/usr/share/octave/packages/3.2/linear-algebra-2.1.0/smwsolve.m is in octave-linear-algebra 2.1.0-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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##
## Author: Jaroslav Hajek <highegg@gmail.com>
##
## This program 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.
##
## This program 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 this program; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn{Function File} {@var{x} =} smwsolve (@var{a}, @var{u}, @var{v}, @var{b})
## @deftypefnx{Function File} {} smwsolve (@var{solver}, @var{u}, @var{v}, @var{b})
## Solves the square system @code{(A + U*V')*X == B}, where @var{u} and @var{v} are
## matrices with several columns, using the Sherman-Morrison-Woodbury formula,
## so that a system with @var{a} as left-hand side is actually solved. This is
## especially advantageous if @var{a} is diagonal, sparse, triangular or
## positive definite.
## @var{a} can be sparse or full, the other matrices are expected to be full.
## Instead of a matrix @var{a}, a user may alternatively provide a function
## @var{solver} that performs the left division operation.
## @end deftypefn
function x = smwsolve (a, u, v, b)
if (nargin != 4)
print_usage ();
endif
n = columns (u);
if (n != columns (v) || rows (a) != rows (u) || columns (a) != rows (v))
error ("smwsolve: dimension mismatch");
elseif (! issquare (a))
error ("smwsolve: need a square matrix");
endif
nc = columns (b);
n = columns (u);
if (ismatrix (a))
xx = a \ [b, u];
elseif (isa (a, "function_handle"))
xx = a ([b, u]);
if (rows (xx) != rows (a) || columns (xx) != (nc + n))
error ("smwsolve: invalid result from a solver function");
endif
else
error ("smwsolve: a must be a matrix or function handle");
endif
x = xx(:,1:nc);
y = xx(:,nc+1:nc+n);
vxx = v' * xx;
vx = vxx(:,1:nc);
vy = vxx(:,nc+1:nc+n);
x = x - y * ((eye (n) + vy) \ vx);
endfunction
%!test
%! A = 2.1*eye (10);
%! u = rand (10, 2); u /= diag (norm (u, "cols"));
%! v = rand (10, 2); v /= diag (norm (v, "cols"));
%! b = rand (10, 2);
%! x1 = (A + u*v') \ b;
%! x2 = smwsolve (A, u, v, b);
%! assert (x1, x2, 1e-13);
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