subspace (a, b), eps);
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