function [L, w] = chol_downdate (L, w)
%CHOL_DOWNDATE downdate a Cholesky factorization.
% Example
%   [L, w] = chol_downdate (L, w)
% See also: cs_demo

% CXSparse, Copyright (c) 2006-2022, Timothy A. Davis. All Rights Reserved.
% SPDX-License-Identifier: LGPL-2.1+

beta = 1 ;
n = size (L,1) ;
for j = 1:n
    alpha = w (j) / L (j,j) ;
    beta2 = sqrt (beta^2 - alpha^2) ;
    if (~isreal (beta2))
        error ('not positive definite') ;
    end
    gamma = alpha / (beta2 * beta) ;
    delta = beta2 / beta ;
    L (j,j) = delta * L (j,j) ;
    w (j) = alpha ;
    beta = beta2 ;
    if (j == n)
        return
    end
    w (j+1:n) = w (j+1:n) - alpha * L (j+1:n,j) ;
    L (j+1:n,j) = delta * L (j+1:n,j) - gamma * w (j+1:n) ;
end