function [X, k] = care_sign(A, G, Q)

n = size(A);
J = [zeros(n) eye(n); -eye(n) zeros(n)];
Z = [-Q -A'; -A G];
err = inf;
k = 0;
while err >= 1e-15
    Zold = Z;
    Z = 1/2*(Z + J*inv(Z)*J);
    % the products with J could be replaced
    % with direct block reordering, for
    % better performance
    err = norm(Zold - Z) / norm(Z);
    k = k + 1;
end
U = null(Z + J);
X = U(n+1:2*n, 1:n) / U(1:n, 1:n);