function [LIK, lik] = diffuse_kalman_filter(T,R,Q,H,Pinf,Pstar,Y,start,Z,kalman_tol,riccati_tol)
% Computes the diffuse likelihood of a state space model.
%
% INPUTS
%    T           [double]      mm*mm transition matrix
%    R           [double]      mm*rr matrix
%    Q           [double]      rr*rr covariance matrix of the structural innovations.
%    H           [double]      pp*pp covariance matrix of the measurement errors (if H is equal to zero (scalar) there is no measurement error). 
%    Pinf        [double]      mm*mm matrix used to initialize the covariance matrix of the state vector.
%    Pstar       [double]      mm*mm matrix used to initialize the covariance matrix of the state vector.
%    Y           [double]      pp*smpl matrix of (detrended) data, where pp is the number of observed variables.
%    start       [integer]     scalar, likelihood evaluation starts at 'start'.
%    Z           [double]      pp*mm matrix, selection matrix or pp linear independant combinations of the state vector.
%    kalman_tol  [double]      scalar, tolerance parameter (rcond).
%    riccati_tol [double]      scalar, tolerance parameter (riccati iteration).
%             
% OUTPUTS
%    LIK:    likelihood
%    lik:    density vector in each period
%        
% REFERENCES
%   See "Filtering and Smoothing of State Vector for Diffuse State Space
%   Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series 
%   Analysis, vol. 24(1), pp. 85-98). 

% Copyright (C) 2004-2008 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 <http://www.gnu.org/licenses/>.

[pp,smpl] = size(Y);
mm   = size(T,2);
a    = zeros(mm,1);
dF   = 1;
QQ   = R*Q*transpose(R);
t    = 0;
oldK = 0;
lik  = zeros(smpl,1);
LIK  = Inf;
%   lik(smpl+1) = smpl*pp*log(2*pi);
notsteady   = 1;
reste       = 0;

while rank(Pinf,kalman_tol) && (t<smpl)
    t = t+1;
    v = Y(:,t)-Z*a;
    Finf  = Z*Pinf*Z' ;
    if rcond(Finf) < kalman_tol
        if ~all(abs(Finf(:)) < kalman_tol)
            % The univariate diffuse kalman filter should be used.
            return
        else
            Fstar  = Z*Pstar*Z' + H;
            if rcond(Fstar) < kalman_tol
                if ~all(abs(Fstar(:))<kalman_tol)
                    % The univariate diffuse kalman filter should be used.
                    return
                else
                    a = T*a;
                    Pstar = T*Pstar*transpose(T)+QQ;
                    Pinf  = T*Pinf*transpose(T);
                end
            else
                iFstar = inv(Fstar);
                dFstar = det(Fstar);
                Kstar  = Pstar*Z'*iFstar;
                lik(t) = log(dFstar) + v'*iFstar*v;
                Pinf   = T*Pinf*transpose(T);
                Pstar  = T*(Pstar-Pstar*Z'*Kstar')*T'+QQ;
                a	 = T*(a+Kstar*v);
            end
        end
    else
        lik(t) = log(det(Finf));
        iFinf	 = inv(Finf);
        Kinf	 = Pinf*Z'*iFinf;
        Fstar	 = Z*Pstar*Z' + H;
        Kstar	 = (Pstar*Z'-Kinf*Fstar)*iFinf;
        Pstar	 = T*(Pstar-Pstar*Z'*Kinf'-Pinf*Z'*Kstar')*T'+QQ;
        Pinf	 = T*(Pinf-Pinf*Z'*Kinf')*T';
        a	 = T*(a+Kinf*v);
    end
end

if t == smpl
    error(['There isn''t enough information to estimate the initial conditions of the nonstationary variables']);                   
end

F_singular = 1;
while notsteady && (t<smpl)
    t = t+1;
    v = Y(:,t)-Z*a;
    F = Z*Pstar*Z' + H;
    oldPstar  = Pstar;
    dF = det(F);
    if rcond(F) < kalman_tol
        if ~all(abs(F(:))<kalman_tol)
            return
        else
            a     = T*a;
            Pstar = T*Pstar*T'+QQ;
        end
    else
        F_singular = 0;
        iF     = inv(F);
        lik(t) = log(dF)+v'*iF*v;
        K      = Pstar*Z'*iF;
        a      = T*(a+K*v);
        Pstar  = T*(Pstar-K*Z*Pstar)*T'+QQ;
    end
    notsteady = ~(max(max(abs(K-oldK)))<riccati_tol);
    oldK = K;
end

if F_singular == 1
    error(['The variance of the forecast error remains singular until the end of the sample'])
end

if t < smpl
    t0 = t;
    while t<smpl
        t = t+1;
        v = Y(:,t)-Z*a;
        a = T*(a+K*v);
        lik(t) = v'*iF*v;
    end
    lik(t0:smpl) = lik(t0:smpl) + log(dF);
end

lik = (lik + pp*log(2*pi))/2;

LIK = sum(lik(start:end)); % Minus the log-likelihood.