From 60feef4a0a2746c7ccdd8c1371b6290b548e145c Mon Sep 17 00:00:00 2001 From: george Date: Mon, 2 Feb 2009 12:55:13 +0000 Subject: [PATCH] Prototype DR1 subset for running k_order_perturbation git-svn-id: https://www.dynare.org/svn/dynare/trunk@2387 ac1d8469-bf42-47a9-8791-bf33cf982152 --- matlab/kordpert/dr1_k_order.m | 507 ++++++++++++++++++++++++++++++++++ 1 file changed, 507 insertions(+) create mode 100644 matlab/kordpert/dr1_k_order.m diff --git a/matlab/kordpert/dr1_k_order.m b/matlab/kordpert/dr1_k_order.m new file mode 100644 index 000000000..a4356321e --- /dev/null +++ b/matlab/kordpert/dr1_k_order.m @@ -0,0 +1,507 @@ +function [dr,info,M_,options_,oo_] = dr1(dr,task,M_,options_,oo_) +% Computes the reduced form solution of a rational expectation model (first or second order +% approximation of the stochastic model around the deterministic steady state). +% +% INPUTS +% dr [matlab structure] Decision rules for stochastic simulations. +% task [integer] if task = 0 then dr1 computes decision rules. +% if task = 1 then dr1 computes eigenvalues. +% M_ [matlab structure] Definition of the model. +% options_ [matlab structure] Global options. +% oo_ [matlab structure] Results +% +% OUTPUTS +% dr [matlab structure] Decision rules for stochastic simulations. +% info [integer] info=1: the model doesn't define current variables uniquely +% info=2: problem in mjdgges.dll info(2) contains error code. +% info=3: BK order condition not satisfied info(2) contains "distance" +% absence of stable trajectory. +% info=4: BK order condition not satisfied info(2) contains "distance" +% indeterminacy. +% info=5: BK rank condition not satisfied. +% M_ [matlab structure] +% options_ [matlab structure] +% oo_ [matlab structure] +% +% ALGORITHM +% ... +% +% SPECIAL REQUIREMENTS +% none. +% + +% Copyright (C) 1996-2008 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 . + + info = 0; + + options_ = set_default_option(options_,'loglinear',0); + options_ = set_default_option(options_,'noprint',0); + options_ = set_default_option(options_,'olr',0); + options_ = set_default_option(options_,'olr_beta',1); + options_ = set_default_option(options_,'qz_criterium',1.000001); + + xlen = M_.maximum_endo_lead + M_.maximum_endo_lag + 1; + klen = M_.maximum_endo_lag + M_.maximum_endo_lead + 1; + iyv = M_.lead_lag_incidence'; + iyv = iyv(:); + iyr0 = find(iyv) ; + it_ = M_.maximum_lag + 1 ; + + if M_.exo_nbr == 0 + oo_.exo_steady_state = [] ; + end + + % expanding system for Optimal Linear Regulator + if options_.ramsey_policy + if isfield(M_,'orig_model') + orig_model = M_.orig_model; + M_.endo_nbr = orig_model.endo_nbr; + M_.endo_names = orig_model.endo_names; + M_.lead_lag_incidence = orig_model.lead_lag_incidence; + M_.maximum_lead = orig_model.maximum_lead; + M_.maximum_endo_lead = orig_model.maximum_endo_lead; + M_.maximum_lag = orig_model.maximum_lag; + M_.maximum_endo_lag = orig_model.maximum_endo_lag; + end + old_solve_algo = options_.solve_algo; + % options_.solve_algo = 1; + oo_.steady_state = dynare_solve('ramsey_static',oo_.steady_state,0,M_,options_,oo_,it_); + options_.solve_algo = old_solve_algo; + [junk,junk,multbar] = ramsey_static(oo_.steady_state,M_,options_,oo_,it_); + [jacobia_,M_] = ramsey_dynamic(oo_.steady_state,multbar,M_,options_,oo_,it_); + klen = M_.maximum_lag + M_.maximum_lead + 1; + dr.ys = [oo_.steady_state;zeros(M_.exo_nbr,1);multbar]; + + else + klen = M_.maximum_lag + M_.maximum_lead + 1; + iyv = M_.lead_lag_incidence'; + iyv = iyv(:); + iyr0 = find(iyv) ; + it_ = M_.maximum_lag + 1 ; + + if M_.exo_nbr == 0 + oo_.exo_steady_state = [] ; + end + + it_ = M_.maximum_lag + 1; + z = repmat(dr.ys,1,klen); + z = z(iyr0) ; + + end + + if options_.debug + save([M_.fname '_debug.mat'],'jacobia_') + end + + dr=set_state_space(dr,M_); + kstate = dr.kstate; + kad = dr.kad; + kae = dr.kae; + nstatic = dr.nstatic; + nfwrd = dr.nfwrd; + npred = dr.npred; + nboth = dr.nboth; + order_var = dr.order_var; + nd = size(kstate,1); + nz = nnz(M_.lead_lag_incidence); + + sdyn = M_.endo_nbr - nstatic; + + k0 = M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var); + k1 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_endo_lag+1),:); + + + + if options_.order == 1 + M_.var_order_endo_names=M_.endo_names(dr.order_var,:); +% z = repmat(dr.ys,1,klen); +% z = z(iyr0) ; +% oo_.dyn_ys=z; % extended ys + try + [ysteady, gx, gu]=k_order_perturbation(dr,task,M_,options_, oo_ ); + load(M_.fname); + ghxu = eval([M_.fname '_g_1']); + sss= size(ghxu,2); + dr.ghx= ghxu(:,1:sss-M_.exo_nbr); + dr.ghu= ghxu(:,sss-M_.exo_nbr+1:end); + dr.ys=eval([M_.fname '_ss']); + catch + disp('*************************************************************************************'); +% disp('Problem with using k_order perturbation solver - Using Dynare solver instead'); +% warning('Problem with using k_order perturbation solver - Using Dynare solver instead'); + error('Problem with using k_order perturbation solver '); + disp('*****************************************************************************'); + options_.use_k_order=0; % and then try mjdgges instead + info(1) = 4; + info(2) = 1000; + return + end + + elseif options_.order > 1 + error(' can not use order > 1 with K-Order yet!') + % or ??? + disp('********************************************************************'); + disp(' can not use order > 1 with K-Order yet - Using Dynare solver instead'); + disp('********************************************************************'); + options_.use_k_order= 0; % and then try mjdgges instead + info(1) = 4; + info(2) = 1000; + return + end + + + if M_.maximum_endo_lead == 0; % backward models + % If required, try Gary Anderson and G Moore AIM solver if not + % check only and if 1st order (added by GP July'08) + + dr.eigval = eig(transition_matrix(dr)); + dr.rank = 0; + if any(abs(dr.eigval) > options_.qz_criterium) + temp = sort(abs(dr.eigval)); + nba = nnz(abs(dr.eigval) > options_.qz_criterium); + temp = temp(nd-nba+1:nd)-1-options_.qz_criterium; + info(1) = 3; + info(2) = temp'*temp; + end + return; + end + + %forward--looking models + [A,B] =transition_matrix(dr); + dr.eigval = eig(A); +% if any(abs(dr.eigval) > options_.qz_criterium) +% temp = sort(abs(dr.eigval)); +% nba = nnz(abs(dr.eigval) > options_.qz_criterium); +% temp = temp(nd-nba+1:nd)-1-options_.qz_criterium; +% info(1) = 3; +% info(2) = temp'*temp; +% return +% end + sdim = sum( abs(dr.eigval) < options_.qz_criterium ); + nba = nd-sdim; + + nyf = sum(kstate(:,2) > M_.maximum_endo_lag+1); + if nba ~= nyf + temp = sort(abs(dr.eigval)); + if nba > nyf + temp = temp(nd-nba+1:nd-nyf)-1-options_.qz_criterium; + info(1) = 3; + elseif nba < nyf; + temp = temp(nd-nyf+1:nd-nba)-1-options_.qz_criterium; + info(1) = 4; + end + info(2) = temp'*temp; + return + end + + + if options_.loglinear == 1 + k = find(dr.kstate(:,2) <= M_.maximum_endo_lag+1); + klag = dr.kstate(k,[1 2]); + k1 = dr.order_var; + + dr.ghx = repmat(1./dr.ys(k1),1,size(dr.ghx,2)).