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## -*- 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}.
##
## Reference:
## @nospell{Andrew V. Knyazev, Merico E. Argentati},
## "Principal Angles between Subspaces in an A-Based Scalar Product:
## Algorithms and Perturbation Estimates",
## @cite{SIAM Journal on Scientific Computing}, @w{Vol.@: 23}, @w{No.@: 6},
## @w{pp.@: 2008}--2040.
## @end deftypefn


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);
