function [y, T, oo]= solve_two_boundaries(fname, y, x, params, steady_state, T, y_index, nze, periods, y_kmin_l, y_kmax_l, is_linear, Block_Num, y_kmin, maxit_, solve_tolf, lambda, cutoff, stack_solve_algo,options,M, oo)
% Computes the deterministic simulation of a block of equation containing
% both lead and lag variables using relaxation methods
%
% INPUTS
%   fname               [string]        name of the file containing the block
%                                       to simulate
%   y                   [matrix]        All the endogenous variables of the model
%   x                   [matrix]        All the exogenous variables of the model
%   params              [vector]        All the parameters of the model
%   steady_state        [vector]        steady state of the model
%   T                   [matrix]        Temporary terms
%   y_index             [vector of int] The index of the endogenous variables of
%                                       the block
%   nze                 [integer]       number of non-zero elements in the
%                                       jacobian matrix
%   periods             [integer]       number of simulation periods
%   y_kmin_l            [integer]       maximum number of lag in the block
%   y_kmax_l            [integer]       maximum number of lead in the block
%   is_linear           [logical]       Whether the block is linear
%   Block_Num           [integer]       block number
%   y_kmin              [integer]       maximum number of lag in the model
%   maxit_              [integer]       maximum number of iteration in Newton
%   solve_tolf          [double]        convergence criteria
%   lambda              [double]        initial value of step size in
%   Newton
%   cutoff              [double]        cutoff to correct the direction in Newton in case
%                                       of singular jacobian matrix
%   stack_solve_algo    [integer]       linear solver method used in the
%                                       Newton algorithm :
%                                            - 1 sprse LU
%                                            - 2 GMRES
%                                            - 3 BicGStab
%                                            - 4 Optimal path length
%   M                   [structure]     Model description
%   oo                  [structure]     Results
%
% OUTPUTS
%   y                   [matrix]        All endogenous variables of the model
%   T                   [matrix]        Temporary terms
%   oo                  [structure]     Results
%
% ALGORITHM
%   Newton with LU or GMRES or BicGstab
%
% SPECIAL REQUIREMENTS
%   none.
%

% Copyright (C) 1996-2022 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/>.

verbose = options.verbosity;

cvg=0;
iter=0;
Per_u_=0;
Blck_size=size(y_index,2);
correcting_factor=0.01;
ilu_setup.droptol=1e-10;
ilu_setup.type = 'ilutp';
%ilu_setup.milu = 'col';
ilu_setup.milu = 'off';
ilu_setup.thresh = 1;
ilu_setup.udiag = 0;
max_resa=1e100;
reduced = 0;
while ~(cvg==1 || iter>maxit_)
    r = NaN(Blck_size, periods);
    g1a = spalloc(Blck_size*periods, Blck_size*periods, nze*periods);
    for it_ = y_kmin+(1:periods)
        [r(:, it_-y_kmin), yy, T(:, it_), g1]=feval(fname, Block_Num, dynvars_from_endo_simul(y, it_, M), x, params, steady_state, T(:, it_), it_, false);
        y(:, it_) = yy(M.lead_lag_incidence(M.maximum_endo_lag+1,:));
        if periods == 1
            g1a = g1(:, Blck_size+(1:Blck_size));
        elseif it_ == y_kmin+1
            g1a(1:Blck_size, 1:Blck_size*2) = g1(:, Blck_size+1:end);
        elseif it_ == y_kmin+periods
            g1a((periods-1)*Blck_size+1:end, (periods-2)*Blck_size+1:end) = g1(:, 1:2*Blck_size);
        else
            g1a((it_-y_kmin-1)*Blck_size+(1:Blck_size), (it_-y_kmin-2)*Blck_size+(1:3*Blck_size)) = g1;
        end
    end
    preconditioner = 2;
    ya = reshape(y(y_index, y_kmin+(1:periods)), 1, periods*Blck_size)';
    ra = reshape(r, periods*Blck_size, 1);
    b=-ra+g1a*ya;
    [max_res, max_indx]=max(max(abs(r')));
    if ~isreal(r)
        max_res = (-max_res^2)^0.5;
    end
    if ~isreal(max_res) || isnan(max_res)
