function [p,S,mu] = myPolyfit(x,y,n) 
%   POLYFIT Fit polynomial to data. 
%   POLYFIT(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. 
% 
%   [P,S] = POLYFIT(X,Y,N) returns the polynomial coefficients P and a 
%   structure S for use with POLYVAL to obtain error estimates on 
%   predictions.  If the errors in the data, Y, are independent normal 
%   with constant variance, POLYVAL will produce error bounds which 
%   contain at least 50% of the predictions. 
% 
%   The structure S contains the Cholesky factor of the Vandermonde 
%   matrix (R), the degrees of freedom (df), and the norm of the 
%   residuals (normr) as fields.    
% 
%   [P,S,MU] = POLYFIT(X,Y,N) finds the coefficients of a polynomial 
%   in XHAT = (X-MU(1))/MU(2) where MU(1) = mean(X) and MU(2) = std(X). 
%   This centering and scaling transformation improves the numerical 
%   properties of both the polynomial and the fitting algorithm. 
% 
%   Warning messages result if N is >= length(X), if X has repeated, or 
%   nearly repeated, points, or if X might need centering and scaling. 
% 
%   See also POLY, POLYVAL, ROOTS. 
 
%   Copyright 1984-2002 The MathWorks, Inc. 
%   $Revision: 5.17 $  $Date: 2002/04/09 00:14:25 $ 
 
% The regression problem is formulated in matrix format as: 
% 
%    y = V*p    or 
% 
%          3  2 
%    y = [x  x  x  1] [p3 
%                      p2 
%                      p1 
%                      p0] 
% 
% where the vector p contains the coefficients to be found.  For a 
% 7th order polynomial, matrix V would be: 
% 
% V = [x.^7 x.^6 x.^5 x.^4 x.^3 x.^2 x ones(size(x))]; 
 
if ~isequal(size(x),size(y)) 
    error('X and Y vectors must be the same size.') 
end 
 
x = x(:); 
y = y(:); 
 
if nargout > 2 
   mu = [mean(x); std(x)]; 
   x = (x - mu(1))/mu(2); 
end 
 
% Construct Vandermonde matrix. 
V(:,n+1) = ones(length(x),1); 
for j = n:-1:1 
   V(:,j) = x.*V(:,j+1); 
end 
 
% Solve least squares problem, and save the Cholesky factor. 
[Q,R] = qr(V,0); 
ws = warning('off','all');  
p = R\(Q'*y);    % Same as p = V\y; 
warning(ws); 
if size(R,2) > size(R,1) 
   warning('MATLAB:polyfit:PolyNotUnique', ... 
       'Polynomial is not unique; degree >= number of data points.') 
elseif condest(R) > 1.0e10 
    if nargout > 2 
        warning('MATLAB:polyfit:RepeatedPoints', ... 
            'Polynomial is badly conditioned. Remove repeated data points.') 
    else 
        warning('MATLAB:polyfit:RepeatedPointsOrRescale', ... 
            ['Polynomial is badly conditioned. Remove repeated data points\n' ... 
            '         or try centering and scaling as described in HELP POLYFIT.']) 
    end 
end 
r = y - V*p; 
p = p.';          % Polynomial coefficients are row vectors by convention. 
 
% S is a structure containing three elements: the Cholesky factor of the 
% Vandermonde matrix, the degrees of freedom and the norm of the residuals. 
 
S.R = R; 
S.df = length(y) - (n+1); 
S.normr = norm(r);