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function [y, T, success, err, iter] = solve_two_boundaries_lbj(fh, y, x, steady_state, T, blk, options_, M_)
% Computes the deterministic simulation of a block of equations containing
% both lead and lag variables, using the LBJ algorithm.
%
% INPUTS
% fh [handle] function handle to the dynamic file for the block
% y [matrix] All the endogenous variables of the model
% x [matrix] All the exogenous variables of the model
% steady_state [vector] steady state of the model
% T [matrix] Temporary terms
% blk [integer] block number
% options_ [structure] storing the options
% M_ [structure] Model description
%
% OUTPUTS
% y [matrix] All endogenous variables of the model
% T [matrix] Temporary terms
% success [logical] Whether a solution was found
% err [double] ∞-norm of Δy
% iter [integer] Number of iterations
%
% ALGORITHM
% Laffargue, Boucekkine, Juillard (LBJ)
% see Juillard (1996) Dynare: A program for the resolution and
% simulation of dynamic models with forward variables through the use
% of a relaxation algorithm. CEPREMAP. Couverture Orange. 9602.
% Copyright © 2023 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 <https://www.gnu.org/licenses/>.
sparse_rowval = M_.block_structure.block(blk).g1_sparse_rowval;
sparse_colval = M_.block_structure.block(blk).g1_sparse_colval;
sparse_colptr = M_.block_structure.block(blk).g1_sparse_colptr;
periods = options_.periods;
% NB: notations are deliberately similar to those of sim1_lbj.m
ny = M_.block_structure.block(blk).mfs;
% Compute which columns, in the 3×n-wide Jacobian, have non-zero elements
% corresponding to the forward (iyf) or backward (iyp) variables
iyp = find(sparse_colptr(2:ny+1)-sparse_colptr(1:ny));
iyf = find(sparse_colptr(2*ny+2:end)-sparse_colptr(2*ny+1:end-1));
y_index = M_.block_structure.block(blk).variable(end-ny+1:end);
success = false;
for iter = 1:options_.simul.maxit
h = clock;
c = zeros(ny*periods, length(iyf)+1); % Stores the D and d of Sébastien’s presentation
it_ = M_.maximum_lag+1;
[yy, T(:, it_), r, g1] = fh(dynendo(y, it_, M_), x(it_, :), M_.params, steady_state, ...
sparse_rowval, sparse_colval, sparse_colptr, T(:, it_));
y(:, it_) = yy(M_.endo_nbr+(1:M_.endo_nbr));
ic = 1:ny;
icp = iyp;
c(ic, :) = full(g1(:, ny+(1:ny))) \ [ full(g1(:, 2*ny+iyf)) -r ];
for it_ = M_.maximum_lag+(2:periods)
[yy, T(:, it_), r, g1] = fh(dynendo(y, it_, M_), x(it_, :), M_.params, steady_state, ...
sparse_rowval, sparse_colval, sparse_colptr, T(:, it_));
y(:, it_) = yy(M_.endo_nbr+(1:M_.endo_nbr));
j = [ full(g1(:, ny+(1:ny))) -r ];
j(:, [ iyf ny+1 ]) = j(:, [ iyf ny+1 ]) - full(g1(:, iyp)) * c(icp, :);
ic = ic + ny;
icp = icp + ny;
c(ic, :) = j(:, 1:ny) \ [ full(g1(:, 2*ny+iyf)) j(:, ny+1) ];
end
dy = back_subst_lbj(c, ny, iyf, periods);
y(y_index, M_.maximum_lag+(1:periods)) = y(y_index, M_.maximum_lag+(1:periods)) + dy;
err = max(max(abs(dy)));
if options_.verbosity
fprintf('Iter: %s,\t err. = %s, \t time = %s\n', num2str(iter), num2str(err), num2str(etime(clock, h)));
end
if err < options_.dynatol.x
success = true;
break
end
end
function y3n = dynendo(y, it_, M_)
y3n = reshape(y(:, it_+(-1:1)), 3*M_.endo_nbr, 1);
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