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function dr = dyn_second_order_solver(jacobia,hessian,dr,M_,threads_ABC,threads_BC)
%@info:
%! @deftypefn {Function File} {@var{dr} =} dyn_second_order_solver (@var{jacobia},@var{hessian},@var{dr},@var{M_},@var{threads_ABC},@var{threads_BC})
%! @anchor{dyn_first_order_solver}
%! @sp 1
%! Computes the first order reduced form of the DSGE model
%! @sp 2
%! @strong{Inputs}
%! @sp 1
%! @table @ @var
%! @item jacobia
%! Matrix containing the Jacobian of the model
%! @item hessian
%! Matrix containing the second order derivatives of the model
%! @item dr
%! Matlab's structure describing the reduced form solution of the model.
%! @item M_
%! Matlab's structure describing the model (initialized by @code{dynare}).
%! @item threads_ABC
%! Integer controlling number of threads in A_times_B_kronecker_C
%! @item threads_BC
%! Integer controlling number of threads in sparse_hessian_times_B_kronecker_C
%! @end table
%! @sp 2
%! @strong{Outputs}
%! @sp 1
%! @table @ @var
%! @item dr
%! Matlab's structure describing the reduced form solution of the model.
%! @end table
%! @end deftypefn
%@eod:
% Copyright (C) 2001-2011 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/>.
dr.ghxx = [];
dr.ghuu = [];
dr.ghxu = [];
dr.ghs2 = [];
Gy = dr.Gy;
kstate = dr.kstate;
kad = dr.kad;
kae = dr.kae;
nstatic = dr.nstatic;
nfwrd = dr.nfwrd;
npred = dr.npred;
nboth = dr.nboth;
nyf = nfwrd+nboth;
order_var = dr.order_var;
nd = size(kstate,1);
lead_lag_incidence = M_.lead_lag_incidence;
np = nd - nyf;
n2 = np + 1;
n3 = nyf;
n4 = n3 + 1;
k1 = nonzeros(lead_lag_incidence(:,order_var)');
kk = [k1; length(k1)+(1:M_.exo_nbr+M_.exo_det_nbr)'];
nk = size(kk,1);
kk1 = reshape([1:nk^2],nk,nk);
kk1 = kk1(kk,kk);
% reordering second order derivatives
hessian = hessian(:,kk1(:));
zx = zeros(np,np);
zu=zeros(np,M_.exo_nbr);
zx(1:np,:)=eye(np);
k0 = [1:M_.endo_nbr];
gx1 = dr.ghx;
hu = dr.ghu(nstatic+[1:npred],:);
k0 = find(lead_lag_incidence(M_.maximum_endo_lag+1,order_var)');
zx = [zx; gx1(k0,:)];
zu = [zu; dr.ghu(k0,:)];
k1 = find(lead_lag_incidence(M_.maximum_endo_lag+2,order_var)');
zu = [zu; gx1(k1,:)*hu];
zx = [zx; gx1(k1,:)*Gy];
zx=[zx; zeros(M_.exo_nbr,np);zeros(M_.exo_det_nbr,np)];
zu=[zu; eye(M_.exo_nbr);zeros(M_.exo_det_nbr,M_.exo_nbr)];
[nrzx,nczx] = size(zx);
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zx,threads_BC);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
rhs = -rhs;
%lhs
n = M_.endo_nbr+sum(kstate(:,2) > M_.maximum_endo_lag+1 & kstate(:,2) < M_.maximum_endo_lag+M_.maximum_endo_lead+1);
A = zeros(M_.endo_nbr,M_.endo_nbr);
B = zeros(M_.endo_nbr,M_.endo_nbr);
A(:,k0) = jacobia(:,nonzeros(lead_lag_incidence(M_.maximum_endo_lag+1,order_var)));
% variables with the highest lead
k1 = find(kstate(:,2) == M_.maximum_endo_lag+2);
% Jacobian with respect to the variables with the highest lead
fyp = jacobia(:,kstate(k1,3)+nnz(M_.lead_lag_incidence(M_.maximum_endo_lag+1,:)));
B(:,nstatic+npred-dr.nboth+1:end) = fyp;
offset = M_.endo_nbr;
gx1 = dr.ghx;
[junk,k1,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+M_.maximum_endo_lead+1,order_var));
A(1:M_.endo_nbr,nstatic+1:nstatic+npred)=...
