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dynare/matlab/pruned_state_space_system.m

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Finished identification order=1|2|3 Note that I still need to do a code clean up (provide some licenses for functions from other people) and to double check order=3. There is also much room for speed and memory improvement, but the code works fine for now. I will also provide more information to the merge request soon about the detailed changes for future reference.
2019-12-20 00:40:00 +01:00
function dr = pruned_state_space_system(M, options, dr)
% This can be set up into the following ABCD representation:
% z = c + A*z(-1) + B*xi
% y = ybar + d + C*z(-1) + D*xi
% Denote
% hx = dr.ghx(indx,:); hu = dr.ghu(indx,:);
% hxx = dr.ghxx(indx,:); hxu = dr.ghxu(indx,:); huu = dr.ghuu(indx,:); hs2 = dr.ghs2(indx,:);
% gx = dr.ghx(indy,:); gu = dr.ghu(indy,:);
% gxx = dr.ghxx(indy,:); gxu = dr.ghxu(indy,:); guu = dr.ghuu(indy,:); gs2 = dr.ghs2(indy,:);
% hxss = dr.ghxss(indx,:); huss = dr.ghuss(indx,:); hxxx = dr.ghxxx(indx,:); huuu = dr.ghuuu(indx,:); hxxu = dr.ghxxu(indx,:); hxuu = dr.ghxxu(indx,:);
% gxss = dr.ghxss(indy,:); guss = dr.ghuss(indy,:); gxxx = dr.ghxxx(indy,:); guuu = dr.ghuuu(indy,:); gxxu = dr.ghxxu(indy,:); gxuu = dr.ghxxu(indy,:);
% Law of motion for first-order effects states xf and selected endogenous first-order effects variables yf:
% xf = hx*xf(-1) + hu*u
% yf = gx*xf(-1) + gu*u
% Law of motion for second-order effects states xs and selected endogenous second-order effects variables ys:
% xs = hx*xs(-1) + 1/2*hxx*kron(xf(-1),xf(-1)) + 1/2*huu*kron(u,u) + hxu*kron(xf(-1),u) + 1/2*hs2
% ys = gx*xs(-1) + 1/2*gxx*kron(xf(-1),xf(-1)) + 1/2*guu*kron(u,u) + gxu*kron(xf(-1),u) + 1/2*gs2
% Law of motion for third-order effects states xrd and selected endogenous second-order effects variables yrd:
% xrd = hx*xrd(-1) + hxx*kron(xf(-1),xs(-1)) + hxu*kron(xs(-1),u) + 3/6*hxss*xf(-1) + 3/6*huss*u + 1/6*hxxx*kron(xf(-1),kron(xf(-1),xf(-1))) + 1/6*huuu*kron(u,kron(u,u)) + 3/6*hxxu*kron(xf(-1),kron(xf(-1),u)) + 3/6*hxuu*kron(xf(-1),kron(u,u))
% yrd = gx*xrd(-1) + gxx*kron(xf(-1),xs(-1)) + gxu*kron(xs(-1),u) + 3/6*gxss*xf(-1) + 3/6*guss*u + 1/6*gxxx*kron(xf(-1),kron(xf(-1),xf(-1))) + 1/6*guuu*kron(u,kron(u,u)) + 3/6*gxxu*kron(xf(-1),kron(xf(-1),u)) + 3/6*gxuu*kron(xf(-1),kron(u,u))
% Selected variables: y = ybar + yf + ys
% Pruned state vector: z = [xf; xs; kron(xf,xf)]
% Pruned innovation vector: xi = [u; kron(u,u)-vec(Sigma_u); kron(xf(-1),u); kron(u,xf(-1))]
% Second moments of xi
% Sig_xi = E[u*u' , u*kron(u,u)' , zeros(u_nbr,x_nbr*u_nbr), zeros(u_nbr,u_nbr*x_nbr);
% kron(u,u)*u', kron(u,u)*kron(u,u)'-Sig_u(:)*Sig_u(:)', zeros(u_nbr^2,x_nbr*u_nbr), zeros(u_nbr^2,u_nbr*x_nbr);
% zeros(x_nbr*u_nbr,u_nbr), zeros(x_nbr*u_nbr,u_nbr^2), kron(xf(-1),u)*kron(xf(-1),u)', kron(xf(-1),u)*kron(u,xf(-1))';
% zeros(u_nbr*x_nbr,u_nbr), zeros(u_nbr*x_nbr,u_nbr^2), kron(u,xf(-1))*kron(xf(-1),u)', kron(u,xf(-1))*kron(u,xf(-1))';]
% That is, we only need to compute:
% - Sig_xi_11 = E[u*u']=Sig_u
% - Sig_xi_21 = E[kron(u,u)*u']=0 for symmetric distributions
% - Sig_xi_12 = transpose(Sig_xi_21)= 0 for symmetric distributions
% - Sig_xi_22 = E[kron(u,u)*kron(u,u)']-Sig_u(:)*Sig_u(:)' which requires the fourth-order product moments of u
% - Sig_xi_34 = E[kron(xf(-1),u)*kron(xf(-1),u)'] which requires the second-order moments of xf and second-order moments of u
% - Sig_xi_43 = transpose(Sig_xi_34)
% - Sig_xi_33 =
% - Sig_xi_44 =
%% Auxiliary indices, numbers and flags
order = options.order;
indx = [M.nstatic+(1:M.nspred) M.endo_nbr+(1:size(dr.ghx,2)-M.nspred)]';
indy = (1:M.endo_nbr)'; %by default select all variables
compute_derivs = false;
if isfield(options,'options_ident')
compute_derivs = true;
stderrparam_nbr = length(options.options_ident.indpstderr);
corrparam_nbr = size(options.options_ident.indpcorr,1);
modparam_nbr = length(options.options_ident.indpmodel);
totparam_nbr = stderrparam_nbr+corrparam_nbr+modparam_nbr;
indy = options.options_ident.indvobs;
end
Yss = dr.ys(dr.order_var); if compute_derivs; dYss = dr.derivs.dYss; end
E_uu = M.Sigma_e; if compute_derivs; dE_uu = dr.derivs.dSigma_e; end
Correlation_matrix = M.Correlation_matrix; if compute_derivs; dCorrelation_matrix = dr.derivs.dCorrelation_matrix; end
ghx = dr.ghx; if compute_derivs; dghx = dr.derivs.dghx; end
ghu = dr.ghu; if compute_derivs; dghu = dr.derivs.dghu; end
Om = ghu*E_uu*transpose(ghu); if compute_derivs; dOm = dr.derivs.dOm; end
y_nbr = length(indy);
x_nbr = length(indx);
u_nbr = M.exo_nbr;
% indices for extended state vector z and extended shock vector e
id_z1_xf = (1:x_nbr);
id_e1_u = (1:u_nbr);
if options.order > 1
id_z2_xs = id_z1_xf(end) + (1:x_nbr);
id_z3_xf_xf = id_z2_xs(end) + (1:x_nbr*x_nbr);
id_e2_u_u = id_e1_u(end) + (1:u_nbr*u_nbr);
id_e3_xf_u = id_e2_u_u(end) + (1:x_nbr*u_nbr);
id_e4_u_xf = id_e3_xf_u(end) + (1:u_nbr*x_nbr);
ghxx = dr.ghxx; if compute_derivs; dghxx = dr.derivs.dghxx; end
ghxu = dr.ghxu; if compute_derivs; dghxu = dr.derivs.dghxu; end
ghuu = dr.ghuu; if compute_derivs; dghuu = dr.derivs.dghuu; end
ghs2 = dr.ghs2; if compute_derivs; dghs2 = dr.derivs.dghs2; end
end
if options.order > 2
id_z4_xrd = id_z3_xf_xf(end) + (1:x_nbr);
id_z5_xf_xs = id_z4_xrd(end) + (1:x_nbr*x_nbr);
id_z6_xf_xf_xf = id_z5_xf_xs(end) + (1:x_nbr*x_nbr*x_nbr);
id_e5_xs_u = id_e4_u_xf(end) + (1:x_nbr*u_nbr);
id_e6_u_xs = id_e5_xs_u(end) + (1:u_nbr*x_nbr);
id_e7_xf_xf_u = id_e6_u_xs(end) + (1:x_nbr*x_nbr*u_nbr);
id_e8_xf_u_xf = id_e7_xf_xf_u(end) + (1:x_nbr*u_nbr*x_nbr);
id_e9_u_xf_xf = id_e8_xf_u_xf(end) + (1:u_nbr*x_nbr*x_nbr);
id_e10_xf_u_u = id_e9_u_xf_xf(end) + (1:x_nbr*u_nbr*u_nbr);
id_e11_u_xf_u = id_e10_xf_u_u(end) + (1:u_nbr*x_nbr*u_nbr);
id_e12_u_u_xf = id_e11_u_xf_u(end) + (1:u_nbr*u_nbr*x_nbr);
id_e13_u_u_u = id_e12_u_u_xf(end) + (1:u_nbr*u_nbr*u_nbr);
ghxxx = dr.ghxxx; if compute_derivs; dghxxx = dr.derivs.dghxxx; end
ghxxu = dr.ghxxu; if compute_derivs; dghxxu = dr.derivs.dghxxu; end
ghxuu = dr.ghxuu; if compute_derivs; dghxuu = dr.derivs.dghxuu; end
ghuuu = dr.ghuuu; if compute_derivs; dghuuu = dr.derivs.dghuuu; end