        cvg = 0;
    elseif is_linear && iter>0
        cvg = 1;
    else
        cvg=(max_res<solve_tolf);
    end
    if ~cvg
        if iter>0
            if ~isreal(max_res) || isnan(max_res) || (max_resa<max_res && iter>1)
                if verbose && ~isreal(max_res)
                    disp(['Variable ' M.endo_names{max_indx} ' (' int2str(max_indx) ') returns an undefined value']);
                end
                if isnan(max_res)
                    detJ=det(g1aa);
                    if abs(detJ)<1e-7
                        max_factor=max(max(abs(g1aa)));
                        ze_elem=sum(diag(g1aa)<cutoff);
                        if verbose
                            disp([num2str(full(ze_elem),'%d') ' elements on the Jacobian diagonal are below the cutoff (' num2str(cutoff,'%f') ')']);
                        end
                        if correcting_factor<max_factor
                            correcting_factor=correcting_factor*4;
                            if verbose
                                disp(['The Jacobain matrix is singular, det(Jacobian)=' num2str(detJ,'%f') '.']);
                                disp(['    trying to correct the Jacobian matrix:']);
                                disp(['    correcting_factor=' num2str(correcting_factor,'%f') ' max(Jacobian)=' num2str(full(max_factor),'%f')]);
                            end
                            dx = (g1aa+correcting_factor*speye(periods*Blck_size))\ba- ya_save;
                            y(y_index, y_kmin+(1:periods))=reshape((ya_save+lambda*dx)',length(y_index),periods);
                            continue
                        else
                            disp('The singularity of the jacobian matrix could not be corrected');
                            oo.deterministic_simulation.status = false;
                            oo.deterministic_simulation.error = max_res;
                            oo.deterministic_simulation.iterations = iter;
                            oo.deterministic_simulation.block(Block_Num).status = false;% Convergency failed.
                            oo.deterministic_simulation.block(Block_Num).error = max_res;
                            oo.deterministic_simulation.block(Block_Num).iterations = iter;
                            return
                        end
                    end
                elseif lambda>1e-8
                    lambda=lambda/2;
                    reduced = 1;
                    if verbose
                        disp(['reducing the path length: lambda=' num2str(lambda,'%f')]);
                    end
                    y(y_index, y_kmin+(1:periods))=reshape((ya_save+lambda*dx)',length(y_index),periods);
                    continue
                else
                    if verbose
                        if cutoff==0
                            fprintf('Convergence not achieved in block %d, after %d iterations.\n Increase "maxit".\n',Block_Num, iter);
                        else
                            fprintf('Convergence not achieved in block %d, after %d iterations.\n Increase "maxit" or set "cutoff=0" in model options.\n',Block_Num, iter);
                        end
                    end
                    oo.deterministic_simulation.status = false;
                    oo.deterministic_simulation.error = max_res;
                    oo.deterministic_simulation.iterations = iter;
                    oo.deterministic_simulation.block(Block_Num).status = false;% Convergency failed.
                    oo.deterministic_simulation.block(Block_Num).error = max_res;
                    oo.deterministic_simulation.block(Block_Num).iterations = iter;
                    return
                end
            else
                if lambda<1
                    lambda=max(lambda*2, 1);
                end
            end
        end
        ya_save=ya;
        g1aa=g1a;
        ba=b;
        max_resa=max_res;
        if stack_solve_algo==0
            dx = g1a\b- ya;
            ya = ya + lambda*dx;
            y(y_index, y_kmin+(1:periods))=reshape(ya',length(y_index),periods);
        elseif stack_solve_algo==1
