function a_coeff = polynomial_fit(x, y, n)

%POLYNOMIAL_FIT Fit polynomial to data.
%   POLYNOMIALFIT(X,Y,N) finds the coefficients of a polynomial P(X) of
%   degree N that fits the data, P(X(I))~=Y(I), in a least-squares sense.
%
%   Inputs:
%      X:     a vector of data points x (the independent variable)
%      Y:     a vector of data points y (the dependent variable) 
%      N:     the order of the polynomial
%
%   Outputs:
%      A_COEFF:  a vector of coefficients [a_n a_n-1 ... a_1 a_0] representing
%             the n-th order polynomial (a_i is the coefficient of x^i)
%
%  Assumptions:
%     -  if the degree of the polynomial is large and the data is not well
%     centered around the origin, then the algorithm may fail due to a
%     near-singular set of equations.  In this case, one should use the
%     polyfit function (see documentation)
%     - the variables x and y should have the same size.
%     - the degree of the polynomial must be >0


%% ME 2016 
%%
%% AUTHOR:  Chris Paredis
%%

% make sure x and y are both column vectors
x=x(:);
y=y(:);

% create the matrix, z, such that the set of equations:
% a_n*(x_i^n) + ... + a_1*x_i + a_0 = y_i
% can be represented as z*a=y
z = ones(length(x),n+1);
for i = 1:n
    z(:,i) = x.^(n+1-i);
end

% solve for the coefficients by solving the set of equations in a
% least-squares sense
a_coeff = (z\y)';