ghxss = dr.ghxss; if compute_derivs; dghxss = dr.derivs.dghxss; end
ghuss = dr.ghuss; if compute_derivs; dghuss = dr.derivs.dghuss; end
end
%% First-order
z_nbr = x_nbr;
e_nbr = M.exo_nbr;
A = ghx(indx,:);
B = ghu(indx,:);
C = ghx(indy,:);
D = ghu(indy,:);
c = zeros(x_nbr,1);
d = zeros(y_nbr,1);
Varinov = E_uu;
E_z = zeros(x_nbr,1); %this is E_xf
if compute_derivs == 1
dA = dghx(indx,:,:);
dB = dghu(indx,:,:);
dC = dghx(indy,:,:);
dD = dghu(indy,:,:);
dc = zeros(x_nbr,totparam_nbr);
dd = zeros(y_nbr,totparam_nbr);
dVarinov = dE_uu;
dE_z = zeros(x_nbr,totparam_nbr);
end
if order > 1
% Some common and useful objects for order > 1
% Compute E[xf*xf']
E_xfxf = lyapunov_symm(ghx(indx,:), Om(indx,indx), options.lyapunov_fixed_point_tol, options.qz_criterium, options.lyapunov_complex_threshold, 1, options.debug); %use 1 to initialize persistent variables
K_ux = commutation(u_nbr,x_nbr); %commutation matrix
K_xu = commutation(x_nbr,u_nbr); %commutation matrix
QPu = quadruplication(u_nbr); %quadruplication matrix as in Meijer (2005), similar to Magnus-Neudecker definition of duplication matrix (i.e. dyn_vec) but for fourth-order product moments
%Compute unique product moments of E[kron(u*u',uu')] = E[kron(u,u)*kron(u,u)']
COMBOS4 = flipud(allVL1(u_nbr, 4)); %all possible combinations of powers that sum up to four for fourth-order product moments of u
E_u_u_u_u = zeros(u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4,1); %only unique entries of E[kron(u,kron(u,kron(u,u)))]
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u = zeros(u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4,stderrparam_nbr+corrparam_nbr);
end
for j = 1:size(COMBOS4,1)
E_u_u_u_u(j) = prodmom(E_uu, 1:u_nbr, COMBOS4(j,:));
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u(j,1:(stderrparam_nbr+corrparam_nbr)) = prodmom_deriv(E_uu, 1:u_nbr, COMBOS4(j,:), dE_uu(:,:,1:(stderrparam_nbr+corrparam_nbr)), dCorrelation_matrix(:,:,1:(stderrparam_nbr+corrparam_nbr)));
end
end
E_u_u_u_u = QPu*E_u_u_u_u; %add duplicate product moments, i.e. this is E[kron(u,kron(u,kron(u,u)))]
E_uu_uu = reshape(E_u_u_u_u,u_nbr^2,u_nbr^2); %E[kron(u*u',uu')] = E[kron(u,u)*kron(u,u)']
% E[kron((xf*xf'),(u*u'))]
E_xfxf_E_uu = kron(E_xfxf,E_uu);
if compute_derivs
dE_xfxf = zeros(size(E_xfxf,1),size(E_xfxf,2),totparam_nbr);
dE_xfxf_E_uu = zeros(size(E_xfxf_E_uu,1),size(E_xfxf_E_uu,2),totparam_nbr);
for jp = 1:totparam_nbr
if jp <= (stderrparam_nbr+corrparam_nbr)
%Jacobian of E_xfxf wrt exogenous paramters: dE_xfxf(:,:,jp)-hx*dE_xfxf(:,:,jp)*hx'=dOm(:,:,jp), because dhx is zero by construction for stderr and corr parameters
dE_xfxf(:,:,jp) = lyapunov_symm(ghx(indx,:), dOm(indx,indx,jp),options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,2,options.debug);
else
%Jacobian of E_xfxf wrt model parameters: dE_xfxf(:,:,jp) - hx*dE_xfxf(:,:,jp)*hx' = d(hu*Sig_u*hu')(:,:,jp) + dhx(:,:,jp)*E_xfxf*hx'+ hx*E_xfxf*dhx(:,:,jp)'
dE_xfxf(:,:,jp) = lyapunov_symm(ghx(indx,:), dghx(indx,:,jp)*E_xfxf*ghx(indx,:)'+ghx(indx,:)*E_xfxf*dghx(indx,:,jp)'+dOm(indx,indx,jp),options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,2,options.debug);
%here method=2 is used to spare a lot of computing time while not repeating Schur every time
end
dE_xfxf_E_uu(:,:,jp) = kron(dE_xfxf(:,:,jp),E_uu) + kron(E_xfxf,dE_uu(:,:,jp));
end
end
% Second-order pruned state space system
z_nbr = x_nbr+x_nbr+x_nbr*x_nbr;
e_nbr = u_nbr+u_nbr*u_nbr+x_nbr*u_nbr+u_nbr*x_nbr;
A = zeros(z_nbr, z_nbr);
A(id_z1_xf , id_z1_xf ) = ghx(indx,:);
A(id_z2_xs , id_z2_xs ) = ghx(indx,:);
A(id_z2_xs , id_z3_xf_xf) = 0.5*ghxx(indx,:);
A(id_z3_xf_xf , id_z3_xf_xf) = kron(ghx(indx,:),ghx(indx,:));
B = zeros(z_nbr, e_nbr);
B(id_z1_xf , id_e1_u ) = ghu(indx,:);
B(id_z2_xs , id_e2_u_u ) = 0.5*ghuu(indx,:);
B(id_z2_xs , id_e3_xf_u) = ghxu(indx,:);
B(id_z3_xf_xf , id_e2_u_u ) = kron(ghu(indx,:),ghu(indx,:));
B(id_z3_xf_xf , id_e3_xf_u) = kron(ghx(indx,:),ghu(indx,:));
B(id_z3_xf_xf , id_e4_u_xf) = kron(ghu(indx,:),ghx(indx,:));
C = zeros(y_nbr, z_nbr);
C(1:y_nbr , id_z1_xf ) = ghx(indy,:);
C(1:y_nbr , id_z2_xs ) = ghx(indy,:);
C(1:y_nbr , id_z3_xf_xf) = 0.5*ghxx(indy,:);
D = zeros(y_nbr, e_nbr);
D(1:y_nbr , id_e1_u ) = ghu(indy,:);
D(1:y_nbr , id_e2_u_u ) = 0.5*ghuu(indy,:);
D(1:y_nbr , id_e3_xf_u) = ghxu(indy,:);
c = zeros(z_nbr, 1);
c(id_z2_xs , 1) = 0.5*ghs2(indx,:) + 0.5*ghuu(indx,:)*E_uu(:);
c(id_z3_xf_xf , 1) = kron(ghu(indx,:),ghu(indx,:))*E_uu(:);
d = zeros(y_nbr, 1);
d(1:y_nbr, 1) = 0.5*ghs2(indy,:) + 0.5*ghuu(indy,:)*E_uu(:);
Varinov = zeros(e_nbr,e_nbr);
Varinov(id_e1_u , id_e1_u) = E_uu;
Varinov(id_e2_u_u , id_e2_u_u) = E_uu_uu-E_uu(:)*transpose(E_uu(:));
Varinov(id_e3_xf_u , id_e3_xf_u) = E_xfxf_E_uu;
Varinov(id_e4_u_xf , id_e3_xf_u) = K_ux*E_xfxf_E_uu;
Varinov(id_e3_xf_u , id_e4_u_xf) = transpose(Varinov(id_e4_u_xf,id_e3_xf_u));
Varinov(id_e4_u_xf , id_e4_u_xf) = K_ux*E_xfxf_E_uu*transpose(K_ux);
I_hx = speye(x_nbr)-ghx(indx,:);
I_hxinv = I_hx\speye(x_nbr);
E_xs = I_hxinv*(0.5*ghxx(indx,:)*E_xfxf(:) + c(x_nbr+(1:x_nbr),1));
E_z = [zeros(x_nbr,1); E_xs; E_xfxf(:)];
if compute_derivs
dA = zeros(size(A,1),size(A,2),totparam_nbr);
dB = zeros(size(B,1),size(B,2),totparam_nbr);
dC = zeros(size(C,1),size(C,2),totparam_nbr);
dD = zeros(size(D,1),size(D,2),totparam_nbr);
dc = zeros(size(c,1),totparam_nbr);
dd = zeros(size(d,1),totparam_nbr);
dVarinov = zeros(size(Varinov,1),size(Varinov,2),totparam_nbr);
dE_xs = zeros(size(E_xs,1),size(E_xs,2),totparam_nbr);
dE_z = zeros(size(E_z,1),size(E_z,2),totparam_nbr);
for jp = 1:totparam_nbr
dA(id_z1_xf , id_z1_xf , jp) = dghx(indx,:,jp);
dA(id_z2_xs , id_z2_xs , jp) = dghx(indx,:,jp);
dA(id_z2_xs , id_z3_xf_xf , jp) = 0.5*dghxx(indx,:,jp);
dA(id_z3_xf_xf , id_z3_xf_xf , jp) = kron(dghx(indx,:,jp),ghx(indx,:)) + kron(ghx(indx,:),dghx(indx,:,jp));
dB(id_z1_xf , id_e1_u , jp) = dghu(indx,:,jp);
dB(id_z2_xs , id_e2_u_u , jp) = 0.5*dghuu(indx,:,jp);
dB(id_z2_xs , id_e3_xf_u , jp) = dghxu(indx,:,jp);
dB(id_z3_xf_xf , id_e2_u_u , jp) = kron(dghu(indx,:,jp),ghu(indx,:)) + kron(ghu(indx,:),dghu(indx,:,jp));
dB(id_z3_xf_xf , id_e3_xf_u , jp) = kron(dghx(indx,:,jp),ghu(indx,:)) + kron(ghx(indx,:),dghu(indx,:,jp));