A(1:M_.endo_nbr,nstatic+[1:npred])+fyp*gx1(k1,1:npred);
C = Gy;
D = [rhs; zeros(n-M_.endo_nbr,size(rhs,2))];
[err, dr.ghxx] = gensylv(2,A,B,C,D);
mexErrCheck('gensylv', err);
%ghxu
%rhs
hu = dr.ghu(nstatic+1:nstatic+npred,:);
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zx,zu,threads_BC);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
hu1 = [hu;zeros(np-npred,M_.exo_nbr)];
[nrhx,nchx] = size(Gy);
[nrhu1,nchu1] = size(hu1);
[abcOut,err] = A_times_B_kronecker_C(dr.ghxx,Gy,hu1,threads_ABC);
mexErrCheck('A_times_B_kronecker_C', err);
B1 = B*abcOut;
rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1;
%lhs
dr.ghxu = A\rhs;
%ghuu
%rhs
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zu,threads_BC);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
[B1, err] = A_times_B_kronecker_C(B*dr.ghxx,hu1,threads_ABC);
mexErrCheck('A_times_B_kronecker_C', err);
rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1;
%lhs
dr.ghuu = A\rhs;
dr.ghxx = dr.ghxx(1:M_.endo_nbr,:);
dr.ghxu = dr.ghxu(1:M_.endo_nbr,:);
rdr.ghuu = dr.ghuu(1:M_.endo_nbr,:);
% dr.ghs2
% derivatives of F with respect to forward variables
% reordering predetermined variables in diminishing lag order
O1 = zeros(M_.endo_nbr,nstatic);
O2 = zeros(M_.endo_nbr,M_.endo_nbr-nstatic-npred);
LHS = zeros(M_.endo_nbr,M_.endo_nbr);
LHS(:,k0) = jacobia(:,nonzeros(lead_lag_incidence(M_.maximum_endo_lag+1,order_var)));
RHS = zeros(M_.endo_nbr,M_.exo_nbr^2);
kk = find(kstate(:,2) == M_.maximum_endo_lag+2);
gu = dr.ghu;
guu = dr.ghuu;
Gu = [dr.ghu(nstatic+[1:npred],:); zeros(np-npred,M_.exo_nbr)];
Guu = [dr.ghuu(nstatic+[1:npred],:); zeros(np-npred,M_.exo_nbr*M_.exo_nbr)];
E = eye(M_.endo_nbr);
kh = reshape([1:nk^2],nk,nk);
kp = sum(kstate(:,2) <= M_.maximum_endo_lag+1);
E1 = [eye(npred); zeros(kp-npred,npred)];
H = E1;
hxx = dr.ghxx(nstatic+[1:npred],:);
[junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+2,order_var));
k3 = nnz(M_.lead_lag_incidence(1:M_.maximum_endo_lag+1,:))+(1:dr.nsfwrd)';
[B1, err] = sparse_hessian_times_B_kronecker_C(hessian(:,kh(k3,k3)),gu(k2a,:),threads_BC);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
RHS = RHS + jacobia(:,k2)*guu(k2a,:)+B1;
% LHS
LHS = LHS + jacobia(:,k2)*(E(k2a,:)+[O1(k2a,:) dr.ghx(k2a,:)*H O2(k2a,:)]);
RHS = RHS*M_.Sigma_e(:);
dr.fuu = RHS;
%RHS = -RHS-dr.fbias;
RHS = -RHS;
dr.ghs2 = LHS\RHS;
% deterministic exogenous variables
if M_.exo_det_nbr > 0
hud = dr.ghud{1}(nstatic+1:nstatic+npred,:);
zud=[zeros(np,M_.exo_det_nbr);dr.ghud{1};gx(:,1:npred)*hud;zeros(M_.exo_nbr,M_.exo_det_nbr);eye(M_.exo_det_nbr)];
R1 = hessian*kron(zx,zud);
dr.ghxud = cell(M_.exo_det_length,1);
kf = [M_.endo_nbr-nyf+1:M_.endo_nbr];
kp = nstatic+[1:npred];
dr.ghxud{1} = -M1*(R1+f1*dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{1}(kp,:)));
Eud = eye(M_.exo_det_nbr);
for i = 2:M_.exo_det_length
hudi = dr.ghud{i}(kp,:);
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zx,zudi);
dr.ghxud{i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(Gy,Eud)+dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{i}(kp,:)))-M1*R2;
end
R1 = hessian*kron(zu,zud);
dr.ghudud = cell(M_.exo_det_length,1);
kf = [M_.endo_nbr-nyf+1:M_.endo_nbr];
dr.ghuud{1} = -M1*(R1+f1*dr.ghxx(kf,:)*kron(dr.ghu(kp,:),dr.ghud{1}(kp,:)));
Eud = eye(M_.exo_det_nbr);
for i = 2:M_.exo_det_length
hudi = dr.ghud{i}(kp,:);
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zu,zudi);
dr.ghuud{i} = -M2*dr.ghxud{i-1}(kf,:)*kron(hu,Eud)-M1*R2;
end
R1 = hessian*kron(zud,zud);
dr.ghudud = cell(M_.exo_det_length,M_.exo_det_length);
dr.ghudud{1,1} = -M1*R1-M2*dr.ghxx(kf,:)*kron(hud,hud);
for i = 2:M_.exo_det_length
hudi = dr.ghud{i}(nstatic+1:nstatic+npred,:);
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi+dr.ghud{i-1}(kf,:);zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zudi,zudi);
dr.ghudud{i,i} = -M2*(dr.ghudud{i-1,i-1}(kf,:)+...
2*dr.ghxud{i-1}(kf,:)*kron(hudi,Eud) ...
+dr.ghxx(kf,:)*kron(hudi,hudi))-M1*R2;
R2 = hessian*kron(zud,zudi);
dr.ghudud{1,i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(hud,Eud)+...
dr.ghxx(kf,:)*kron(hud,hudi))...
-M1*R2;
for j=2:i-1
hudj = dr.ghud{j}(kp,:);
zudj=[zeros(np,M_.exo_det_nbr);dr.ghud{j};gx(:,1:npred)*hudj;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zudj,zudi);
dr.ghudud{j,i} = -M2*(dr.ghudud{j-1,i-1}(kf,:)+dr.ghxud{j-1}(kf,:)* ...
kron(hudi,Eud)+dr.ghxud{i-1}(kf,:)* ...
kron(hudj,Eud)+dr.ghxx(kf,:)*kron(hudj,hudi))-M1*R2;
end
end
end
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