function [p,S,mu] = polyfit(x,y,n)
% POLYFIT Fit polynomial to data.
% P = POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of
% degree N that fits the data Y best in a least-squares sense. P is a
% row vector of length N+1 containing the polynomial coefficients in
% descending powers, P(1)*X^N + P(2)*X^(N-1) +...+ P(N)*X + P(N+1).
%
% [P,S] = POLYFIT(X,Y,N) returns the polynomial coefficients P and a
% structure S for use with POLYVAL to obtain error estimates for
% predictions. S contains fields for the triangular factor (R) from a QR
% decomposition of the Vandermonde matrix of X, the degrees of freedom
% (df), and the norm of the residuals (normr). If the data Y are random,
% an estimate of the covariance matrix of P is (Rinv*Rinv')*normr^2/df,
% where Rinv is the inverse of R.
%
% [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.
%
% Class support for inputs X,Y:
% float: double, single
%
% See also POLY, POLYVAL, ROOTS.
% Copyright 1984-2005 The MathWorks, Inc.
% $Revision: 5.17.4.8 $ $Date: 2006/06/20 20:11:56 $
% 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('MATLAB:polyfit:XYSizeMismatch',...
'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,class(x));
for j = n:-1:1
V(:,j) = x.*V(:,j+1);
end
% Solve least squares problem.
[Q,R] = qr(V,0);
p = R\(Q'*y); % Same as p = V\y;
r = y - V*p;
p = p.'; % Polynomial coefficients are row vectors by convention.
% S is a structure containing three elements: the triangular factor from a
% QR decomposition of the Vandermonde matrix, the degrees of freedom and
% the norm of the residuals.
S.R = R;
S.df = max(0,length(y) - (n+1));
S.normr = norm(r);