            for t=1:periods
                first_elem = (t-1)*Blck_size+1;
                last_elem = t*Blck_size;
                next_elem = (t+1)*Blck_size;
                Elem = first_elem:last_elem;
                Elem_1 = last_elem+1:next_elem;
                B1_inv = inv(g1a(Elem, Elem));
                if (t < periods)
                    S1 = B1_inv * g1a(Elem, Elem_1);
                    g1a(Elem, Elem_1) = S1;
                end
                b(Elem) = B1_inv * b(Elem);
                g1a(Elem, Elem) = ones(Blck_size, Blck_size);
                if t<periods
                    g1a(Elem_1, Elem_1) = g1a(Elem_1, Elem_1) - g1a(Elem_1, Elem) * S1;
                    b(Elem_1) = b(Elem_1) - g1a(Elem_1, Elem) * b(Elem);
                    g1a(Elem_1, Elem) = zeros(Blck_size, Blck_size);
                end
            end
            za = b(Elem);
            zaa = za;
            y_Elem = (periods - 1) * Blck_size + 1:(periods) * Blck_size;
            dx = ya;
            dx(y_Elem) = za - ya(y_Elem);
            ya(y_Elem) = ya(y_Elem) + lambda*dx(y_Elem);
            y(y_index, y_kmin + periods) = ya(y_Elem);
            for t=periods-1:-1:1
                first_elem = (t-1)*Blck_size+1;
                last_elem = t*Blck_size;
                next_elem = (t+1)*Blck_size;
                Elem_1 = last_elem+1:next_elem;
                Elem = first_elem:last_elem;
                za = b(Elem) - g1a(Elem, Elem_1) * zaa;
                zaa = za;
                y_Elem = Blck_size * (t-1)+1:Blck_size * (t);
                dx(y_Elem) = za - ya(y_Elem);
                ya(y_Elem) = ya(y_Elem) + lambda*dx(y_Elem);
                y(y_index, y_kmin + t) = ya(y_Elem);
            end
        elseif stack_solve_algo==2
            flag1=1;
            while flag1>0
                if preconditioner==2
                    [L1, U1]=ilu(g1a,ilu_setup);
                elseif preconditioner==3
                    Size = Blck_size;
                    gss1 =  g1a(Size + 1: 2*Size,Size + 1: 2*Size) + g1a(Size + 1: 2*Size,2*Size+1: 3*Size);
                    [L1, U1]=lu(gss1);
                    L(1:Size,1:Size) = L1;
                    U(1:Size,1:Size) = U1;
                    gss2 = g1a(Size + 1: 2*Size,1: Size) + g1a(Size + 1: 2*Size,Size+1: 2*Size) + g1a(Size + 1: 2*Size,2*Size+1: 3*Size);
                    [L2, U2]=lu(gss2);
                    L(Size+1:(periods-1)*Size,Size+1:(periods-1)*Size) = kron(eye(periods-2), L2);
                    U(Size+1:(periods-1)*Size,Size+1:(periods-1)*Size) = kron(eye(periods-2), U2);
                    gss2 = g1a(Size + 1: 2*Size,1: Size) + g1a(Size + 1: 2*Size,Size+1: 2*Size);
                    [L3, U3]=lu(gss2);
                    L((periods-1)*Size+1:periods*Size,(periods-1)*Size+1:periods*Size) = L3;
                    U((periods-1)*Size+1:periods*Size,(periods-1)*Size+1:periods*Size) = U3;
                    L1 = L;
                    U1 = U;
                elseif preconditioner==4
                    Size = Blck_size;
                    gss1 =  g1a(1: 3*Size, 1: 3*Size);
                    [L, U] = lu(gss1);
                    L1 = kron(eye(ceil(periods/3)),L);
                    U1 = kron(eye(ceil(periods/3)),U);
                    L1 = L1(1:periods * Size, 1:periods * Size);
                    U1 = U1(1:periods * Size, 1:periods * Size);
                end
                [za,flag1] = gmres(g1a,b,Blck_size,1e-6,Blck_size*periods,L1,U1);
                if (flag1>0 || reduced)
                    if verbose
                        if flag1==1
                            disp(['Error in simul: No convergence inside GMRES after ' num2str(periods*10,'%6d') ' iterations, in block ' num2str(Blck_size,'%3d')]);
                        elseif flag1==2
                            disp(['Error in simul: Preconditioner is ill-conditioned, in block ' num2str(Blck_size,'%3d')]);
                        elseif flag1==3
                            disp(['Error in simul: GMRES stagnated (Two consecutive iterates were the same.), in block ' num2str(Blck_size,'%3d')]);
                        end
                    end
