function [M] = solve_direct(M)
% We solve the system directly for a grid of 5*5 where there are 4
% equations for points of F and one equation for a point of G.
% The boundaries are fixed and moreover they are zero since by now we are
% dealing just with the error (zero boundary)
%
% In total we get 4 dependent equations for F and adding the G equation we
% get a fully determined system with 5 equations.

s = size(M); rows = s(1); cols = s(2);
h = .25;

if (rows ~= 5)
    'Error: can not solve directly.'
else
% We solve the equations here directly 
%=======================
% The F,G points
a = M(2,2) ; b = M(2,4) ; c = M(4,2) ; d = M(4,4) ; e = M(3,3) ; 
if (abs(a+b+c+d) > exp(-5))
    'Error[In solve_direct.m]: abs(a+b+c+d) > exp(-5)'
    
end
% The expanded equations
% =======================
M(3,2) = 2*(a/2 + b/4   - c/4   + e/4);
M(3,4) = 2*(a/2 + b*.75 + c/4   - e/4);
M(2,3) = 2*(a/2 - b/4   + c/4   - e/4);
M(4,3) = 2*(a/2 + b/4   + c*.75 + e/4);
end








%    [M(3,2),M(3,4),M(2,3),m(4,3)] = solve('M(3,2) + M(2,3) = M(2,2)*2*h', 'M(3,4) - M(2,3) = M(2,4)*2*h', '-M(3,2)+ M(4,3) = M(4,2)*2*h', '-M(3,4)- M(4,3) = M(4,4)*2*h', 'M(4,3) - M(2,3) -M(3,4) + M(3,2) = M(3,3)*2*h');
%    M_2_2 = M(2,2) ; M_2_4 = M(2,4) ; M_4_2 = M(4,2) ; M_4_4 = M(4,4) ; M_3_3 = M(3,3);
%   solve('M_3_2 + M_2_3 = M_2_2*2*h', 'M_3_4 - M_2_3 = M_2_4*2*h', '-M_3_2+ M_4_3 = M_4_2*2*h', '-M_3_4- M_4_3 = M_4_4*2*h', 'M_4_3 - M_2_3 -M_3_4 + M_3_2 = M_3_3*2*h','M_3_2,M_3_4,M_2_3,M_4_3' )