function [V, Q, policy] = mdp_bellman_operator_with_Q(P, PR, discount, Vprev)


% mdp_bellman_operator Applies the Bellman operator on the value function Vprev
%                      Returns a new value function and a Vprev-improving policy
% Arguments ---------------------------------------------------------------
% Let S = number of states, A = number of actions
%   P(SxSxA) = transition matrix
%              P could be an array with 3 dimensions or 
%              a cell array (1xA), each cell containing a matrix (SxS) possibly sparse
%   PR(SxA) = reward matrix
%              PR could be an array with 2 dimensions or 
%              a sparse matrix
%   discount = discount rate, in ]0, 1]
%   Vprev(S) = value function
% Evaluation --------------------------------------------------------------
%   V(S)   = new value function
%   policy(S) = Vprev-improving policy

% MDPtoolbox: Markov Decision Processes Toolbox
% Copyright (C) 2009  INRA
% Redistribution and use in source and binary forms, with or without modification, 
% are permitted provided that the following conditions are met:
%    * Redistributions of source code must retain the above copyright notice, 
%      this list of conditions and the following disclaimer.
%    * Redistributions in binary form must reproduce the above copyright notice, 
%      this list of conditions and the following disclaimer in the documentation 
%      and/or other materials provided with the distribution.
%    * Neither the name of the <ORGANIZATION> nor the names of its contributors 
%      may be used to endorse or promote products derived from this software 
%      without specific prior written permission.
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND 
% ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED 
% WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
% IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT,
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% BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 
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% OF THE POSSIBILITY OF SUCH DAMAGE.


if iscell(P); S = size(P{1},1); else S = size(P,1); end;
if discount <= 0 || discount > 1
     disp('--------------------------------------------------------')
     disp('MDP Toolbox ERROR: Discount rate must be in ]0; 1]')
     disp('--------------------------------------------------------')
elseif size(Vprev,1) ~= S
    disp('--------------------------------------------------------')
    disp('MDP Toolbox ERROR: Vprev must have the same dimension as P')
    disp('--------------------------------------------------------')
else
        
    if iscell(P)
        A = length(P);
        for a=1:A           
            Q(:,a) = PR(:,a) + discount*P{a}*Vprev;
        end
    else
        A = size(PR,2);
        for a=1:A
            Q(:,a) = PR(:,a) + discount*P(:,:,a)*Vprev;
        end
    end
    [V, policy] = max(Q,[],2);
 
end; 