dB(id_z3_xf_xf , id_e4_u_xf , jp) = kron(dghu(indx,:,jp),ghx(indx,:)) + kron(ghu(indx,:),dghx(indx,:,jp));
dC(1:y_nbr , id_z1_xf , jp) = dghx(indy,:,jp);
dC(1:y_nbr , id_z2_xs , jp) = dghx(indy,:,jp);
dC(1:y_nbr , id_z3_xf_xf , jp) = 0.5*dghxx(indy,:,jp);
dD(1:y_nbr , id_e1_u , jp) = dghu(indy,:,jp);
dD(1:y_nbr , id_e2_u_u , jp) = 0.5*dghuu(indy,:,jp);
dD(1:y_nbr , id_e3_xf_u , jp) = dghxu(indy,:,jp);
dc(id_z2_xs , jp) = 0.5*dghs2(indx,jp) + 0.5*(dghuu(indx,:,jp)*E_uu(:) + ghuu(indx,:)*vec(dE_uu(:,:,jp)));
dc(id_z3_xf_xf , jp) = (kron(dghu(indx,:,jp),ghu(indx,:)) + kron(ghu(indx,:),dghu(indx,:,jp)))*E_uu(:) + kron(ghu(indx,:),ghu(indx,:))*vec(dE_uu(:,:,jp));
dd(1:y_nbr , jp) = 0.5*dghs2(indy,jp) + 0.5*(dghuu(indy,:,jp)*E_uu(:) + ghuu(indy,:)*vec(dE_uu(:,:,jp)));
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e1_u , id_e1_u , jp) = dE_uu(:,:,jp);
dVarinov(id_e2_u_u , id_e2_u_u , jp) = reshape(QPu*dE_u_u_u_u(:,jp), u_nbr^2, u_nbr^2) - (reshape(dE_uu(:,:,jp),u_nbr^2,1)*transpose(E_uu(:)) + E_uu(:)*transpose(reshape(dE_uu(:,:,jp),u_nbr^2,1)));
end
dVarinov(id_e3_xf_u , id_e3_xf_u , jp) = dE_xfxf_E_uu(:,:,jp);
dVarinov(id_e4_u_xf , id_e3_xf_u , jp) = K_ux*dE_xfxf_E_uu(:,:,jp);
dVarinov(id_e3_xf_u , id_e4_u_xf , jp) = transpose(dVarinov(id_e4_u_xf,id_e3_xf_u,jp));
dVarinov(id_e4_u_xf , id_e4_u_xf , jp) = dVarinov(id_e4_u_xf , id_e3_xf_u , jp)*transpose(K_ux);
dE_xs(:,jp) = I_hxinv*( dghx(indx,:,jp)*E_xs + 1/2*(dghxx(indx,:,jp)*E_xfxf(:) + ghxx(indx,:)*vec(dE_xfxf(:,:,jp)) + dc(x_nbr+(1:x_nbr),jp)) );
dE_z(:,jp) = [zeros(x_nbr,1); dE_xs(:,jp); vec(dE_xfxf(:,:,jp))];
end
end
if order > 2
Q6Pu = Q6_plication(u_nbr); %quadruplication matrix as in Meijer (2005), similar to Magnus-Neudecker definition of duplication matrix (i.e. dyn_vec) but for fourth-order product moments
%Compute unique product moments of E[kron(u*u',uu')] = E[kron(u,u)*kron(u,u)']
COMBOS6 = flipud(allVL1(u_nbr, 6)); %all possible combinations of powers that sum up to four for fourth-order product moments of u
E_u_u_u_u_u_u = zeros(u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4*(u_nbr+4)/5*(u_nbr+5)/6,1); %only unique entries of E[kron(u,kron(u,kron(u,kron(u,kron(u,u)))))]
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u_u_u = zeros(size(E_u_u_u_u_u_u,1),stderrparam_nbr+corrparam_nbr);
end
for j = 1:size(COMBOS6,1)
E_u_u_u_u_u_u(j) = prodmom(E_uu, 1:u_nbr, COMBOS6(j,:));
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u_u_u(j,1:(stderrparam_nbr+corrparam_nbr)) = prodmom_deriv(E_uu, 1:u_nbr, COMBOS6(j,:), dE_uu(:,:,1:(stderrparam_nbr+corrparam_nbr)), dCorrelation_matrix(:,:,1:(stderrparam_nbr+corrparam_nbr)));
end
end
Var_z = lyapunov_symm(A, B*Varinov*transpose(B), options.lyapunov_fixed_point_tol, options.qz_criterium, options.lyapunov_complex_threshold, 1, options.debug);
%E_xfxf = Var_z(id_z1_xf,id_z1_xf); %this is E_xfxf, as E_xf=0, we already have that
E_xfxs = Var_z(id_z1_xf,id_z2_xs); %as E_xf=0
E_xf_xf_xf = vec(Var_z(id_z1_xf,id_z3_xf_xf)); %as E_xf=0, this is vec(E[xf*kron(xf,xf)'])
E_xsxf = Var_z(id_z2_xs,id_z1_xf); %as E_xf=0
E_xsxs = Var_z(id_z2_xs,id_z2_xs) + E_xs*transpose(E_xs);
E_xs_xf_xf = vec(Var_z(id_z2_xs,id_z3_xf_xf)+E_xs*E_xfxf(:)'); %this is vec(E[xs*kron(xf,xf)'])
%E_xf_xf_xf = vec(Var_z(id_z3_xf_xf,id_z1_xf));
E_xf_xf_xs = vec(Var_z(id_z3_xf_xf,id_z2_xs) + E_xfxf(:)*E_xs');
E_xf_xf_xf_xf = vec(Var_z(id_z3_xf_xf,id_z3_xf_xf) + E_xfxf(:)*E_xfxf(:)');
%E_xs_E_uu = kron(E_xs,E_uu);
E_xfxs_E_uu = kron(E_xfxs,E_uu);
z_nbr = x_nbr+x_nbr+x_nbr^2+x_nbr+x_nbr*x_nbr+x_nbr^3;
e_nbr = u_nbr+u_nbr^2+x_nbr*u_nbr+u_nbr*x_nbr+x_nbr*u_nbr+u_nbr*x_nbr+x_nbr*x_nbr*u_nbr+x_nbr*u_nbr*x_nbr+u_nbr*x_nbr*x_nbr+u_nbr*u_nbr*x_nbr+u_nbr*x_nbr*u_nbr+x_nbr*u_nbr*u_nbr+u_nbr*u_nbr*u_nbr;
hx_hx = kron(ghx(indx,:),ghx(indx,:));
hu_hu = kron(ghu(indx,:),ghu(indx,:));
hx_hu = kron(ghx(indx,:),ghu(indx,:));
hu_hx = kron(ghu(indx,:),ghx(indx,:));
if compute_derivs
dVar_z = zeros(size(Var_z,1),size(Var_z,2),totparam_nbr);
dE_xfxs = zeros(size(E_xfxs,1),size(E_xfxs,2),totparam_nbr);
dE_xf_xf_xf = zeros(size(E_xf_xf_xf,1),totparam_nbr);
dE_xsxf = zeros(size(E_xsxf,1),size(E_xsxf,2),totparam_nbr);
dE_xsxs = zeros(size(E_xsxs,1),size(E_xsxs,2),totparam_nbr);
dE_xs_xf_xf = zeros(size(E_xs_xf_xf,1),totparam_nbr);
dE_xf_xf_xs = zeros(size(E_xf_xf_xs,1),totparam_nbr);
dE_xf_xf_xf_xf = zeros(size(E_xf_xf_xf_xf,1),totparam_nbr);
dE_xfxs_E_uu = zeros(size(E_xfxs_E_uu,1),size(E_xfxs_E_uu,2),totparam_nbr);
dhx_hx = zeros(size(hx_hx,1),size(hx_hx,2),totparam_nbr);
dhu_hu = zeros(size(hu_hu,1),size(hu_hu,2),totparam_nbr);
dhx_hu = zeros(size(hx_hu,1),size(hx_hu,2),totparam_nbr);
dhu_hx = zeros(size(hu_hx,1),size(hu_hx,2),totparam_nbr);
for jp=1:totparam_nbr
dVar_z(:,:,jp) = lyapunov_symm(A, dB(:,:,jp)*Varinov*transpose(B) + B*dVarinov(:,:,jp)*transpose(B) +B*Varinov*transpose(dB(:,:,jp)) + dA(:,:,jp)*Var_z*A' + A*Var_z*transpose(dA(:,:,jp)),... ,...
options.lyapunov_fixed_point_tol, options.qz_criterium, options.lyapunov_complex_threshold, 2, options.debug); %2 is used as we use 1 above
dE_xfxs(:,:,jp) = dVar_z(id_z1_xf,id_z2_xs,jp);
dE_xf_xf_xf(:,jp) = vec(dVar_z(id_z1_xf,id_z3_xf_xf,jp));
dE_xsxf(:,:,jp) = dVar_z(id_z2_xs,id_z1_xf,jp);
dE_xsxs(:,:,jp) = dVar_z(id_z2_xs,id_z2_xs,jp) + dE_xs(:,jp)*transpose(E_xs) + E_xs*transpose(dE_xs(:,jp));
dE_xs_xf_xf(:,jp) = vec(dVar_z(id_z2_xs,id_z3_xf_xf,jp) + dE_xs(:,jp)*E_xfxf(:)' + E_xs*vec(dE_xfxf(:,:,jp))');
dE_xf_xf_xs(:,jp) = vec(dVar_z(id_z3_xf_xf,id_z2_xs,jp) + vec(dE_xfxf(:,:,jp))*E_xs' + E_xfxf(:)*dE_xs(:,jp)');
dE_xf_xf_xf_xf(:,jp) = vec(dVar_z(id_z3_xf_xf,id_z3_xf_xf,jp) + vec(dE_xfxf(:,:,jp))*E_xfxf(:)' + E_xfxf(:)*vec(dE_xfxf(:,:,jp))');
dE_xfxs_E_uu(:,:,jp) = kron(dE_xfxs(:,:,jp),E_uu) + kron(E_xfxs,dE_uu(:,:,jp));
dhx_hx(:,:,jp) = kron(dghx(indx,:,jp),ghx(indx,:)) + kron(ghx(indx,:),dghx(indx,:,jp));
dhu_hu(:,:,jp) = kron(dghu(indx,:,jp),ghu(indx,:)) + kron(ghu(indx,:),dghu(indx,:,jp));
dhx_hu(:,:,jp) = kron(dghx(indx,:,jp),ghu(indx,:)) + kron(ghx(indx,:),dghu(indx,:,jp));
dhu_hx(:,:,jp) = kron(dghu(indx,:,jp),ghx(indx,:)) + kron(ghu(indx,:),dghx(indx,:,jp));
end
end
A = zeros(z_nbr,z_nbr);
A(id_z1_xf , id_z1_xf ) = ghx(indx,:);
A(id_z2_xs , id_z2_xs ) = ghx(indx,:);
