95 lines
2.7 KiB
Matlab
95 lines
2.7 KiB
Matlab
function [x,u]=lyapunov_symm(a,b,qz_criterium)
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% Solves the Lyapunov equation x-a*x*a' = b, for b (and then x) symmetrical
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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: coefficient matrix (n x n)
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% b: coefficient square matrix (n x n)
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%
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% OUTPUTS
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% x: solution matrix (m x m)
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% ns_var: vector of indices of non-stationary variables (p x 1)
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% (m + p = n)
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% u: 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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% needs Matlab >= 7.0.1 for ordeig function (otherwise uses my_ordeig)
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%
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% part of DYNARE, copyright Dynare Team (2006-2008)
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% Gnu Public License
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ns_var = [];
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u = [];
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info = 0;
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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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[u,t] = schur(a);
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if exist('OCTAVE_VERSION') || matlab_ver_less_than('7.0.1')
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e1 = abs(my_ordeig(t)) > 2-qz_criterium;
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else
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e1 = abs(ordeig(t)) > 2-qz_criterium;
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end
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k = sum(e1);
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if exist('ordschur','builtin')
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% selects stable roots
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[u,t] = ordschur(u,t,e1);
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n = length(e1)-k;
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b=u(:,k+1:end)'*b*u(:,k+1:end);
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t = t(k+1:end,k+1:end);
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elseif k > 0
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% problem for Matlab version that don't have ordschur
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error(['lyapunov_sym: you need a Matlab version > 6.5 to handle models' ...
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' with unit roots'])
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else
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% no unit root
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n = length(e1);
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b=u'*b*u;
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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 t(i,i-1) == 0
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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); |