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function [x,u] = lyapunov_symm(a,b,lyapunov_fixed_point_tol,qz_criterium,lyapunov_complex_threshold,method,debug)
% Solves the Lyapunov equation x-a*x*a' = b, for b and x symmetric matrices.
% If a has some unit roots, the function computes only the solution of the stable subsystem.
%
% INPUTS:
% a [double] n*n matrix.
% b [double] n*n matrix.
% qz_criterium [double] unit root threshold for eigenvalues
% lyapunov_fixed_point_tol [double] convergence criteria for fixed_point algorithm.
% lyapunov_complex_threshold [double] scalar, complex block threshold for the upper triangular matrix T.
% method [integer] Scalar, if method=0 [default] then U, T, n and k are not persistent.
% method=1 then U, T, n and k are declared as persistent
% variables and the Schur decomposition is triggered.
% method=2 then U, T, n and k are declared as persistent
% variables and the Schur decomposition is not performed.
% method=3 fixed point method
% OUTPUTS
% x: [double] m*m solution matrix of the lyapunov equation, where m is the dimension of the stable subsystem.
% u: [double] Schur vectors associated with unit roots
%
% ALGORITHM
% Uses reordered Schur decomposition (Bartels-Stewart algorithm)
% [method<3] or a fixed point algorithm (method==3)
%
% SPECIAL REQUIREMENTS
% None
% Copyright © 2006-2023 Dynare Team
%
% This file is part of Dynare.
%
% Dynare 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.
%
% Dynare 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 Dynare. If not, see <https://www.gnu.org/licenses/>.
if nargin<6 || isempty(method)
method = 0;
end
if nargin<7
debug = 0;
end
if method == 3
persistent X method1;
if ~isempty(method1)
method = method1;
end
if debug
fprintf('lyapunov_symm:: [method=%d] \n',method);
end
if method == 3
%tol = 1e-10;
it_fp = 0;
evol = 100;
if isempty(X) || length(X)~=length(b)
X = b;
max_it_fp = 2000;
else
max_it_fp = 300;
end
at = a';
%fixed point iterations
while evol > lyapunov_fixed_point_tol && it_fp < max_it_fp
X_old = X;
X = a * X * at + b;
evol = max(sum(abs(X - X_old))); %norm_1
%evol = max(sum(abs(X - X_old)')); %norm_inf
it_fp = it_fp + 1;
end
if debug
fprintf('lyapunov_symm:: lyapunov fixed_point iterations=%d norm=%g\n',it_fp,evol);
end
if it_fp >= max_it_fp
disp(['lyapunov_symm:: convergence not achieved in solution of Lyapunov equation after ' int2str(it_fp) ' iterations, switching method from 3 to 0']);
method1 = 0;
method = 0;
else
method1 = 3;
x = X;
return
end
end
end
if method
persistent U T k n
else
% if exist('U','var')
% clear('U','T','k','n')
% end
end
u = [];
if size(a,1) == 1
x=b/(1-a*a);
return
end
if method<2
[U,T] = schur(full(a));
e1 = abs(ordeig(T)) > 2-qz_criterium;
k = sum(e1); % Number of unit roots.
n = length(e1)-k; % Number of stationary variables.
if k > 0
% Selects stable roots
[U,T] = ordschur(U,T,e1);
T = T(k+1:end,k+1:end);
end
end
B = U(:,k+1:end)'*b*U(:,k+1:end);
x = zeros(n,n);
i = n;
while i >= 2
if abs(T(i,i-1))<lyapunov_complex_threshold
if i == n
c = zeros(n,1);
else
c = T(1:i,:)*(x(:,i+1:end)*T(i,i+1:end)') + ...
T(i,i)*T(1:i,i+1:end)*x(i+1:end,i);
end
q = eye(i)-T(1:i,1:i)*T(i,i);
x(1:i,i) = q\(B(1:i,i)+c);
x(i,1:i-1) = x(1:i-1,i)';
i = i - 1;
else
if i == n
c = zeros(n,1);
c1 = zeros(n,1);
else
c = T(1:i,:)*(x(:,i+1:end)*T(i,i+1:end)') + ...
T(i,i)*T(1:i,i+1:end)*x(i+1:end,i) + ...
T(i,i-1)*T(1:i,i+1:end)*x(i+1:end,i-1);
c1 = T(1:i,:)*(x(:,i+1:end)*T(i-1,i+1:end)') + ...
T(i-1,i-1)*T(1:i,i+1:end)*x(i+1:end,i-1) + ...
T(i-1,i)*T(1:i,i+1:end)*x(i+1:end,i);
end
q = [ eye(i)-T(1:i,1:i)*T(i,i) , -T(1:i,1:i)*T(i,i-1) ; ...
-T(1:i,1:i)*T(i-1,i) , eye(i)-T(1:i,1:i)*T(i-1,i-1) ];
z = q\[ B(1:i,i)+c ; B(1:i,i-1) + c1 ];
x(1:i,i) = z(1:i);
x(1:i,i-1) = z(i+1:end);
x(i,1:i-1) = x(1:i-1,i)';
x(i-1,1:i-2) = x(1:i-2,i-1)';
i = i - 2;
end
end
if i == 1
c = T(1,:)*(x(:,2:end)*T(1,2:end)') + T(1,1)*T(1,2:end)*x(2:end,1);
x(1,1) = (B(1,1)+c)/(1-T(1,1)*T(1,1));
end
x = U(:,k+1:end)*x*U(:,k+1:end)';
u = U(:,1:k);
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