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function [Da,DP1,DLIK,D2a,D2P1,Hesst] = computeDLIK(k,tmp,Z,Zflag,v,T,K,P,iF,Da,DYss,DT,DOm,DP,DH,notsteady,D2a,D2Yss,D2T,D2Om,D2P)
% Copyright (C) 2012 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 <http://www.gnu.org/licenses/>.
% AUTHOR(S) marco.ratto@jrc.ec.europa.eu
persistent DK DF D2K D2F
if notsteady
if Zflag
[DK,DF,DP1] = computeDKalmanZ(T,DT,DOm,P,DP,DH,Z,iF,K);
if nargout>4,
[D2K,D2F,D2P1] = computeD2KalmanZ(T,DT,D2T,D2Om,P,DP,D2P,DH,Z,iF,K,DK);
end
else
[DK,DF,DP1] = computeDKalman(T,DT,DOm,P,DP,DH,Z,iF,K);
if nargout>4,
[D2K,D2F,D2P1] = computeD2Kalman(T,DT,D2T,D2Om,P,DP,D2P,DH,Z,iF,K,DK);
end
end
else
DP1=DP;
if nargout>4,
D2P1=D2P;
end
end
Dv=zeros(length(v),k);
% D2v=zeros(length(v),k,k);
for ii = 1:k
if Zflag
Dv(:,ii) = -Z*Da(:,ii) - Z*DYss(:,ii);
% if nargout>4,
% for jj = 1:ii
% D2v(:,jj,ii) = -Z*D2Yss(:,jj,ii) - Z*D2a(:,jj,ii);
% D2v(:,ii,jj) = D2v(:,jj,ii);
% end
% end
else
Dv(:,ii) = -Da(Z,ii) - DYss(Z,ii);
% if nargout>4,
% for jj = 1:ii
% D2v(:,jj,ii) = -D2Yss(Z,jj,ii) - D2a(Z,jj,ii);
% D2v(:,ii,jj) = D2v(:,jj,ii);
% end
% end
end
end
Hesst = zeros(k,k);
DLIK=zeros(k,1);
for ii = 1:k
% dai = da(:,:,ii);
dKi = DK(:,:,ii);
dtmp(:,ii) = Da(:,ii)+dKi*v+K*Dv(:,ii);
if nargout>4,
diFi = -iF*DF(:,:,ii)*iF;
for jj = 1:ii
dFj = DF(:,:,jj);
diFj = -iF*DF(:,:,jj)*iF;
dKj = DK(:,:,jj);
d2Kij = D2K(:,:,jj,ii);
d2Fij = D2F(:,:,jj,ii);
d2iFij = -diFi*dFj*iF -iF*d2Fij*iF -iF*dFj*diFi;
% dtmpj = Da(:,jj)+dKj*v+K*Dv(:,jj);
% d2vij = D2v(:,ii,jj);
if Zflag
d2vij = -Z*D2Yss(:,jj,ii) - Z*D2a(:,jj,ii);
else
d2vij = -D2Yss(Z,jj,ii) - D2a(Z,jj,ii);
end
d2tmpij = D2a(:,jj,ii) + d2Kij*v + dKj*Dv(:,ii) + dKi*Dv(:,jj) + K*d2vij;
D2a(:,jj,ii) = D2T(:,:,jj,ii)*tmp + DT(:,:,jj)*dtmp(:,ii) + DT(:,:,ii)*dtmp(:,jj) + T*d2tmpij;
D2a(:,ii,jj) = D2a(:,jj,ii);
if nargout==6,
Hesst(ii,jj) = getHesst_ij(v,Dv(:,ii),Dv(:,jj),d2vij,iF,diFi,diFj,d2iFij,dFj,d2Fij);
end
end
end
Da(:,ii) = DT(:,:,ii)*tmp + T*dtmp(:,ii);
DLIK(ii,1) = trace( iF*DF(:,:,ii) ) + 2*Dv(:,ii)'*iF*v - v'*(iF*DF(:,:,ii)*iF)*v;
end
if nargout==4,
% Hesst(ii,jj) = getHesst_ij(v,Dv(:,ii),Dv(:,jj),0,iF,diFi,diFj,0,dFj,0);
vecDPmf = reshape(DF,[],k);
D2a = 2*Dv'*iF*Dv + (vecDPmf' * kron(iF,iF) * vecDPmf);
% for ii = 1:k
%