A(id_z2_xs , id_z3_xf_xf ) = 1/2*ghxx(indx,:);
A(id_z3_xf_xf , id_z3_xf_xf ) = hx_hx;
A(id_z4_xrd , id_z1_xf ) = 3/6*ghxss(indx,:);
A(id_z4_xrd , id_z4_xrd ) = ghx(indx,:);
A(id_z4_xrd , id_z5_xf_xs ) = ghxx(indx,:);
A(id_z4_xrd , id_z6_xf_xf_xf) = 1/6*ghxxx(indx,:);
A(id_z5_xf_xs , id_z1_xf ) = kron(ghx(indx,:),1/2*ghs2(indx,:));
A(id_z5_xf_xs , id_z5_xf_xs ) = hx_hx;
A(id_z5_xf_xs , id_z6_xf_xf_xf) = kron(ghx(indx,:),1/2*ghxx(indx,:));
A(id_z6_xf_xf_xf , id_z6_xf_xf_xf) = kron(ghx(indx,:),hx_hx);
B = zeros(z_nbr,e_nbr);
B(id_z1_xf , id_e1_u ) = ghu(indx,:);
B(id_z2_xs , id_e2_u_u ) = 1/2*ghuu(indx,:);
B(id_z2_xs , id_e3_xf_u ) = ghxu(indx,:);
B(id_z3_xf_xf , id_e2_u_u ) = hu_hu;
B(id_z3_xf_xf , id_e3_xf_u ) = hx_hu;
B(id_z3_xf_xf , id_e4_u_xf ) = hu_hx;
B(id_z4_xrd , id_e1_u ) = 3/6*ghuss(indx,:);
B(id_z4_xrd , id_e5_xs_u ) = ghxu(indx,:);
B(id_z4_xrd , id_e7_xf_xf_u) = 3/6*ghxxu(indx,:);
B(id_z4_xrd , id_e10_xf_u_u) = 3/6*ghxuu(indx,:);
B(id_z4_xrd , id_e13_u_u_u ) = 1/6*ghuuu(indx,:);
B(id_z5_xf_xs , id_e1_u ) = kron(ghu(indx,:),1/2*ghs2(indx,:));
B(id_z5_xf_xs , id_e6_u_xs ) = hu_hx;
B(id_z5_xf_xs , id_e7_xf_xf_u) = kron(ghx(indx,:),ghxu(indx,:));
B(id_z5_xf_xs , id_e9_u_xf_xf) = kron(ghu(indx,:),1/2*ghxx(indx,:));
B(id_z5_xf_xs , id_e10_xf_u_u) = kron(ghx(indx,:),1/2*ghuu(indx,:));
B(id_z5_xf_xs , id_e11_u_xf_u) = kron(ghu(indx,:),ghxu(indx,:));
B(id_z5_xf_xs , id_e13_u_u_u ) = kron(ghu(indx,:),1/2*ghuu(indx,:));
B(id_z6_xf_xf_xf , id_e7_xf_xf_u) = kron(hx_hx,ghu(indx,:));
B(id_z6_xf_xf_xf , id_e8_xf_u_xf) = kron(ghx(indx,:),hu_hx);
B(id_z6_xf_xf_xf , id_e9_u_xf_xf) = kron(ghu(indx,:),hx_hx);
B(id_z6_xf_xf_xf , id_e10_xf_u_u) = kron(hx_hu,ghu(indx,:));
B(id_z6_xf_xf_xf , id_e11_u_xf_u) = kron(ghu(indx,:),hx_hu);
B(id_z6_xf_xf_xf , id_e12_u_u_xf) = kron(hu_hu,ghx(indx,:));
B(id_z6_xf_xf_xf , id_e13_u_u_u ) = kron(ghu(indx,:),hu_hu);
C = zeros(y_nbr,z_nbr);
C(1:y_nbr , id_z1_xf ) = ghx(indy,:) + 1/2*ghxss(indy,:);
C(1:y_nbr , id_z2_xs ) = ghx(indy,:);
C(1:y_nbr , id_z3_xf_xf ) = 1/2*ghxx(indy,:);
C(1:y_nbr , id_z4_xrd ) = ghx(indy,:);
C(1:y_nbr , id_z5_xf_xs ) = ghxx(indy,:);
C(1:y_nbr , id_z6_xf_xf_xf) = 1/6*ghxxx(indy,:);
D = zeros(y_nbr,e_nbr);
D(1:y_nbr , id_e1_u ) = ghu(indy,:) + 1/2*ghuss(indy,:);
D(1:y_nbr , id_e2_u_u ) = 1/2*ghuu(indy,:);
D(1:y_nbr , id_e3_xf_u ) = ghxu(indy,:);
D(1:y_nbr , id_e5_xs_u ) = ghxu(indy,:);
D(1:y_nbr , id_e7_xf_xf_u) = 1/2*ghxxu(indy,:);
D(1:y_nbr , id_e10_xf_u_u) = 1/2*ghxuu(indy,:);
D(1:y_nbr , id_e13_u_u_u ) = 1/6*ghuuu(indy,:);
c = zeros(z_nbr,1);
c(id_z2_xs , 1) = 1/2*ghs2(indx,:) + 1/2*ghuu(indx,:)*E_uu(:);
c(id_z3_xf_xf , 1) = hu_hu*E_uu(:);
d = zeros(y_nbr,1);
d(1:y_nbr , 1) = 0.5*ghs2(indy,:) + 0.5*ghuu(indy,:)*E_uu(:);
Varinov = zeros(e_nbr,e_nbr);
Varinov(id_e1_u , id_e1_u ) = E_uu;
Varinov(id_e1_u , id_e5_xs_u ) = kron(E_xs',E_uu);
Varinov(id_e1_u , id_e6_u_xs ) = kron(E_uu,E_xs');
Varinov(id_e1_u , id_e7_xf_xf_u) = kron(E_xfxf(:)',E_uu);
Varinov(id_e1_u , id_e8_xf_u_xf) = kron(E_uu,E_xfxf(:)')*kron(K_xu,speye(x_nbr));
Varinov(id_e1_u , id_e9_u_xf_xf) = kron(E_uu,E_xfxf(:)');
Varinov(id_e1_u , id_e13_u_u_u ) = reshape(E_u_u_u_u,u_nbr,u_nbr^3);
Varinov(id_e2_u_u , id_e2_u_u ) = E_uu_uu-E_uu(:)*transpose(E_uu(:));
Varinov(id_e3_xf_u , id_e3_xf_u ) = E_xfxf_E_uu;
Varinov(id_e3_xf_u , id_e4_u_xf ) = E_xfxf_E_uu*K_ux';
Varinov(id_e3_xf_u , id_e5_xs_u ) = E_xfxs_E_uu;
Varinov(id_e3_xf_u , id_e6_u_xs ) = E_xfxs_E_uu*K_ux';
Varinov(id_e3_xf_u , id_e7_xf_xf_u) = kron(reshape(E_xf_xf_xf,x_nbr,x_nbr^2), E_uu);
Varinov(id_e3_xf_u , id_e8_xf_u_xf) = kron(reshape(E_xf_xf_xf,x_nbr,x_nbr^2), E_uu)*kron(speye(x_nbr),K_ux)';
Varinov(id_e3_xf_u , id_e9_u_xf_xf) = K_xu*kron(E_uu, reshape(E_xf_xf_xf, x_nbr, x_nbr^2));
Varinov(id_e4_u_xf , id_e3_xf_u ) = K_ux*E_xfxf_E_uu;
Varinov(id_e4_u_xf , id_e4_u_xf ) = kron(E_uu,E_xfxf);
Varinov(id_e4_u_xf , id_e5_xs_u ) = K_ux*kron(E_xfxs,E_uu);
Varinov(id_e4_u_xf , id_e6_u_xs ) = kron(E_uu, E_xfxs);
Varinov(id_e4_u_xf , id_e7_xf_xf_u) = K_ux*kron(reshape(E_xf_xf_xf,x_nbr,x_nbr^2),E_uu);
Varinov(id_e4_u_xf , id_e8_xf_u_xf) = kron(E_uu,reshape(E_xf_xf_xf,x_nbr,x_nbr^2))*kron(K_xu,speye(x_nbr))';
Varinov(id_e4_u_xf , id_e9_u_xf_xf) = kron(E_uu,reshape(E_xf_xf_xf,x_nbr,x_nbr^2));
Varinov(id_e5_xs_u , id_e1_u ) = kron(E_xs, E_uu);
Varinov(id_e5_xs_u , id_e3_xf_u ) = kron(E_xsxf, E_uu);
Varinov(id_e5_xs_u , id_e4_u_xf ) = kron(E_xsxf, E_uu)*K_ux';
Varinov(id_e5_xs_u , id_e5_xs_u ) = kron(E_xsxs, E_uu);
Varinov(id_e5_xs_u , id_e6_u_xs ) = kron(E_xsxs, E_uu)*K_ux';
Varinov(id_e5_xs_u , id_e7_xf_xf_u) = kron(reshape(E_xs_xf_xf,x_nbr,x_nbr^2),E_uu);
Varinov(id_e5_xs_u , id_e8_xf_u_xf) = kron(reshape(E_xs_xf_xf,x_nbr,x_nbr^2),E_uu)*kron(speye(x_nbr),K_ux)';
Varinov(id_e5_xs_u , id_e9_u_xf_xf) = K_xu*kron(E_uu,reshape(E_xs_xf_xf,x_nbr,x_nbr^2));
Varinov(id_e5_xs_u , id_e13_u_u_u ) = kron(E_xs,reshape(E_u_u_u_u,u_nbr,u_nbr^3));
Varinov(id_e6_u_xs , id_e1_u ) = kron(E_uu,E_xs);
Varinov(id_e6_u_xs , id_e3_xf_u ) = K_ux*kron(E_xsxf, E_uu);
Varinov(id_e6_u_xs , id_e4_u_xf ) = kron(E_uu, E_xsxf);
Varinov(id_e6_u_xs , id_e5_xs_u ) = K_ux*kron(E_xsxs,E_uu);
Varinov(id_e6_u_xs , id_e6_u_xs ) = kron(E_uu, E_xsxs);
Varinov(id_e6_u_xs , id_e7_xf_xf_u) = K_ux*kron(reshape(E_xs_xf_xf,x_nbr,x_nbr^2), E_uu);
Varinov(id_e6_u_xs , id_e8_xf_u_xf) = kron(E_uu, reshape(E_xs_xf_xf,x_nbr,x_nbr^2))*kron(K_xu,speye(x_nbr))';
Varinov(id_e6_u_xs , id_e9_u_xf_xf) = kron(E_uu, reshape(E_xs_xf_xf,x_nbr,x_nbr^2));
Varinov(id_e6_u_xs , id_e13_u_u_u ) = K_ux*kron(E_xs,reshape(E_u_u_u_u,u_nbr,u_nbr^3));
Varinov(id_e7_xf_xf_u , id_e1_u ) = kron(E_xfxf(:),E_uu);
Varinov(id_e7_xf_xf_u , id_e3_xf_u ) = kron(reshape(E_xf_xf_xf,x_nbr^2,x_nbr),E_uu);
Varinov(id_e7_xf_xf_u , id_e4_u_xf ) = kron(reshape(E_xf_xf_xf,x_nbr^2,x_nbr),E_uu)*K_ux';
Varinov(id_e7_xf_xf_u , id_e5_xs_u ) = kron(reshape(E_xf_xf_xs,x_nbr^2,x_nbr),E_uu);
Varinov(id_e7_xf_xf_u , id_e6_u_xs ) = kron(reshape(E_xf_xf_xs,x_nbr^2,x_nbr),E_uu)*K_ux';
Varinov(id_e7_xf_xf_u , id_e7_xf_xf_u) = kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),E_uu);
Varinov(id_e7_xf_xf_u , id_e8_xf_u_xf) = kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),E_uu)*kron(speye(x_nbr),K_ux)';
