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