*dr.ghx.* ... + repmat(dr.ys(k1(klag(:,1)))',size(dr.ghx,1),1); + dr.ghu = repmat(1./dr.ys(k1),1,size(dr.ghu,2)).*dr.ghu; + end + + dr.ghx = real(dr.ghx); + dr.ghu = real(dr.ghu); + +return +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %exogenous deterministic variables + if M_.exo_det_nbr > 0 + f1 = sparse(jacobia_(:,nonzeros(M_.lead_lag_incidence(M_.maximum_endo_lag+2:end,order_var)))); + f0 = sparse(jacobia_(:,nonzeros(M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var)))); + fudet = sparse(jacobia_(:,nz+M_.exo_nbr+1:end)); + M1 = inv(f0+[zeros(M_.endo_nbr,nstatic) f1*gx zeros(M_.endo_nbr,nyf-nboth)]); + M2 = M1*f1; + dr.ghud = cell(M_.exo_det_length,1); + dr.ghud{1} = -M1*fudet; + for i = 2:M_.exo_det_length + dr.ghud{i} = -M2*dr.ghud{i-1}(end-nyf+1:end,:); + end + end + + if options_.order == 1 + return + end + + % Second order + %tempex = oo_.exo_simul ; + [junk,jacobia_,hessian] = feval([M_.fname '_dynamic'],z,... + [oo_.exo_simul ... + oo_.exo_det_simul], M_.params, it_); + + %hessian = real(hessext('ff1_',[z; oo_.exo_steady_state]))' ; + kk = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1)); + if M_.maximum_endo_lag > 0 + kk = [cumsum(M_.lead_lag_incidence(1:M_.maximum_endo_lag,order_var),1); kk]; + end + kk = kk'; + kk = find(kk(:)); + nk = size(kk,1) + M_.exo_nbr + M_.exo_det_nbr; + k1 = M_.lead_lag_incidence(:,order_var); + k1 = k1'; + k1 = k1(:); + k1 = k1(kk); + k2 = find(k1); + kk1(k1(k2)) = k2; + kk1 = [kk1 length(k1)+1:length(k1)+M_.exo_nbr+M_.exo_det_nbr]; + kk = reshape([1:nk^2],nk,nk); + kk1 = kk(kk1,kk1); + %[junk,junk,hessian] = feval([M_.fname '_dynamic'],z, oo_.exo_steady_state); + hessian(:,kk1(:)) = hessian; + + %oo_.exo_simul = tempex ; + %clear tempex + + n1 = 0; + n2 = np; + zx = zeros(np,np); + zu=zeros(np,M_.exo_nbr); + for i=2:M_.maximum_endo_lag+1 + k1 = sum(kstate(:,2) == i); + zx(n1+1:n1+k1,n2-k1+1:n2)=eye(k1); + n1 = n1+k1; + n2 = n2-k1; + end + kk = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1)); + k0 = [1:M_.endo_nbr]; + gx1 = dr.ghx; + hu = dr.ghu(nstatic+[1:npred],:); + zx = [zx; gx1]; + zu = [zu; dr.ghu]; + for i=1:M_.maximum_endo_lead + k1 = find(kk(i+1,k0) > 0); + zu = [zu; gx1(k1,1:npred)*hu]; + gx1 = gx1(k1,:)*hx; + zx = [zx; gx1]; + kk = kk(:,k0); + k0 = k1; + end + 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 = -sparse_hessian_times_B_kronecker_C(hessian,zx); + + %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(n,n); + B = zeros(n,n); + A(1:M_.endo_nbr,1:M_.endo_nbr) = jacobia_(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var)); + % variables with the highest lead + k1 = find(kstate(:,2) == M_.maximum_endo_lag+M_.maximum_endo_lead+1); + if M_.maximum_endo_lead > 1 + k2 = find(kstate(:,2) == M_.maximum_endo_lag+M_.maximum_endo_lead); + [junk,junk,k3] = intersect(kstate(k1,1),kstate(k2,1)); + else + k2 = [1:M_.endo_nbr]; + k3 = kstate(k1,1); + end + % Jacobian with respect to the variables with the highest lead + B(1:M_.endo_nbr,end-length(k2)+k3) = jacobia_(:,kstate(k1,3)+M_.endo_nbr); + offset = M_.endo_nbr; + k0 = [1:M_.endo_nbr]; + gx1 = dr.ghx; + for i=1:M_.maximum_endo_lead-1 + k1 = find(kstate(:,2) == M_.maximum_endo_lag+i+1); + [k2,junk,k3] = find(kstate(k1,3)); + A(1:M_.endo_nbr,offset+k2) = jacobia_(:,k3+M_.endo_nbr); + n1 = length(k1); + A(offset+[1:n1],nstatic+[1:npred]) = -gx1(kstate(k1,1),1:npred); + gx1 = gx1*hx; + A(offset+[1:n1],offset+[1:n1]) = eye(n1); + n0 = length(k0); + E = eye(n0); + if i == 1 + [junk,junk,k4]=intersect(kstate(k1,1),[1:M_.endo_nbr]); + else + [junk,junk,k4]=intersect(kstate(k1,1),kstate(k0,1)); + end + i1 = offset-n0+n1; + B(offset+[1:n1],offset-n0+[1:n0]) = -E(k4,:); + k0 = k1; + offset = offset + n1; + end + [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])+jacobia_(:,k2)*gx1(k1,1:npred); + C = hx; + D = [rhs; zeros(n-M_.endo_nbr,size(rhs,2))]; + + + dr.ghxx = gensylv(2,A,B,C,D); + + %ghxu + %rhs + hu = dr.ghu(nstatic+1:nstatic+npred,:); + %kk = reshape([1:np*np],np,np); + %kk = kk(1:npred,1:npred); + %rhs = -hessian*kron(zx,zu)-f1*dr.ghxx(end-nyf+1:end,kk(:))*kron(hx(1:npred,:),hu(1:npred,:)); + + rhs = sparse_hessian_times_B_kronecker_C(hessian,zx,zu); + + nyf1 = sum(kstate(:,2) == M_.maximum_endo_lag+2); + hu1 = [hu;zeros(np-npred,M_.exo_nbr)]; + %B1 = [B(1:M_.endo_nbr,:);zeros(size(A,1)-M_.endo_nbr,size(B,2))]; + [nrhx,nchx] = size(hx); + [nrhu1,nchu1] = size(hu1); + + B1 = B*A_times_B_kronecker_C(dr.ghxx,hx,hu1); + rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1; + + + %lhs + dr.ghxu = A\rhs; + + %ghuu + %rhs + kk = reshape([1:np*np],np,np); + kk = kk(1:npred,1:npred); + + rhs = sparse_hessian_times_B_kronecker_C(hessian,zu); + + + B1 = A_times_B_kronecker_C(B*dr.ghxx,hu1); + 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,:); + dr.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 = jacobia_(:,M_.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); + M_.lead_lag_incidenceordered = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1)); + if M_.maximum_endo_lag > 0 + M_.lead_lag_incidenceordered = [cumsum(M_.lead_lag_incidence(1:M_.maximum_endo_lag,order_var),1); M_.lead_lag_incidenceordered]; + end + M_.lead_lag_incidenceordered = M_.lead_lag_incidenceordered'; + M_.lead_lag_incidenceordered = M_.lead_lag_incidenceordered(:); + k1 = find(M_.lead_lag_incidenceordered); + M_.lead_lag_incidenceordered(k1) = [1:length(k1)]'; + M_.lead_lag_incidenceordered =reshape(M_.lead_lag_incidenceordered,M_.endo_nbr,M_.maximum_endo_lag+M_.maximum_endo_lead+1)'; + 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],:); + for i=1:M_.maximum_endo_lead + for j=i:M_.maximum_endo_lead + [junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+j+1,order_var)); + [junk,k3a,k3] = ... + find(M_.lead_lag_incidenceordered(M_.maximum_endo_lag+j+1,:)); + nk3a = length(k3a); + B1 = sparse_hessian_times_B_kronecker_C(hessian(:,kh(k3,k3)),gu(k3a,:)); + RHS = RHS + jacobia_(:,k2)*guu(k2a,:)+B1; + end + % LHS + [junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+i+1,order_var)); + LHS = LHS + jacobia_(:,k2)*(E(k2a,:)+[O1(k2a,:) dr.ghx(k2a,:)*H O2(k2a,:)]); + + if i == M_.maximum_endo_lead + break + end + + kk = find(kstate(:,2) == M_.maximum_endo_lag+i+1); + gu = dr.ghx*Gu; + [nrGu,ncGu] = size(Gu); + G1 = A_times_B_kronecker_C(dr.ghxx,Gu); + G2 = A_times_B_kronecker_C(hxx,Gu); + guu = dr.ghx*Guu+G1; + Gu = hx*Gu; + Guu = hx*Guu; + Guu(end-npred+1:end,:) = Guu(end-npred+1:end,:) + G2; + H = E1 + hx*H; + end + 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(hx,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 \ No newline at end of file