Varinov(id_e7_xf_xf_u , id_e9_u_xf_xf) = kron(speye(x_nbr),K_ux)*kron(K_ux,speye(x_nbr))*kron(E_uu, reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2));
Varinov(id_e7_xf_xf_u , id_e13_u_u_u ) = kron(E_xfxf(:),reshape(E_u_u_u_u,u_nbr,u_nbr^3));
Varinov(id_e8_xf_u_xf , id_e1_u ) = kron(K_xu,speye(x_nbr))*kron(E_uu,E_xfxf(:));
Varinov(id_e8_xf_u_xf , id_e3_xf_u ) = kron(speye(x_nbr),K_xu)*kron(reshape(E_xf_xf_xf,x_nbr^2,x_nbr),E_uu);
Varinov(id_e8_xf_u_xf , id_e4_u_xf ) = kron(K_xu,speye(x_nbr))*kron(E_uu,reshape(E_xf_xf_xf,x_nbr^2,x_nbr));
Varinov(id_e8_xf_u_xf , id_e5_xs_u ) = kron(speye(x_nbr),K_ux)*kron(reshape(E_xf_xf_xs,x_nbr^2,x_nbr),E_uu);
Varinov(id_e8_xf_u_xf , id_e6_u_xs ) = kron(K_xu,speye(x_nbr))*kron(E_uu,reshape(E_xf_xf_xs,x_nbr^2,x_nbr));
Varinov(id_e8_xf_u_xf , id_e7_xf_xf_u) = kron(speye(x_nbr),K_ux)*kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),E_uu);
Varinov(id_e8_xf_u_xf , id_e8_xf_u_xf) = kron(K_xu,speye(x_nbr))*kron(E_uu,reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2))*kron(K_xu,speye(x_nbr))';
Varinov(id_e8_xf_u_xf , id_e9_u_xf_xf) = kron(K_xu,speye(x_nbr))*kron(E_uu,reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2));
Varinov(id_e8_xf_u_xf , id_e13_u_u_u ) = kron(K_xu,speye(x_nbr))*kron(reshape(E_u_u_u_u,u_nbr,u_nbr^3),E_xfxf(:));
Varinov(id_e9_u_xf_xf , id_e1_u ) = kron(E_uu, E_xfxf(:));
Varinov(id_e9_u_xf_xf , id_e3_xf_u ) = kron(E_uu, reshape(E_xf_xf_xf,x_nbr^2,x_nbr))*K_xu';
Varinov(id_e9_u_xf_xf , id_e4_u_xf ) = kron(E_uu, reshape(E_xf_xf_xf,x_nbr^2,x_nbr));
Varinov(id_e9_u_xf_xf , id_e5_xs_u ) = kron(E_uu, reshape(E_xf_xf_xs,x_nbr^2,x_nbr))*K_xu';
Varinov(id_e9_u_xf_xf , id_e6_u_xs ) = kron(E_uu, reshape(E_xf_xf_xs,x_nbr^2,x_nbr));
Varinov(id_e9_u_xf_xf , id_e7_xf_xf_u) = kron(speye(x_nbr),K_ux)*kron(K_ux,speye(x_nbr))*kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),E_uu);
Varinov(id_e9_u_xf_xf , id_e8_xf_u_xf) = kron(E_uu,reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2))*kron(speye(x_nbr),K_ux)';
Varinov(id_e9_u_xf_xf , id_e9_u_xf_xf) = kron(E_uu,reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2));
Varinov(id_e9_u_xf_xf , id_e13_u_u_u ) = kron(reshape(E_u_u_u_u,u_nbr,u_nbr^3),E_xfxf(:));
Varinov(id_e10_xf_u_u , id_e10_xf_u_u) = kron(E_xfxf,reshape(E_u_u_u_u,u_nbr^2,u_nbr^2));
Varinov(id_e10_xf_u_u , id_e11_u_xf_u) = kron(E_xfxf,reshape(E_u_u_u_u,u_nbr^2,u_nbr^2))*kron(K_ux,speye(u_nbr))';
Varinov(id_e10_xf_u_u , id_e12_u_u_xf) = kron(E_xfxf,reshape(E_u_u_u_u,u_nbr^2,u_nbr^2))*kron(K_ux,speye(u_nbr))'*kron(speye(u_nbr),K_ux)';
Varinov(id_e11_u_xf_u , id_e10_xf_u_u) = kron(K_ux,speye(u_nbr))*kron(E_xfxf,reshape(E_u_u_u_u,u_nbr^2,u_nbr^2));
Varinov(id_e11_u_xf_u , id_e11_u_xf_u) = kron(K_ux,speye(u_nbr))*kron(E_xfxf,reshape(E_u_u_u_u,u_nbr^2,u_nbr^2))*kron(K_ux,speye(u_nbr))';
Varinov(id_e11_u_xf_u , id_e12_u_u_xf) = kron(speye(u_nbr),K_ux)*kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),E_xfxf);
Varinov(id_e12_u_u_xf , id_e10_xf_u_u) = kron(K_ux,speye(u_nbr))*kron(speye(u_nbr),K_ux)*kron(E_xfxf,reshape(E_u_u_u_u,u_nbr^2,u_nbr^2));
Varinov(id_e12_u_u_xf , id_e11_u_xf_u) = kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),E_xfxf)*kron(speye(u_nbr),K_xu)';
Varinov(id_e12_u_u_xf , id_e12_u_u_xf) = kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),E_xfxf);
Varinov(id_e13_u_u_u , id_e1_u ) = reshape(E_u_u_u_u,u_nbr^3,u_nbr);
Varinov(id_e13_u_u_u , id_e5_xs_u ) = kron(E_xs', reshape(E_u_u_u_u,u_nbr^3,u_nbr));
Varinov(id_e13_u_u_u , id_e6_u_xs ) = kron(reshape(E_u_u_u_u,u_nbr^3,u_nbr),E_xs');
Varinov(id_e13_u_u_u , id_e7_xf_xf_u ) = kron(E_xfxf(:)',reshape(E_u_u_u_u,u_nbr^3,u_nbr));
Varinov(id_e13_u_u_u , id_e8_xf_u_xf ) = kron(E_xfxf(:)',reshape(E_u_u_u_u,u_nbr^3,u_nbr))*kron(speye(x_nbr),K_ux)';
Varinov(id_e13_u_u_u , id_e9_u_xf_xf ) = kron(reshape(E_u_u_u_u,u_nbr^3,u_nbr), E_xfxf(:)');
Varinov(id_e13_u_u_u , id_e13_u_u_u ) = reshape(Q6Pu*E_u_u_u_u_u_u,u_nbr^3,u_nbr^3);
E_z = (speye(z_nbr)-A)\c;
if compute_derivs
dA = zeros(z_nbr,z_nbr,totparam_nbr);
dB = zeros(z_nbr,e_nbr,totparam_nbr);
dC = zeros(y_nbr,z_nbr,totparam_nbr);
dD = zeros(y_nbr,e_nbr,totparam_nbr);
dc = zeros(z_nbr,totparam_nbr);
dd = zeros(y_nbr,totparam_nbr);
dVarinov = zeros(e_nbr,e_nbr,totparam_nbr);
dE_z =zeros(z_nbr,totparam_nbr);
for jp=1:totparam_nbr
dA(id_z1_xf , id_z1_xf ,jp) = dghx(indx,:,jp);
dA(id_z2_xs , id_z2_xs ,jp) = dghx(indx,:,jp);
dA(id_z2_xs , id_z3_xf_xf ,jp) = 1/2*dghxx(indx,:,jp);
dA(id_z3_xf_xf , id_z3_xf_xf ,jp) = dhx_hx(:,:,jp);
dA(id_z4_xrd , id_z1_xf ,jp) = 3/6*dghxss(indx,:,jp);
dA(id_z4_xrd , id_z4_xrd ,jp) = dghx(indx,:,jp);
dA(id_z4_xrd , id_z5_xf_xs ,jp) = dghxx(indx,:,jp);
dA(id_z4_xrd , id_z6_xf_xf_xf ,jp) = 1/6*dghxxx(indx,:,jp);
dA(id_z5_xf_xs , id_z1_xf ,jp) = kron(dghx(indx,:,jp),1/2*ghs2(indx,:)) + kron(ghx(indx,:),1/2*dghs2(indx,jp));
dA(id_z5_xf_xs , id_z5_xf_xs ,jp) = dhx_hx(:,:,jp);
dA(id_z5_xf_xs , id_z6_xf_xf_xf ,jp) = kron(dghx(indx,:,jp),1/2*ghxx(indx,:)) + kron(ghx(indx,:),1/2*dghxx(indx,:,jp));
dA(id_z6_xf_xf_xf , id_z6_xf_xf_xf ,jp) = kron(dghx(indx,:,jp),hx_hx) + kron(ghx(indx,:),dhx_hx(:,:,jp));
dB(id_z1_xf , id_e1_u , jp) = dghu(indx,:,jp);
dB(id_z2_xs , id_e2_u_u , jp) = 1/2*dghuu(indx,:,jp);
dB(id_z2_xs , id_e3_xf_u , jp) = dghxu(indx,:,jp);
dB(id_z3_xf_xf , id_e2_u_u , jp) = dhu_hu(:,:,jp);
dB(id_z3_xf_xf , id_e3_xf_u , jp) = dhx_hu(:,:,jp);
dB(id_z3_xf_xf , id_e4_u_xf , jp) = dhu_hx(:,:,jp);
dB(id_z4_xrd , id_e1_u , jp) = 3/6*dghuss(indx,:,jp);
dB(id_z4_xrd , id_e5_xs_u , jp) = dghxu(indx,:,jp);
dB(id_z4_xrd , id_e7_xf_xf_u , jp) = 3/6*dghxxu(indx,:,jp);
dB(id_z4_xrd , id_e10_xf_u_u , jp) = 3/6*dghxuu(indx,:,jp);
dB(id_z4_xrd , id_e13_u_u_u , jp) = 1/6*dghuuu(indx,:,jp);
dB(id_z5_xf_xs , id_e1_u , jp) = kron(dghu(indx,:,jp),1/2*ghs2(indx,:)) + kron(ghu(indx,:),1/2*dghs2(indx,jp));
dB(id_z5_xf_xs , id_e6_u_xs , jp) = dhu_hx(:,:,jp);
dB(id_z5_xf_xs , id_e7_xf_xf_u , jp) = kron(dghx(indx,:,jp),ghxu(indx,:)) + kron(ghx(indx,:),dghxu(indx,:,jp));
dB(id_z5_xf_xs , id_e9_u_xf_xf , jp) = kron(dghu(indx,:,jp),1/2*ghxx(indx,:)) + kron(ghu(indx,:),1/2*dghxx(indx,:,jp));
dB(id_z5_xf_xs , id_e10_xf_u_u , jp) = kron(dghx(indx,:,jp),1/2*ghuu(indx,:)) + kron(ghx(indx,:),1/2*dghuu(indx,:,jp));