                    ilu_setup.droptol = ilu_setup.droptol/10;
                    reduced = 0;
                else
                    dx = za - ya;
                    ya = ya + lambda*dx;
                    y(y_index, y_kmin+(1:periods))=reshape(ya',length(y_index),periods);
                end
            end
        elseif stack_solve_algo==3
            flag1=1;
            while flag1>0
                if preconditioner==2
                    [L1, U1]=ilu(g1a,ilu_setup);
                    [za,flag1] = bicgstab(g1a,b,1e-7,Blck_size*periods,L1,U1);
                elseif preconditioner==3
                    Size = Blck_size;
                    gss0 = g1a(Size + 1: 2*Size,1: Size) + g1a(Size + 1: 2*Size,Size+1: 2*Size) + g1a(Size + 1: 2*Size,2*Size+1: 3*Size);
                    [L1, U1]=lu(gss0);
                    P1 = eye(size(gss0));
                    Q1 = eye(size(gss0));
                    L = kron(eye(periods),L1);
                    U = kron(eye(periods),U1);
                    P = kron(eye(periods),P1);
                    Q = kron(eye(periods),Q1);
                    [za,flag1] = bicgstab1(g1a,b,1e-7,Blck_size*periods,L,U, P, Q);
                else
                    Size = Blck_size;
                    gss0 = g1a(Size + 1: 2*Size,1: Size) + g1a(Size + 1: 2*Size,Size+1: 2*Size) + g1a(Size + 1: 2*Size,2*Size+1: 3*Size);
                    [L1, U1]=lu(gss0);
                    L1 = kron(eye(periods),L1);
                    U1 = kron(eye(periods),U1);
                    [za,flag1] = bicgstab(g1a,b,1e-7,Blck_size*periods,L1,U1);
                end
                if flag1>0 || reduced
                    if verbose
                        if flag1==1
                            disp(['Error in simul: No convergence inside BICGSTAB after ' num2str(periods*10,'%6d') ' iterations, in block ' num2str(Blck_size,'%3d')]);
                        elseif flag1==2
                            disp(['Error in simul: Preconditioner is ill-conditioned, in block ' num2str(Blck_size,'%3d')]);
                        elseif flag1==3
                            disp(['Error in simul: GMRES stagnated (Two consecutive iterates were the same.), in block ' num2str(Blck_size,'%3d')]);
                        end
                    end
                    ilu_setup.droptol = ilu_setup.droptol/10;
                    reduced = 0;
                else
                    dx = za - ya;
                    ya = ya + lambda*dx;
                    y(y_index, y_kmin+(1:periods))=reshape(ya',length(y_index),periods);
                end
            end
        elseif stack_solve_algo==4
            stpmx = 100 ;
            stpmax = stpmx*max([sqrt(ya'*ya);size(y_index,2)]);
            nn=1:size(ra,1);
            g = (ra'*g1a)';
            f = 0.5*ra'*ra;
            p = -g1a\ra;
            [yn,f,ra,check]=lnsrch1(ya,f,g,p,stpmax,'lnsrch1_wrapper_two_boundaries',nn,nn, options.solve_tolx, fname, Block_Num, y, y_index,x, params, steady_state, T, periods, Blck_size, M);
            dx = ya - yn;
            y(y_index, y_kmin+(1:periods))=reshape(yn',length(y_index),periods);
        end
    end
    iter=iter+1;
    if verbose
        disp(['iteration: ' num2str(iter,'%d') ' error: ' num2str(max_res,'%e')]);
    end
end

if (iter>maxit_)
    if verbose
        printline(41)
        %disp(['No convergence after ' num2str(iter,'%4d') ' iterations in Block ' num2str(Block_Num,'%d')])
    end
    oo.deterministic_simulation.status = false;
    oo.deterministic_simulation.error = max_res;
    oo.deterministic_simulation.iterations = iter;
    oo.deterministic_simulation.block(Block_Num).status = false;% Convergency failed.
    oo.deterministic_simulation.block(Block_Num).error = max_res;
    oo.deterministic_simulation.block(Block_Num).iterations = iter;
    return
end

oo.deterministic_simulation.status = true;
oo.deterministic_simulation.error = max_res;
oo.deterministic_simulation.iterations = iter;
oo.deterministic_simulation.block(Block_Num).status = true;% Convergency obtained.
oo.deterministic_simulation.block(Block_Num).error = max_res;
oo.deterministic_simulation.block(Block_Num).iterations = iter;
