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+## Copyright (C) 1996-2012 John W. Eaton
+##
+## This file is part of Octave.
+##
+## Octave is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; see the file COPYING. If not, see
+## .
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{p} =} polyfit (@var{x}, @var{y}, @var{n})
+## @deftypefnx {Function File} {[@var{p}, @var{s}] =} polyfit (@var{x}, @var{y}, @var{n})
+## @deftypefnx {Function File} {[@var{p}, @var{s}, @var{mu}] =} polyfit (@var{x}, @var{y}, @var{n})
+## Return the coefficients of a polynomial @var{p}(@var{x}) of degree
+## @var{n} that minimizes the least-squares-error of the fit to the points
+## @code{[@var{x}, @var{y}]}.
+##
+## The polynomial coefficients are returned in a row vector.
+##
+## The optional output @var{s} is a structure containing the following fields:
+##
+## @table @samp
+## @item R
+## Triangular factor R from the QR@tie{}decomposition.
+##
+## @item X
+## The Vandermonde matrix used to compute the polynomial coefficients.
+##
+## @item df
+## The degrees of freedom.
+##
+## @item normr
+## The norm of the residuals.
+##
+## @item yf
+## The values of the polynomial for each value of @var{x}.
+## @end table
+##
+## The second output may be used by @code{polyval} to calculate the
+## statistical error limits of the predicted values.
+##
+## When the third output, @var{mu}, is present the
+## coefficients, @var{p}, are associated with a polynomial in
+## @var{xhat} = (@var{x}-@var{mu}(1))/@var{mu}(2).
+## Where @var{mu}(1) = mean (@var{x}), and @var{mu}(2) = std (@var{x}).
+## This linear transformation of @var{x} improves the numerical
+## stability of the fit.
+## @seealso{polyval, polyaffine, roots, vander, zscore}
+## @end deftypefn
+
+## Author: KH
+## Created: 13 December 1994
+## Adapted-By: jwe
+
+function [p, s, mu] = polyfit (x, y, n)
+
+ if (nargin < 3 || nargin > 4)
+ print_usage ();
+ endif
+
+ if (nargout > 2)
+ ## Normalized the x values.
+ mu = [mean(x), std(x)];
+ x = (x - mu(1)) / mu(2);
+ endif
+
+ if (! size_equal (x, y))
+ error ("polyfit: X and Y must be vectors of the same size");
+ endif
+
+ if (! (isscalar (n) && n >= 0 && ! isinf (n) && n == fix (n)))
+ error ("polyfit: N must be a non-negative integer");
+ endif
+
+ y_is_row_vector = (rows (y) == 1);
+
+ ## Reshape x & y into column vectors.
+ l = numel (x);
+ x = x(:);
+ y = y(:);
+
+ ## Construct the Vandermonde matrix.
+ v = vander (x, n+1);
+
+ ## Solve by QR decomposition.
+ [q, r, k] = qr (v, 0);
+ p = r \ (q' * y);
+ p(k) = p;
+
+ if (nargout > 1)
+ yf = v*p;
+
+ if (y_is_row_vector)
+ s.yf = yf.';
+ else
+ s.yf = yf;
+ endif
+
+ s.R = r;
+ s.X = v;
+ s.df = l - n - 1;
+ s.normr = norm (yf - y);
+ endif
+
+ ## Return a row vector.
+ p = p.';
+
+endfunction
+
+%!test
+%! x = [-2, -1, 0, 1, 2];
+%! assert(all (all (abs (polyfit (x, x.^2+x+1, 2) - [1, 1, 1]) < sqrt (eps))));
+
+%!error(polyfit ([1, 2; 3, 4], [1, 2, 3, 4], 2))
+
+%!test
+%! x = [-2, -1, 0, 1, 2];
+%! assert(all (all (abs (polyfit (x, x.^2+x+1, 3) - [0, 1, 1, 1]) < sqrt (eps))));
+
+%!test
+%! x = [-2, -1, 0, 1, 2];
+%! fail("polyfit (x, x.^2+x+1)");
+
+%!test
+%! x = [-2, -1, 0, 1, 2];
+%! fail("polyfit (x, x.^2+x+1, [])");
+
+## Test difficult case where scaling is really needed. This example
+## demonstrates the rather poor result which occurs when the dependent
+## variable is not normalized properly.
+## Also check the usage of 2nd & 3rd output arguments.
+%!test
+%! x = [ -1196.4, -1195.2, -1194, -1192.8, -1191.6, -1190.4, -1189.2, -1188, \
+%! -1186.8, -1185.6, -1184.4, -1183.2, -1182];
+%! y = [ 315571.7086, 315575.9618, 315579.4195, 315582.6206, 315585.4966, \
+%! 315588.3172, 315590.9326, 315593.5934, 315596.0455, 315598.4201, \
+%! 315600.7143, 315602.9508, 315605.1765 ];
+%! [p1, s1] = polyfit (x, y, 10);
+%! [p2, s2, mu] = polyfit (x, y, 10);
+%! assert (s2.normr < s1.normr)
+
+%!test
+%! x = 1:4;
+%! p0 = [1i, 0, 2i, 4];
+%! y0 = polyval (p0, x);
+%! p = polyfit (x, y0, numel(p0)-1);
+%! assert (p, p0, 1000*eps)
+
+%!test
+%! x = 1000 + (-5:5);
+%! xn = (x - mean (x)) / std (x);
+%! pn = ones (1,5);
+%! y = polyval (pn, xn);
+%! [p, s, mu] = polyfit (x, y, numel(pn)-1);
+%! [p2, s2] = polyfit (x, y, numel(pn)-1);
+%! assert (p, pn, s.normr)
+%! assert (s.yf, y, s.normr)
+%! assert (mu, [mean(x), std(x)])
+%! assert (s.normr/s2.normr < sqrt(eps))
+
+%!test
+%! x = [1, 2, 3; 4, 5, 6];
+%! y = [0, 0, 1; 1, 0, 0];
+%! p = polyfit (x, y, 5);
+%! expected = [0, 1, -14, 65, -112, 60]/12;
+%! assert (p, expected, sqrt(eps))
+
+