time   calls  line
                  1 function [p,S,mu] = polyfit(x,y,n)
                  2 %POLYFIT Fit polynomial to data.
                  3 %   P = POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of
                  4 %   degree N that fits the data Y best in a least-squares sense. P is a
                  5 %   row vector of length N+1 containing the polynomial coefficients in
                  6 %   descending powers, P(1)*X^N + P(2)*X^(N-1) +...+ P(N)*X + P(N+1).
                  7 %
                  8 %   [P,S] = POLYFIT(X,Y,N) returns the polynomial coefficients P and a
                  9 %   structure S for use with POLYVAL to obtain error estimates for
                 10 %   predictions.  S contains fields for the triangular factor (R) from a QR
                 11 %   decomposition of the Vandermonde matrix of X, the degrees of freedom
                 12 %   (df), and the norm of the residuals (normr).  If the data Y are random,
                 13 %   an estimate of the covariance matrix of P is (Rinv*Rinv')*normr^2/df,
                 14 %   where Rinv is the inverse of R.
                 15 %
                 16 %   [P,S,MU] = POLYFIT(X,Y,N) finds the coefficients of a polynomial in
                 17 %   XHAT = (X-MU(1))/MU(2) where MU(1) = MEAN(X) and MU(2) = STD(X). This
                 18 %   centering and scaling transformation improves the numerical properties
                 19 %   of both the polynomial and the fitting algorithm.
                 20 %
                 21 %   Warning messages result if N is >= length(X), if X has repeated, or
                 22 %   nearly repeated, points, or if X might need centering and scaling.
                 23 %
                 24 %   Class support for inputs X,Y:
                 25 %      float: double, single
                 26 %
                 27 %   See also POLY, POLYVAL, ROOTS, LSCOV.
                 28 
                 29 %   Copyright 1984-2010 The MathWorks, Inc.
                 30 %   $Revision: 5.17.4.13 $  $Date: 2010/11/17 11:29:32 $
                 31 
                 32 % The regression problem is formulated in matrix format as:
                 33 %
                 34 %    y = V*p    or
                 35 %
                 36 %          3  2
                 37 %    y = [x  x  x  1] [p3
                 38 %                      p2
                 39 %                      p1
                 40 %                      p0]
                 41 %
                 42 % where the vector p contains the coefficients to be found.  For a
                 43 % 7th order polynomial, matrix V would be:
                 44 %
                 45 % V = [x.^7 x.^6 x.^5 x.^4 x.^3 x.^2 x ones(size(x))];
                 46 
           240   47 if ~isequal(size(x),size(y)) 
                 48     error(message('MATLAB:polyfit:XYSizeMismatch'))
                 49 end
                 50 
  0.01     240   51 x = x(:); 
           240   52 y = y(:); 
                 53 
           240   54 if nargout > 2 
                 55    mu = [mean(x); std(x)];
                 56    x = (x - mu(1))/mu(2);
                 57 end
                 58 
                 59 % Construct Vandermonde matrix.
           240   60 V(:,n+1) = ones(length(x),1,class(x)); 
           240   61 for j = n:-1:1 
                 62    V(:,j) = x.*V(:,j+1);
                 63 end
                 64 
                 65 % Solve least squares problem.
           240   66 [Q,R] = qr(V,0); 
  0.01     240   67 ws = warning('off','all');  
  0.01     240   68 p = R\(Q'*y);    % Same as p = V\y; 
           240   69 warning(ws); 
           240   70 if size(R,2) > size(R,1) 
                 71    warning(message('MATLAB:polyfit:PolyNotUnique'))
  0.16     240   72 elseif warnIfLargeConditionNumber(R) 
                 73     if nargout > 2
                 74         warning(message('MATLAB:polyfit:RepeatedPoints'));
                 75     else
                 76         warning('MATLAB:polyfit:RepeatedPointsOrRescale', ...
                 77                 ['Polynomial is badly conditioned. Add points with distinct X\n' ...
                 78                  '         values, reduce the degree of the polynomial, or try centering\n' ...
                 79                  '         and scaling as described in HELP POLYFIT.']);
                 80     end
                 81 end
                 82 
  0.01     240   83 if nargout > 1 
                 84     r = y - V*p;
                 85     % S is a structure containing three elements: the triangular factor from a
                 86     % QR decomposition of the Vandermonde matrix, the degrees of freedom and
                 87     % the norm of the residuals.
                 88     S.R = R;
                 89     S.df = max(0,length(y) - (n+1));
                 90     S.normr = norm(r);
                 91 end
                 92 
           240   93 p = p.';          % Polynomial coefficients are row vectors by convention. 

Other subfunctions in this file are not included in this listing.