% localPolynomialFiniteDifferenceStencilExample

clear, home, format compact

% weights for approximating the first derivative by
% fitting a second degree polynomial through 3 points
%    p_2(x) = a2*x^2 + a1*x + a0

  stencil = [0 1 2];       %  h=1
  center = 1;              %  must be a point in the stencil
  
   B = vander(stencil)     %  basis functions 
                           %       x^2, x, and 1 (columns)
                           %  evaluated at each point (rows) of the stencil
                           
%    H = [2 0 0];          %  second derivative           
   
   H = [2*center 1 0]            %  differentiated basis function 
                                 %      2x, 1, and 0
                                 %  at the evaluation point (center)
                                 %    p_2^\prime = 2x*a2 + a1 + 0*a0
  
  w = H/B                        %   w = H*inv(B) 
  
  
  % now the weights can be used as f_x(x*) = w*f(stencil)    
  

  format