dB(id_z5_xf_xs , id_e11_u_xf_u , jp) = kron(dghu(indx,:,jp),ghxu(indx,:)) + kron(ghu(indx,:),dghxu(indx,:,jp));
dB(id_z5_xf_xs , id_e13_u_u_u , jp) = kron(dghu(indx,:,jp),1/2*ghuu(indx,:)) + kron(ghu(indx,:),1/2*dghuu(indx,:,jp));
dB(id_z6_xf_xf_xf , id_e7_xf_xf_u , jp) = kron(dhx_hx(:,:,jp),ghu(indx,:)) + kron(hx_hx,dghu(indx,:,jp));
dB(id_z6_xf_xf_xf , id_e8_xf_u_xf , jp) = kron(dghx(indx,:,jp),hu_hx) + kron(ghx(indx,:),dhu_hx(:,:,jp));
dB(id_z6_xf_xf_xf , id_e9_u_xf_xf , jp) = kron(dghu(indx,:,jp),hx_hx) + kron(ghu(indx,:),dhx_hx(:,:,jp));
dB(id_z6_xf_xf_xf , id_e10_xf_u_u , jp) = kron(dhx_hu(:,:,jp),ghu(indx,:)) + kron(hx_hu,dghu(indx,:,jp));
dB(id_z6_xf_xf_xf , id_e11_u_xf_u , jp) = kron(dghu(indx,:,jp),hx_hu) + kron(ghu(indx,:),dhx_hu(:,:,jp));
dB(id_z6_xf_xf_xf , id_e12_u_u_xf , jp) = kron(dhu_hu(:,:,jp),ghx(indx,:)) + kron(hu_hu,dghx(indx,:,jp));
dB(id_z6_xf_xf_xf , id_e13_u_u_u , jp) = kron(dghu(indx,:,jp),hu_hu) + kron(ghu(indx,:),dhu_hu(:,:,jp));
dC(1:y_nbr , id_z1_xf , jp) = dghx(indy,:,jp) + 1/2*dghxss(indy,:,jp);
dC(1:y_nbr , id_z2_xs , jp) = dghx(indy,:,jp);
dC(1:y_nbr , id_z3_xf_xf , jp) = 1/2*dghxx(indy,:,jp);
dC(1:y_nbr , id_z4_xrd , jp) = dghx(indy,:,jp);
dC(1:y_nbr , id_z5_xf_xs , jp) = dghxx(indy,:,jp);
dC(1:y_nbr , id_z6_xf_xf_xf , jp) = 1/6*dghxxx(indy,:,jp);
dD(1:y_nbr , id_e1_u , jp) = dghu(indy,:,jp) + 1/2*dghuss(indy,:,jp);
dD(1:y_nbr , id_e2_u_u , jp) = 1/2*dghuu(indy,:,jp);
dD(1:y_nbr , id_e3_xf_u , jp) = dghxu(indy,:,jp);
dD(1:y_nbr , id_e5_xs_u , jp) = dghxu(indy,:,jp);
dD(1:y_nbr , id_e7_xf_xf_u , jp) = 1/2*dghxxu(indy,:,jp);
dD(1:y_nbr , id_e10_xf_u_u , jp) = 1/2*dghxuu(indy,:,jp);
dD(1:y_nbr , id_e13_u_u_u , jp) = 1/6*dghuuu(indy,:,jp);
dc(id_z2_xs , jp) = 1/2*dghs2(indx,jp) + 1/2*dghuu(indx,:,jp)*E_uu(:) + 1/2*ghuu(indx,:)*vec(dE_uu(:,:,jp));
dc(id_z3_xf_xf , jp) = dhu_hu(:,:,jp)*E_uu(:) + hu_hu*vec(dE_uu(:,:,jp));
dd(1:y_nbr , jp) = 0.5*dghs2(indy,jp) + 0.5*dghuu(indy,:,jp)*E_uu(:) + 0.5*ghuu(indy,:)*vec(dE_uu(:,:,jp));
dVarinov(id_e1_u , id_e1_u , jp) = dE_uu(:,:,jp);
dVarinov(id_e1_u , id_e5_xs_u , jp) = kron(dE_xs(:,jp)',E_uu) + kron(E_xs',dE_uu(:,:,jp));
dVarinov(id_e1_u , id_e6_u_xs , jp) = kron(dE_uu(:,:,jp),E_xs') + kron(E_uu,dE_xs(:,jp)');
dVarinov(id_e1_u , id_e7_xf_xf_u , jp) = kron(vec(dE_xfxf(:,:,jp))',E_uu) + kron(E_xfxf(:)',dE_uu(:,:,jp));
dVarinov(id_e1_u , id_e8_xf_u_xf , jp) = (kron(dE_uu(:,:,jp),E_xfxf(:)') + kron(E_uu,vec(dE_xfxf(:,:,jp))'))*kron(K_xu,speye(x_nbr));
dVarinov(id_e1_u , id_e9_u_xf_xf , jp) = kron(dE_uu(:,:,jp),E_xfxf(:)') + kron(E_uu,vec(dE_xfxf(:,:,jp))');
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e1_u , id_e13_u_u_u , jp) = reshape(QPu*dE_u_u_u_u(:,jp),u_nbr,u_nbr^3);
dVarinov(id_e2_u_u , id_e2_u_u , jp) = reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2) - vec(dE_uu(:,:,jp))*transpose(E_uu(:)) - E_uu(:)*transpose(vec(dE_uu(:,:,jp)));
end
dVarinov(id_e3_xf_u , id_e3_xf_u , jp) = dE_xfxf_E_uu(:,:,jp);
dVarinov(id_e3_xf_u , id_e4_u_xf , jp) = dE_xfxf_E_uu(:,:,jp)*K_ux';
dVarinov(id_e3_xf_u , id_e5_xs_u , jp) = dE_xfxs_E_uu(:,:,jp);
dVarinov(id_e3_xf_u , id_e6_u_xs , jp) = dE_xfxs_E_uu(:,:,jp)*K_ux';
dVarinov(id_e3_xf_u , id_e7_xf_xf_u , jp) = kron(reshape(dE_xf_xf_xf(:,jp),x_nbr,x_nbr^2), E_uu) + kron(reshape(E_xf_xf_xf,x_nbr,x_nbr^2), dE_uu(:,:,jp));
dVarinov(id_e3_xf_u , id_e8_xf_u_xf , jp) = (kron(reshape(dE_xf_xf_xf(:,jp),x_nbr,x_nbr^2), E_uu) + kron(reshape(E_xf_xf_xf,x_nbr,x_nbr^2), dE_uu(:,:,jp)))*kron(speye(x_nbr),K_ux)';
dVarinov(id_e3_xf_u , id_e9_u_xf_xf , jp) = K_xu*(kron(dE_uu(:,:,jp), reshape(E_xf_xf_xf, x_nbr, x_nbr^2)) + kron(E_uu, reshape(dE_xf_xf_xf(:,jp), x_nbr, x_nbr^2)));
dVarinov(id_e4_u_xf , id_e3_xf_u , jp) = K_ux*dE_xfxf_E_uu(:,:,jp);
dVarinov(id_e4_u_xf , id_e4_u_xf , jp) = kron(dE_uu(:,:,jp),E_xfxf) + kron(E_uu,dE_xfxf(:,:,jp));
dVarinov(id_e4_u_xf , id_e5_xs_u , jp) = K_ux*(kron(dE_xfxs(:,:,jp),E_uu) + kron(E_xfxs,dE_uu(:,:,jp)));
dVarinov(id_e4_u_xf , id_e6_u_xs , jp) = kron(dE_uu(:,:,jp), E_xfxs) + kron(E_uu, dE_xfxs(:,:,jp));
dVarinov(id_e4_u_xf , id_e7_xf_xf_u , jp) = K_ux*(kron(reshape(dE_xf_xf_xf(:,jp),x_nbr,x_nbr^2),E_uu) + kron(reshape(E_xf_xf_xf,x_nbr,x_nbr^2),dE_uu(:,:,jp)));
dVarinov(id_e4_u_xf , id_e8_xf_u_xf , jp) = (kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf,x_nbr,x_nbr^2)) + kron(E_uu,reshape(dE_xf_xf_xf(:,jp),x_nbr,x_nbr^2)))*kron(K_xu,speye(x_nbr))';
dVarinov(id_e4_u_xf , id_e9_u_xf_xf , jp) = kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf,x_nbr,x_nbr^2)) + kron(E_uu,reshape(dE_xf_xf_xf(:,jp),x_nbr,x_nbr^2));
dVarinov(id_e5_xs_u , id_e1_u , jp) = kron(dE_xs(:,jp), E_uu) + kron(E_xs, dE_uu(:,:,jp));
dVarinov(id_e5_xs_u , id_e3_xf_u , jp) = kron(dE_xsxf(:,:,jp), E_uu) + kron(E_xsxf, dE_uu(:,:,jp));
dVarinov(id_e5_xs_u , id_e4_u_xf , jp) = (kron(dE_xsxf(:,:,jp), E_uu) + kron(E_xsxf, dE_uu(:,:,jp)))*K_ux';
dVarinov(id_e5_xs_u , id_e5_xs_u , jp) = kron(dE_xsxs(:,:,jp), E_uu) + kron(E_xsxs, dE_uu(:,:,jp));
dVarinov(id_e5_xs_u , id_e6_u_xs , jp) = (kron(dE_xsxs(:,:,jp), E_uu) + kron(E_xsxs, dE_uu(:,:,jp)))*K_ux';
dVarinov(id_e5_xs_u , id_e7_xf_xf_u , jp) = kron(reshape(dE_xs_xf_xf(:,jp),x_nbr,x_nbr^2),E_uu) + kron(reshape(E_xs_xf_xf,x_nbr,x_nbr^2),dE_uu(:,:,jp));
dVarinov(id_e5_xs_u , id_e8_xf_u_xf , jp) = (kron(reshape(dE_xs_xf_xf(:,jp),x_nbr,x_nbr^2),E_uu) + kron(reshape(E_xs_xf_xf,x_nbr,x_nbr^2),dE_uu(:,:,jp)))*kron(speye(x_nbr),K_ux)';
dVarinov(id_e5_xs_u , id_e9_u_xf_xf , jp) = K_xu*(kron(dE_uu(:,:,jp),reshape(E_xs_xf_xf,x_nbr,x_nbr^2)) + kron(E_uu,reshape(dE_xs_xf_xf(:,jp),x_nbr,x_nbr^2)));
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e5_xs_u , id_e13_u_u_u , jp) = kron(dE_xs(:,jp),reshape(E_u_u_u_u,u_nbr,u_nbr^3)) + kron(E_xs,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr,u_nbr^3));
else
dVarinov(id_e5_xs_u , id_e13_u_u_u , jp) = kron(dE_xs(:,jp),reshape(E_u_u_u_u,u_nbr,u_nbr^3));
end
dVarinov(id_e6_u_xs , id_e1_u , jp) = kron(dE_uu(:,:,jp),E_xs) + kron(E_uu,dE_xs(:,jp));
dVarinov(id_e6_u_xs , id_e3_xf_u , jp) = K_ux*(kron(dE_xsxf(:,:,jp), E_uu) + kron(E_xsxf, dE_uu(:,:,jp)));
dVarinov(id_e6_u_xs , id_e4_u_xf , jp) = kron(dE_uu(:,:,jp), E_xsxf) + kron(E_uu, dE_xsxf(:,:,jp));