% diFi = -iF*DF(:,:,ii)*iF;
% for jj = 1:ii
% dFj = DF(:,:,jj);
% diFj = -iF*DF(:,:,jj)*iF;
%
% Hesst(ii,jj) = getHesst_ij(v*0,Dv(:,ii),Dv(:,jj),v*0,iF,diFi,diFj,0,-dFj,0);
% end
% end
end
% end of computeDLIK
function Hesst_ij = getHesst_ij(e,dei,dej,d2eij,iS,diSi,diSj,d2iSij,dSj,d2Sij);
% computes (i,j) term in the Hessian
Hesst_ij = trace(diSi*dSj + iS*d2Sij) + e'*d2iSij*e + 2*(dei'*diSj*e + dei'*iS*dej + e'*diSi*dej + e'*iS*d2eij);
% end of getHesst_ij
function [DK,DF,DP1] = computeDKalman(T,DT,DOm,P,DP,DH,Z,iF,K)
k = size(DT,3);
tmp = P-K*P(Z,:);
DF = zeros([size(iF),k]);
DK = zeros([size(K),k]);
DP1 = zeros([size(P),k]);
for ii = 1:k
DF(:,:,ii) = DP(Z,Z,ii) + DH(:,:,ii);
DiF = -iF*DF(:,:,ii)*iF;
DK(:,:,ii) = DP(:,Z,ii)*iF + P(:,Z)*DiF;
Dtmp = DP(:,:,ii) - DK(:,:,ii)*P(Z,:) - K*DP(Z,:,ii);
DP1(:,:,ii) = DT(:,:,ii)*tmp*T' + T*Dtmp*T' + T*tmp*DT(:,:,ii)' + DOm(:,:,ii);
end
% end of computeDKalman
function [DK,DF,DP1] = computeDKalmanZ(T,DT,DOm,P,DP,DH,Z,iF,K)
k = size(DT,3);
tmp = P-K*Z*P;
DF = zeros([size(iF),k]);
DK = zeros([size(K),k]);
DP1 = zeros([size(P),k]);
for ii = 1:k
DF(:,:,ii) = Z*DP(:,:,ii)*Z + DH(:,:,ii);
DiF = -iF*DF(:,:,ii)*iF;
DK(:,:,ii) = DP(:,:,ii)*Z*iF + P(:,:)*Z*DiF;
Dtmp = DP(:,:,ii) - DK(:,:,ii)*Z*P(:,:) - K*Z*DP(:,:,ii);
DP1(:,:,ii) = DT(:,:,ii)*tmp*T' + T*Dtmp*T' + T*tmp*DT(:,:,ii)' + DOm(:,:,ii);
end
% end of computeDKalmanZ
function [d2K,d2S,d2P1] = computeD2Kalman(A,dA,d2A,d2Om,P0,dP0,d2P0,DH,Z,iF,K0,dK0);
% computes the second derivatives of the Kalman matrices
% note: A=T in main func.
k = size(dA,3);
tmp = P0-K0*P0(Z,:);
[ns,no] = size(K0);
% CPC = C*P0*C'; CPC = .5*(CPC+CPC');iF = inv(CPC);
% APC = A*P0*C';
% APA = A*P0*A';
d2K = zeros(ns,no,k,k);
d2S = zeros(no,no,k,k);
d2P1 = zeros(ns,ns,k,k);
for ii = 1:k
dAi = dA(:,:,ii);
dFi = dP0(Z,Z,ii);
d2Omi = d2Om(:,:,ii);
diFi = -iF*dFi*iF;
dKi = dK0(:,:,ii);
for jj = 1:k
dAj = dA(:,:,jj);
dFj = dP0(Z,Z,jj);
d2Omj = d2Om(:,:,jj);
dFj = dP0(Z,Z,jj);
diFj = -iF*dFj*iF;
dKj = dK0(:,:,jj);
d2Aij = d2A(:,:,jj,ii);
d2Pij = d2P0(:,:,jj,ii);
d2Omij = d2Om(:,:,jj,ii);
% second order
d2Fij = d2Pij(Z,Z) ;
% d2APC = d2Aij*P0*C' + A*d2Pij*C' + A*P0*d2Cij' + dAi*dPj*C' + dAj*dPi*C' + A*dPj*dCi' + A*dPi*dCj' + dAi*P0*dCj' + dAj*P0*dCi';
d2APC = d2Pij(:,Z);
