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## Copyright (C) 1994-2013 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
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{y} =} polyval (@var{p}, @var{x})
## @deftypefnx {Function File} {@var{y} =} polyval (@var{p}, @var{x}, [], @var{mu})
## @deftypefnx {Function File} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{s})
## @deftypefnx {Function File} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{s}, @var{mu})
##
## Evaluate the polynomial @var{p} at the specified values of @var{x}. When
## @var{mu} is present, evaluate the polynomial for
## (@var{x}-@var{mu}(1))/@var{mu}(2).
## If @var{x} is a vector or matrix, the polynomial is evaluated for each of
## the elements of @var{x}.
##
## In addition to evaluating the polynomial, the second output
## represents the prediction interval, @var{y} +/- @var{dy}, which
## contains at least 50% of the future predictions. To calculate the
## prediction interval, the structured variable @var{s}, originating
## from @code{polyfit}, must be supplied.
##
## @seealso{polyvalm, polyaffine, polyfit, roots, poly}
## @end deftypefn
## Author: Tony Richardson <arichard@stark.cc.oh.us>
## Created: June 1994
## Adapted-By: jwe
function [y, dy] = polyval (p, x, s = [], mu)
if (nargin < 2 || nargin > 4 || (nargout == 2 && nargin < 3))
print_usage ();
endif
if (isempty (x))
y = [];
return;
elseif (isempty (p))
y = zeros (size (x));
return;
elseif (! isvector (p))
error ("polyval: first argument must be a vector");
endif
if (nargin > 3)
x = (x - mu(1)) / mu(2);
endif
n = length (p) - 1;
y = p(1) * ones (size (x));
for i = 2:n+1
y = y .* x + p(i);
endfor
if (nargout == 2)
## Note: the F-Distribution is generally considered to be single-sided.
## http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm
## t = finv (1-alpha, s.df, s.df);
## dy = t * sqrt (1 + sumsq (A/s.R, 2)) * s.normr / sqrt (s.df)
## If my inference is correct, then t must equal 1 for polyval.
## This is because finv (0.5, n, n) = 1.0 for any n.
try
k = numel (x);
A = (x(:) * ones (1, n+1)) .^ (ones (k, 1) * (n:-1:0));
dy = sqrt (1 + sumsq (A/s.R, 2)) * s.normr / sqrt (s.df);
dy = reshape (dy, size (x));
catch
if (isempty (s))
error ("polyval: third input is required.");
elseif (isstruct (s)
&& all (ismember ({"R", "normr", "df"}, fieldnames (s))))
error (lasterr ());
elseif (isstruct (s))
error ("polyval: third input is missing the required fields.");
else
error ("polyval: third input is not a structure.");
endif
end_try_catch
endif
endfunction
%!fail ("polyval ([1,0;0,1],0:10)")
%!test
%! r = 0:10:50;
%! p = poly (r);
%! p = p / max (abs (p));
%! x = linspace (0,50,11);
%! y = polyval (p,x) + 0.25*sin (100*x);
%! [pf, s] = polyfit (x, y, numel (r));
%! [y1, delta] = polyval (pf, x, s);
%! expected = [0.37235, 0.35854, 0.32231, 0.32448, 0.31328, ...
%! 0.32036, 0.31328, 0.32448, 0.32231, 0.35854, 0.37235];
%! assert (delta, expected, 0.00001);
%!test
%! x = 10 + (-2:2);
%! y = [0, 0, 1, 0, 2];
%! p = polyfit (x, y, numel (x) - 1);
%! [pn, s, mu] = polyfit (x, y, numel (x) - 1);
%! y1 = polyval (p, x);
%! yn = polyval (pn, x, [], mu);
%! assert (y1, y, sqrt (eps));
%! assert (yn, y, sqrt (eps));
%!test
%! p = [0, 1, 0];
%! x = 1:10;
%! assert (x, polyval (p,x), eps);
%! x = x(:);
%! assert (x, polyval (p,x), eps);
%! x = reshape (x, [2, 5]);
%! assert (x, polyval (p,x), eps);
%! x = reshape (x, [5, 2]);
%! assert (x, polyval (p,x), eps);
%! x = reshape (x, [1, 1, 5, 2]);
%! assert (x, polyval (p,x), eps);
%!test
%! p = [1];
%! x = 1:10;
%! y = ones (size (x));
%! assert (y, polyval (p,x), eps);
%! x = x(:);
%! y = ones (size (x));
%! assert (y, polyval (p,x), eps);
%! x = reshape (x, [2, 5]);
%! y = ones (size (x));
%! assert (y, polyval (p,x), eps);
%! x = reshape (x, [5, 2]);
%! y = ones (size (x));
%! assert (y, polyval (p,x), eps);
%! x = reshape (x, [1, 1, 5, 2]);
%!assert (zeros (1, 10), polyval ([], 1:10))
%!assert ([], polyval (1, []))
%!assert ([], polyval ([], []))
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