function LD = ldlupdate (LD,C,updown)                                 %#ok
%LDLUPDATE multiple-rank update or downdate of a sparse LDL' factorization
%
% On input, LD contains the LDL' factorization of A (L*D*L'=A or A(q,q)).
% The unit-diagonal of L is not stored.  In its place is the diagonal
% matrix D.  LD can be computed using the CHOLMOD mexFunctions:
%
%   LD = ldlchol (A) ;
% or
%   [LD,p,q] = ldlchol (A) ;
%
% With this LD, either of the following MATLAB statements:
%
% Example:
%       LD = ldlupdate (LD,C)
%       LD = ldlupdate (LD,C,'+')
%
% return the LDL' factorization of A+C*C' or A(q,q)-C*C' if LD holds the
% LDL' factorization of A(q,q) on input.  For a downdate:
%
%       LD = ldlupdate (LD,C,'-')
%
% returns the LDL' factorization of A-C*C' or A(q,q)-C*C'.
%
% LD and C must be sparse and real.  LD must be square, and C must have
% the same number of rows as LD.  You must not modify LD in MATLAB (see
% the WARNING below).
%
% Note that if C is sparse with few columns, most of the time spent in
% this routine is taken by copying the input LD to the output LD.  If
% MATLAB allowed mexFunctions to safely modify its inputs, this
% mexFunction would be much faster, since not all of LD changes.
%
% See also ldlchol, ldlsplit, ldlsolve, cholupdate.
%
% ========================================================================
% ============================ WARNING ===================================
% ========================================================================
%
% MATLAB drops zero entries from its sparse matrices.  LD can contain
% numerically zero entries that are symbolically present in the sparse
% matrix data structure.  These are essential for ldlupdate and ldlsolve
% to work properly, since they exploit the graph-theoretic structure of a
% sparse Cholesky factorization. If you modify LD in MATLAB, those zero
% entries may get dropped and the required graph property will be
% destroyed.  In this case, ldlupdate and ldlsolve will fail
% catastrophically (possibly with a segmentation fault, terminating
% MATLAB).  It takes much more time to ensure this property holds than
% the time it takes to do the update/downdate or the solve, so ldlupdate
% and ldlsolve simply assume the propertly holds.

 % Copyright 2006-2023, Timothy A. Davis, All Rights Reserved.
 % SPDX-License-Identifier: GPL-2.0+

error ('ldlupdate mexFunction not found') ;