d2iF = -diFi*dFj*iF -iF*d2Fij*iF -iF*dFj*diFi;
d2Kij= d2Pij(:,Z)*iF + P0(:,Z)*d2iF + dP0(:,Z,jj)*diFi + dP0(:,Z,ii)*diFj;
d2KCP = d2Kij*P0(Z,:) + K0*d2Pij(Z,:) + dKi*dP0(Z,:,jj) + dKj*dP0(Z,:,ii) ;
dtmpi = dP0(:,:,ii) - dK0(:,:,ii)*P0(Z,:) - K0*dP0(Z,:,ii);
dtmpj = dP0(:,:,jj) - dK0(:,:,jj)*P0(Z,:) - K0*dP0(Z,:,jj);
d2tmp = d2Pij - d2KCP;
d2AtmpA = d2Aij*tmp*A' + A*d2tmp*A' + A*tmp*d2Aij' + dAi*dtmpj*A' + dAj*dtmpi*A' + A*dtmpj*dAi' + A*dtmpi*dAj' + dAi*tmp*dAj' + dAj*tmp*dAi';
d2K(:,:,ii,jj) = d2Kij; %#ok<NASGU>
d2P1(:,:,ii,jj) = d2AtmpA + d2Omij; %#ok<*NASGU>
d2S(:,:,ii,jj) = d2Fij;
% d2iS(:,:,ii,jj) = d2iF;
end
end
% end of computeD2Kalman
function [d2K,d2S,d2P1] = computeD2KalmanZ(A,dA,d2A,d2Om,P0,dP0,d2P0,DH,Z,iF,K0,dK0);
% computes the second derivatives of the Kalman matrices
% note: A=T in main func.
k = size(dA,3);
tmp = P0-K0*Z*P0(:,:);
[ns,no] = size(K0);
% CPC = C*P0*C'; CPC = .5*(CPC+CPC');iF = inv(CPC);
% APC = A*P0*C';
% APA = A*P0*A';
d2K = zeros(ns,no,k,k);
d2S = zeros(no,no,k,k);
d2P1 = zeros(ns,ns,k,k);
for ii = 1:k
dAi = dA(:,:,ii);
dFi = Z*dP0(:,:,ii)*Z;
d2Omi = d2Om(:,:,ii);
diFi = -iF*dFi*iF;
dKi = dK0(:,:,ii);
for jj = 1:k
dAj = dA(:,:,jj);
dFj = Z*dP0(:,:,jj)*Z;
d2Omj = d2Om(:,:,jj);
dFj = Z*dP0(:,:,jj)*Z;
diFj = -iF*dFj*iF;
dKj = dK0(:,:,jj);
d2Aij = d2A(:,:,jj,ii);
d2Pij = d2P0(:,:,jj,ii);
d2Omij = d2Om(:,:,jj,ii);
% second order
d2Fij = Z*d2Pij(:,:)*Z ;
% d2APC = d2Aij*P0*C' + A*d2Pij*C' + A*P0*d2Cij' + dAi*dPj*C' + dAj*dPi*C' + A*dPj*dCi' + A*dPi*dCj' + dAi*P0*dCj' + dAj*P0*dCi';
d2APC = d2Pij(:,:)*Z;
d2iF = -diFi*dFj*iF -iF*d2Fij*iF -iF*dFj*diFi;
d2Kij= d2Pij(:,:)*Z*iF + P0(:,:)*Z*d2iF + dP0(:,:,jj)*Z*diFi + dP0(:,:,ii)*Z*diFj;
d2KCP = d2Kij*Z*P0(:,:) + K0*Z*d2Pij(:,:) + dKi*Z*dP0(:,:,jj) + dKj*Z*dP0(:,:,ii) ;
dtmpi = dP0(:,:,ii) - dK0(:,:,ii)*Z*P0(:,:) - K0*Z*dP0(:,:,ii);
dtmpj = dP0(:,:,jj) - dK0(:,:,jj)*Z*P0(:,:) - K0*Z*dP0(:,:,jj);
d2tmp = d2Pij - d2KCP;
d2AtmpA = d2Aij*tmp*A' + A*d2tmp*A' + A*tmp*d2Aij' + dAi*dtmpj*A' + dAj*dtmpi*A' + A*dtmpj*dAi' + A*dtmpi*dAj' + dAi*tmp*dAj' + dAj*tmp*dAi';
d2K(:,:,ii,jj) = d2Kij; %#ok<NASGU>
d2P1(:,:,ii,jj) = d2AtmpA + d2Omij; %#ok<*NASGU>
d2S(:,:,ii,jj) = d2Fij;
% d2iS(:,:,ii,jj) = d2iF;
end
end
% end of computeD2KalmanZ
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