% dmFirst3Pt.m
%          
%  1st order 3 point approximation (differentiated quadratic interpolating polynomial)
%   to the 1st derivative on a uniform grid

%       x    the grid
%      dm    differentiation matrix

% Note: test on a unit spaced grid
%     x = linspace(0,10,11)
%    dm = dmFirst3Pt(x) 

%  the weights come from:
%
%  >> C = finiteDifferenceWeights(1,[0 1 2],1)
% C =
%         0    1.0000         0
%   -0.5000         0    0.5000
%>> C = finiteDifferenceWeights(0,[0 1 2],1)
% C =
%    1.0000         0         0
%   -1.5000    2.0000   -0.5000
%>> C = finiteDifferenceWeights(2,[0 1 2],1)
% C =
%         0         0    1.0000
%    0.5000   -2.0000    1.5000


 function dm = dmFirst3Pt(x) 

     N = length(x);
     h = abs(x(2)-x(1));
     
     up = diag(0.5*ones(1,N),1);
     low = diag(-0.5*ones(1,N),-1);
     
     
     dm = diag(0*ones(1,N)) + up(2:end,2:end) + low(2:end,2:end);
     
     dm(1,1) = -1.5;  dm(1,2) = 2;  dm(1,3) = -0.5;
     dm(N,N-2) = 0.5;  dm(N,N-1) = -2;  dm(N,N) =  1.5;
     
     dm = dm./h;