dVarinov(id_e6_u_xs , id_e5_xs_u , jp) = K_ux*(kron(dE_xsxs(:,:,jp),E_uu) + kron(E_xsxs,dE_uu(:,:,jp)));
dVarinov(id_e6_u_xs , id_e6_u_xs , jp) = kron(dE_uu(:,:,jp), E_xsxs) + kron(E_uu, dE_xsxs(:,:,jp));
dVarinov(id_e6_u_xs , id_e7_xf_xf_u , jp) = K_ux*(kron(reshape(dE_xs_xf_xf(:,jp),x_nbr,x_nbr^2), E_uu) + kron(reshape(E_xs_xf_xf,x_nbr,x_nbr^2), dE_uu(:,:,jp)));
dVarinov(id_e6_u_xs , id_e8_xf_u_xf , jp) = (kron(dE_uu(:,:,jp), reshape(E_xs_xf_xf,x_nbr,x_nbr^2)) + kron(E_uu, reshape(dE_xs_xf_xf(:,jp),x_nbr,x_nbr^2)))*kron(K_xu,speye(x_nbr))';
dVarinov(id_e6_u_xs , id_e9_u_xf_xf , jp) = kron(dE_uu(:,:,jp), reshape(E_xs_xf_xf,x_nbr,x_nbr^2)) + kron(E_uu, reshape(dE_xs_xf_xf(:,jp),x_nbr,x_nbr^2));
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e6_u_xs , id_e13_u_u_u , jp) = K_ux*(kron(dE_xs(:,jp),reshape(E_u_u_u_u,u_nbr,u_nbr^3)) + kron(E_xs,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr,u_nbr^3)));
else
dVarinov(id_e6_u_xs , id_e13_u_u_u , jp) = K_ux*kron(dE_xs(:,jp),reshape(E_u_u_u_u,u_nbr,u_nbr^3));
end
dVarinov(id_e7_xf_xf_u , id_e1_u , jp) = kron(vec(dE_xfxf(:,:,jp)),E_uu) + kron(E_xfxf(:),dE_uu(:,:,jp));
dVarinov(id_e7_xf_xf_u , id_e3_xf_u , jp) = kron(reshape(dE_xf_xf_xf(:,jp),x_nbr^2,x_nbr),E_uu) + kron(reshape(E_xf_xf_xf,x_nbr^2,x_nbr),dE_uu(:,:,jp));
dVarinov(id_e7_xf_xf_u , id_e4_u_xf , jp) = (kron(reshape(dE_xf_xf_xf(:,jp),x_nbr^2,x_nbr),E_uu) + kron(reshape(E_xf_xf_xf,x_nbr^2,x_nbr),dE_uu(:,:,jp)))*K_ux';
dVarinov(id_e7_xf_xf_u , id_e5_xs_u , jp) = kron(reshape(dE_xf_xf_xs(:,jp),x_nbr^2,x_nbr),E_uu) + kron(reshape(E_xf_xf_xs,x_nbr^2,x_nbr),dE_uu(:,:,jp));
dVarinov(id_e7_xf_xf_u , id_e6_u_xs , jp) = (kron(reshape(dE_xf_xf_xs(:,jp),x_nbr^2,x_nbr),E_uu) + kron(reshape(E_xf_xf_xs,x_nbr^2,x_nbr),dE_uu(:,:,jp)))*K_ux';
dVarinov(id_e7_xf_xf_u , id_e7_xf_xf_u , jp) = kron(reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2),E_uu) + kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),dE_uu(:,:,jp));
dVarinov(id_e7_xf_xf_u , id_e8_xf_u_xf , jp) = (kron(reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2),E_uu) + kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),dE_uu(:,:,jp)))*kron(speye(x_nbr),K_ux)';
dVarinov(id_e7_xf_xf_u , id_e9_u_xf_xf , jp) = kron(speye(x_nbr),K_ux)*kron(K_ux,speye(x_nbr))*(kron(dE_uu(:,:,jp), reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2)) + kron(E_uu, reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2)));
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e7_xf_xf_u , id_e13_u_u_u , jp) = kron(vec(dE_xfxf(:,:,jp)),reshape(E_u_u_u_u,u_nbr,u_nbr^3)) + kron(E_xfxf(:),reshape(QPu*dE_u_u_u_u(:,jp),u_nbr,u_nbr^3));
else
dVarinov(id_e7_xf_xf_u , id_e13_u_u_u , jp) = kron(vec(dE_xfxf(:,:,jp)),reshape(E_u_u_u_u,u_nbr,u_nbr^3));
end
dVarinov(id_e8_xf_u_xf , id_e1_u , jp) = kron(K_xu,speye(x_nbr))*(kron(dE_uu(:,:,jp),E_xfxf(:)) + kron(E_uu,vec(dE_xfxf(:,:,jp))));
dVarinov(id_e8_xf_u_xf , id_e3_xf_u , jp) = kron(speye(x_nbr),K_xu)*(kron(reshape(dE_xf_xf_xf(:,jp),x_nbr^2,x_nbr),E_uu) + kron(reshape(E_xf_xf_xf,x_nbr^2,x_nbr),dE_uu(:,:,jp)));
dVarinov(id_e8_xf_u_xf , id_e4_u_xf , jp) = kron(K_xu,speye(x_nbr))*(kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf,x_nbr^2,x_nbr)) + kron(E_uu,reshape(dE_xf_xf_xf(:,jp),x_nbr^2,x_nbr)));
dVarinov(id_e8_xf_u_xf , id_e5_xs_u , jp) = kron(speye(x_nbr),K_ux)*(kron(reshape(dE_xf_xf_xs(:,jp),x_nbr^2,x_nbr),E_uu) + kron(reshape(E_xf_xf_xs,x_nbr^2,x_nbr),dE_uu(:,:,jp)));
dVarinov(id_e8_xf_u_xf , id_e6_u_xs , jp) = kron(K_xu,speye(x_nbr))*(kron(dE_uu(:,:,jp),reshape(E_xf_xf_xs,x_nbr^2,x_nbr)) + kron(E_uu,reshape(dE_xf_xf_xs(:,jp),x_nbr^2,x_nbr)));
dVarinov(id_e8_xf_u_xf , id_e7_xf_xf_u , jp) = kron(speye(x_nbr),K_ux)*(kron(reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2),E_uu) + kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),dE_uu(:,:,jp)));
dVarinov(id_e8_xf_u_xf , id_e8_xf_u_xf , jp) = kron(K_xu,speye(x_nbr))*(kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2)) + kron(E_uu,reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2)))*kron(K_xu,speye(x_nbr))';
dVarinov(id_e8_xf_u_xf , id_e9_u_xf_xf , jp) = kron(K_xu,speye(x_nbr))*(kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2)) + kron(E_uu,reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2)));
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e8_xf_u_xf , id_e13_u_u_u , jp) = kron(K_xu,speye(x_nbr))*(kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr,u_nbr^3),E_xfxf(:)) + kron(reshape(E_u_u_u_u,u_nbr,u_nbr^3),vec(dE_xfxf(:,:,jp))));
else
dVarinov(id_e8_xf_u_xf , id_e13_u_u_u , jp) = kron(K_xu,speye(x_nbr))*kron(reshape(E_u_u_u_u,u_nbr,u_nbr^3),vec(dE_xfxf(:,:,jp)));
end
dVarinov(id_e9_u_xf_xf , id_e1_u , jp) = kron(dE_uu(:,:,jp), E_xfxf(:)) + kron(E_uu, vec(dE_xfxf(:,:,jp)));
dVarinov(id_e9_u_xf_xf , id_e3_xf_u , jp) = (kron(dE_uu(:,:,jp), reshape(E_xf_xf_xf,x_nbr^2,x_nbr)) + kron(E_uu, reshape(dE_xf_xf_xf(:,jp),x_nbr^2,x_nbr)))*K_xu';
dVarinov(id_e9_u_xf_xf , id_e4_u_xf , jp) = kron(dE_uu(:,:,jp), reshape(E_xf_xf_xf,x_nbr^2,x_nbr)) + kron(E_uu, reshape(dE_xf_xf_xf(:,jp),x_nbr^2,x_nbr));
dVarinov(id_e9_u_xf_xf , id_e5_xs_u , jp) = (kron(dE_uu(:,:,jp), reshape(E_xf_xf_xs,x_nbr^2,x_nbr)) + kron(E_uu, reshape(dE_xf_xf_xs(:,jp),x_nbr^2,x_nbr)))*K_xu';
dVarinov(id_e9_u_xf_xf , id_e6_u_xs , jp) = kron(dE_uu(:,:,jp), reshape(E_xf_xf_xs,x_nbr^2,x_nbr)) + kron(E_uu, reshape(dE_xf_xf_xs(:,jp),x_nbr^2,x_nbr));
dVarinov(id_e9_u_xf_xf , id_e7_xf_xf_u , jp) = kron(speye(x_nbr),K_ux)*kron(K_ux,speye(x_nbr))*(kron(reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2),E_uu) + kron(reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2),dE_uu(:,:,jp)));
dVarinov(id_e9_u_xf_xf , id_e8_xf_u_xf , jp) = (kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2)) + kron(E_uu,reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2)))*kron(speye(x_nbr),K_ux)';
dVarinov(id_e9_u_xf_xf , id_e9_u_xf_xf , jp) = kron(dE_uu(:,:,jp),reshape(E_xf_xf_xf_xf,x_nbr^2,x_nbr^2)) + kron(E_uu,reshape(dE_xf_xf_xf_xf(:,jp),x_nbr^2,x_nbr^2));
