function [LD,p,q] = ldlchol (A,beta)                                  %#ok
%LDLCHOL sparse A=LDL' factorization
% Note that L*L' (LCHOL) and L*D*L' (LDLCHOL) factorizations are faster
% than R'*R (CHOL2 and CHOL) and use less memory.  The LL' and LDL'
% factorization methods use tril(A).  A must be sparse.
%
% Example:
%   LD = ldlchol (A)            return the LDL' factorization of A
%   [LD,p] = ldlchol (A)        similar [R,p] = chol(A), but for L*D*L'
%   [LD,p,q] = ldlchol (A)      factorizes A(q,q) into L*D*L', where q is
%                               a fill-reducing ordering
%   LD = ldlchol (A,beta)       LDL' factorization of A*A'+beta*I
%   [LD,p] = ldlchol (A,beta)   like [R,p] = chol(A*A'+beta+I)
%   [LD,p,q] = ldlchol (A,beta) factorizes A(q,:)*A(q,:)'+beta*I = L*D*L'
%
% The output matrix LD contains both L and D.  D is on the diagonal of
% LD, and L is contained in the strictly lower triangular part of LD.
% The unit- diagonal of L is not stored.  You can obtain the L and D
% matrices with [L,D] = ldlsplit (LD).  LD is in the form needed by
% ldlupdate.
%
% Explicit zeros may appear in the LD matrix.  The pattern of LD matches
% the pattern of L as computed by symbfact2, even if some entries in LD
% are explicitly zero.  This is to ensure that ldlupdate and ldlsolve
% work properly.  You must NOT modify LD in MATLAB itself and then use
% ldlupdate or ldlsolve if LD contains explicit zero entries; ldlupdate
% and ldlsolve will fail catastrophically in this case.
%
% You MAY modify LD in MATLAB if you do not pass it back to ldlupdate or
% ldlsolve.  Just be aware that LD contains explicit zero entries,
% contrary to the standard practice in MATLAB of removing those entries
% from all sparse matrices.  LD = sparse (LD) will remove any zero
% entries in LD.
%
% See also ldlupdate, ldlsolve, ldlsplit, chol2, lchol, chol.

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

error ('ldlchol mexFunction not found') ;