176 lines
6.0 KiB
Matlab
176 lines
6.0 KiB
Matlab
function [x,u] = lyapunov_symm(a,b,third_argument,lyapunov_complex_threshold,method, R)
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% Solves the Lyapunov equation x-a*x*a' = b, for b and x symmetric matrices.
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% If a has some unit roots, the function computes only the solution of the stable subsystem.
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%
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% INPUTS:
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% a [double] n*n matrix.
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% b [double] n*n matrix.
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% third_argument [double] scalar, if method <= 2 :
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% qz_criterium = third_argument unit root threshold for eigenvalues in a,
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% elseif method = 3 :
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% tol =third_argument the convergence criteria for fixed_point algorithm.
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% lyapunov_complex_threshold [double] scalar, complex block threshold for the upper triangular matrix T.
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% method [integer] Scalar, if method=0 [default] then U, T, n and k are not persistent.
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% method=1 then U, T, n and k are declared as persistent
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% variables and the schur decomposition is triggered.
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% method=2 then U, T, n and k are declared as persistent
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% variables and the schur decomposition is not performed.
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% method=3 fixed point method
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% OUTPUTS
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% x: [double] m*m solution matrix of the lyapunov equation, where m is the dimension of the stable subsystem.
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% u: [double] Schur vectors associated with unit roots
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%
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% ALGORITHM
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% Uses reordered Schur decomposition
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%
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% SPECIAL REQUIREMENTS
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% None
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% Copyright (C) 2006-2012 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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if nargin<5
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method = 0;
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end
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if method == 3
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persistent X method1;
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if ~isempty(method1)
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method = method1;
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end;
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tol = third_argument;
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fprintf(' [methode=%d] ',method);
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if method == 3
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%tol = 1e-10;
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it_fp = 0;
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evol = 100;
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if isempty(X)
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X = b;
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max_it_fp = 2000;
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else
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max_it_fp = 300;
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end;
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at = a';
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%fixed point iterations
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while evol > tol && it_fp < max_it_fp;
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X_old = X;
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X = a * X * at + b;
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evol = max(sum(abs(X - X_old))); %norm_1
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%evol = max(sum(abs(X - X_old)')); %norm_inf
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it_fp = it_fp + 1;
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end;
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fprintf('lyapunov it_fp=%d evol=%g\n',it_fp,evol);
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if it_fp >= max_it_fp
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disp(['convergence not achieved in solution of Lyapunov equation after ' int2str(it_fp) ' iterations, switching method from 3 to 0']);
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method1 = 0;
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method = 0;
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else
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method1 = 3;
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x = X;
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return;
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end;
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end;
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elseif method == 4
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% works only with Matlab System Control toolbox or octave the control package,
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if isoctave
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if ~user_has_octave_forge_package('control')
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error('lyapunov=square_root_solver not available; you must install the control package from Octave Forge')
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end
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else
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if ~user_has_matlab_license('control_toolbox')
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error('lyapunov=square_root_solver not available; you must install the control system toolbox')
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end
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end
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chol_b = R*chol(b,'lower');
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Rx = dlyapchol(a,chol_b);
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x = Rx' * Rx;
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return;
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end;
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qz_criterium = third_argument;
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if method
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persistent U T k n
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else
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if exist('U','var')
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clear('U','T','k','n')
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end
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end
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u = [];
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if size(a,1) == 1
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x=b/(1-a*a);
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return
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end
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if method<2
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[U,T] = schur(a);
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e1 = abs(ordeig(T)) > 2-qz_criterium;
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k = sum(e1); % Number of unit roots.
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n = length(e1)-k; % Number of stationary variables.
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if k > 0
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% Selects stable roots
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[U,T] = ordschur(U,T,e1);
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T = T(k+1:end,k+1:end);
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end
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end
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B = U(:,k+1:end)'*b*U(:,k+1:end);
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x = zeros(n,n);
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i = n;
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while i >= 2
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if abs(T(i,i-1))<lyapunov_complex_threshold
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if i == n
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c = zeros(n,1);
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else
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c = T(1:i,:)*(x(:,i+1:end)*T(i,i+1:end)') + ...
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T(i,i)*T(1:i,i+1:end)*x(i+1:end,i);
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end
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q = eye(i)-T(1:i,1:i)*T(i,i);
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x(1:i,i) = q\(B(1:i,i)+c);
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x(i,1:i-1) = x(1:i-1,i)';
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i = i - 1;
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else
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if i == n
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c = zeros(n,1);
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c1 = zeros(n,1);
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else
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c = T(1:i,:)*(x(:,i+1:end)*T(i,i+1:end)') + ...
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T(i,i)*T(1:i,i+1:end)*x(i+1:end,i) + ...
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T(i,i-1)*T(1:i,i+1:end)*x(i+1:end,i-1);
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c1 = T(1:i,:)*(x(:,i+1:end)*T(i-1,i+1:end)') + ...
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T(i-1,i-1)*T(1:i,i+1:end)*x(i+1:end,i-1) + ...
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T(i-1,i)*T(1:i,i+1:end)*x(i+1:end,i);
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end
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q = [ eye(i)-T(1:i,1:i)*T(i,i) , -T(1:i,1:i)*T(i,i-1) ; ...
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-T(1:i,1:i)*T(i-1,i) , eye(i)-T(1:i,1:i)*T(i-1,i-1) ];
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z = q\[ B(1:i,i)+c ; B(1:i,i-1) + c1 ];
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x(1:i,i) = z(1:i);
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x(1:i,i-1) = z(i+1:end);
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x(i,1:i-1) = x(1:i-1,i)';
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x(i-1,1:i-2) = x(1:i-2,i-1)';
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i = i - 2;
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end
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end
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if i == 1
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c = T(1,:)*(x(:,2:end)*T(1,2:end)') + T(1,1)*T(1,2:end)*x(2:end,1);
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x(1,1) = (B(1,1)+c)/(1-T(1,1)*T(1,1));
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end
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x = U(:,k+1:end)*x*U(:,k+1:end)';
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u = U(:,1:k);
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