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e9_u_xf_xf , id_e13_u_u_u , jp) = kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr,u_nbr^3),E_xfxf(:)) + kron(reshape(E_u_u_u_u,u_nbr,u_nbr^3),vec(dE_xfxf(:,:,jp)));
else
dVarinov(id_e9_u_xf_xf , id_e13_u_u_u , jp) = kron(reshape(E_u_u_u_u,u_nbr,u_nbr^3),vec(dE_xfxf(:,:,jp)));
end
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e10_xf_u_u , id_e10_xf_u_u , jp) = kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2));
dVarinov(id_e10_xf_u_u , id_e11_u_xf_u , jp) = (kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2)))*kron(K_ux,speye(u_nbr))';
dVarinov(id_e10_xf_u_u , id_e12_u_u_xf , jp) = (kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2)))*kron(K_ux,speye(u_nbr))'*kron(speye(u_nbr),K_ux)';
else
dVarinov(id_e10_xf_u_u , id_e10_xf_u_u , jp) = kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2));
dVarinov(id_e10_xf_u_u , id_e11_u_xf_u , jp) = kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2))*kron(K_ux,speye(u_nbr))';
dVarinov(id_e10_xf_u_u , id_e12_u_u_xf , jp) = kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2))*kron(K_ux,speye(u_nbr))'*kron(speye(u_nbr),K_ux)';
end
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e11_u_xf_u , id_e10_xf_u_u , jp) = kron(K_ux,speye(u_nbr))*(kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2)));
dVarinov(id_e11_u_xf_u , id_e11_u_xf_u , jp) = kron(K_ux,speye(u_nbr))*(kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2)))*kron(K_ux,speye(u_nbr))';
dVarinov(id_e11_u_xf_u , id_e12_u_u_xf , jp) = kron(speye(u_nbr),K_ux)*(kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2),E_xfxf) + kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),dE_xfxf(:,:,jp)));
else
dVarinov(id_e11_u_xf_u , id_e10_xf_u_u , jp) = kron(K_ux,speye(u_nbr))*kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2));
dVarinov(id_e11_u_xf_u , id_e11_u_xf_u , jp) = kron(K_ux,speye(u_nbr))*kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2))*kron(K_ux,speye(u_nbr))';
dVarinov(id_e11_u_xf_u , id_e12_u_u_xf , jp) = kron(speye(u_nbr),K_ux)*kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),dE_xfxf(:,:,jp));
end
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e12_u_u_xf , id_e10_xf_u_u , jp) = kron(K_ux,speye(u_nbr))*kron(speye(u_nbr),K_ux)*(kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2)));
dVarinov(id_e12_u_u_xf , id_e11_u_xf_u , jp) = (kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2),E_xfxf) + kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),dE_xfxf(:,:,jp)))*kron(speye(u_nbr),K_xu)';
dVarinov(id_e12_u_u_xf , id_e12_u_u_xf , jp) = kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^2,u_nbr^2),E_xfxf) + kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),dE_xfxf(:,:,jp));
else
dVarinov(id_e12_u_u_xf , id_e10_xf_u_u , jp) = kron(K_ux,speye(u_nbr))*kron(speye(u_nbr),K_ux)*kron(dE_xfxf(:,:,jp),reshape(E_u_u_u_u,u_nbr^2,u_nbr^2));
dVarinov(id_e12_u_u_xf , id_e11_u_xf_u , jp) = kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),dE_xfxf(:,:,jp))*kron(speye(u_nbr),K_xu)';
dVarinov(id_e12_u_u_xf , id_e12_u_u_xf , jp) = kron(reshape(E_u_u_u_u,u_nbr^2,u_nbr^2),dE_xfxf(:,:,jp));
end
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e13_u_u_u , id_e1_u , jp) = reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^3,u_nbr);
dVarinov(id_e13_u_u_u , id_e5_xs_u , jp) = kron(dE_xs(:,jp)', reshape(E_u_u_u_u,u_nbr^3,u_nbr)) + kron(E_xs', reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^3,u_nbr));
dVarinov(id_e13_u_u_u , id_e6_u_xs , jp) = kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^3,u_nbr),E_xs') + kron(reshape(E_u_u_u_u,u_nbr^3,u_nbr),dE_xs(:,jp)');
dVarinov(id_e13_u_u_u , id_e7_xf_xf_u , jp) = kron(vec(dE_xfxf(:,:,jp))',reshape(E_u_u_u_u,u_nbr^3,u_nbr)) + kron(E_xfxf(:)',reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^3,u_nbr));
dVarinov(id_e13_u_u_u , id_e8_xf_u_xf , jp) = (kron(vec(dE_xfxf(:,:,jp))',reshape(E_u_u_u_u,u_nbr^3,u_nbr)) + kron(E_xfxf(:)',reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^3,u_nbr)))*kron(speye(x_nbr),K_ux)';
dVarinov(id_e13_u_u_u , id_e9_u_xf_xf , jp) = kron(reshape(QPu*dE_u_u_u_u(:,jp),u_nbr^3,u_nbr), E_xfxf(:)') + kron(reshape(E_u_u_u_u,u_nbr^3,u_nbr), vec(dE_xfxf(:,:,jp))');
else
dVarinov(id_e13_u_u_u , id_e5_xs_u , jp) = kron(dE_xs(:,jp)', reshape(E_u_u_u_u,u_nbr^3,u_nbr));
dVarinov(id_e13_u_u_u , id_e6_u_xs , jp) = kron(reshape(E_u_u_u_u,u_nbr^3,u_nbr),dE_xs(:,jp)');
dVarinov(id_e13_u_u_u , id_e7_xf_xf_u , jp) = kron(vec(dE_xfxf(:,:,jp))',reshape(E_u_u_u_u,u_nbr^3,u_nbr));
dVarinov(id_e13_u_u_u , id_e8_xf_u_xf , jp) = kron(vec(dE_xfxf(:,:,jp))',reshape(E_u_u_u_u,u_nbr^3,u_nbr))*kron(speye(x_nbr),K_ux)';
dVarinov(id_e13_u_u_u , id_e9_u_xf_xf , jp) = kron(reshape(E_u_u_u_u,u_nbr^3,u_nbr), vec(dE_xfxf(:,:,jp))');
end
if jp <= (stderrparam_nbr+corrparam_nbr)
dVarinov(id_e13_u_u_u , id_e13_u_u_u , jp) = reshape(Q6Pu*dE_u_u_u_u_u_u(:,jp),u_nbr^3,u_nbr^3);
end
dE_z(:,jp) = (speye(z_nbr)-A)\(dc(:,jp) + dA(:,:,jp)*E_z);
end
end
end
end
E_y = Yss(indy,:) + C*E_z + d;
Om_z = B*Varinov*transpose(B);
Om_y = D*Varinov*transpose(D);
if compute_derivs
dE_y = zeros(y_nbr,totparam_nbr);
dOm_z = zeros(z_nbr,z_nbr,totparam_nbr);
dOm_y = zeros(y_nbr,y_nbr,totparam_nbr);
for jp = 1:totparam_nbr
dE_y(:,jp) = dC(:,:,jp)*E_z + C*dE_z(:,jp) + dd(:,jp);
dOm_z(:,:,jp) = dB(:,:,jp)*Varinov*B' + B*dVarinov(:,:,jp)*B' + B*Varinov*dB(:,:,jp)';
dOm_y(:,:,jp) = dD(:,:,jp)*Varinov*D' + D*dVarinov(:,:,jp)*D' + D*Varinov*dD(:,:,jp)';
if jp > (stderrparam_nbr+corrparam_nbr)
dE_y(:,jp) = dE_y(:,jp) + dYss(indy,jp-stderrparam_nbr-corrparam_nbr); %add steady state
end
end
end
%% Store into output structure
dr.pruned.indx = indx;
dr.pruned.indy = indy;
%dr.pruned.E_xfxf = E_xfxf;
dr.pruned.A = A;
dr.pruned.B = B;
dr.pruned.C = C;
dr.pruned.D = D;
dr.pruned.c = c;
dr.pruned.d = d;
dr.pruned.Om_z = Om_z;
dr.pruned.Om_y = Om_y;
dr.pruned.Varinov = Varinov;
dr.pruned.E_z = E_z;
dr.pruned.E_y = E_y;
if compute_derivs == 1
%dr.pruned.dE_xfxf = dE_xfxf;
dr.pruned.dA = dA;
dr.pruned.dB = dB;
dr.pruned.dC = dC;
dr.pruned.dD = dD;
dr.pruned.dc = dc;
dr.pruned.dd = dd;
dr.pruned.dOm_z = dOm_z;
dr.pruned.dOm_y = dOm_y;
dr.pruned.dVarinov = dVarinov;
dr.pruned.dE_z = dE_z;
dr.pruned.dE_y = dE_y;
end