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## -*- texinfo -*-
## @deftypefn  {} {@var{y} =} polyval (@var{p}, @var{x})
## @deftypefnx {} {@var{y} =} polyval (@var{p}, @var{x}, [], @var{mu})
## @deftypefnx {} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{s})
## @deftypefnx {} {[@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}.
##
## If @var{x} is a vector or matrix, the polynomial is evaluated for each of
## the elements of @var{x}.
##
## When @var{mu} is present, evaluate the polynomial for
## @w{(@var{x} - @var{mu}(1)) / @var{mu}(2)}.
##
## 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

function [y, dy] = polyval (p, x, s = [], mu)

  if (nargin < 2 || (nargout == 2 && nargin < 3))
    print_usage ();
  endif

  ## Algorithm requires floating point values
  if (! isfloat (p) || (! isvector (p) && ! isempty (p)))
    error ("polyval: P must be a numeric floating point vector");
  endif
  if (! isfloat (x))
    error ("polyval: X must be numeric floating point");
  endif

  if (nargout > 1)
    if (isempty (s))
      error ("polyval: S input is required for DY output argument");
    elseif (isstruct (s))
      if (! all (ismember ({"R", "normr", "df"}, fieldnames (s))))
        error ("polyval: S input is missing required fields");
      endif
    else
      error ("polyval: S input must be a structure");
    endif
  endif

  if (nargin == 4 && (! isfloat (mu) || numel (mu) < 2))
    error ("polyval: MU must be numeric floating point with 2 values");
  endif

  if (isempty (p) || isempty (x))
    if (isa (p, "single") || isa (x, "single"))
      y = zeros (size (x), "single");
    else
      y = zeros (size (x));
    endif
    return;
  endif

  if (nargin == 4)
    x = (x - mu(1)) / mu(2);
  endif

  n = numel (p) - 1;
  y = p(1) * ones (size (x), class (x));
  for i = 2:n+1
    y = y .* x + p(i);
  endfor

  if (nargout > 1)
    ## 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.
    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));
  endif

endfunction


%!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]);

## Test empty combinations
%!assert (polyval ([], 1:10), zeros (1, 10))
%!assert (class (polyval (single ([]), 1:10)), "single")
%!assert (class (polyval ([], single (1:10))), "single")
%!assert (polyval (1, []), [])
%!assert (polyval ([], []), [])
%!assert (polyval (1, zeros (0,3)), zeros (0, 3))
%!assert (class (polyval (single (1), [])), "single")
%!assert (class (polyval (1, single ([]))), "single")
%!assert (class (polyval (single ([]), [])), "single")
%!assert (class (polyval ([], single ([]))), "single")

## Test input validation
%!error <Invalid call> polyval ()
%!error <Invalid call> polyval (1)
%!error <Invalid call> [y, dy] = polyval (1, 2)
%!error <P must be a numeric floating point vector> polyval ({1, 0}, 0:10)
%!error <P must be a numeric floating point vector> polyval (int8 ([1]), 0:10)
%!error <P must be a numeric floating point vector> polyval ([1,0;0,1], 0:10)
%!error <X must be numeric floating point> polyval ([1,0], {0:10})
%!error <X must be numeric floating point> polyval ([1,0], int8 (0:10))
%!error <S input is required> [y, dy] = polyval (1, 1, [])
%!error <S input is missing required fields>
%! [y, dy] = polyval (1, 1, struct ("T", 0, "normr", 1, "df", 2));
%!error <S input must be a structure> [y, dy] = polyval (1, 1, 2)
%!error <MU must be numeric floating point with 2 values>
%! polyval (1, 1, [], {1, 2});
%!error <MU must be numeric floating point with 2 values>
%! polyval (1, 1, [], int8 ([1,2]));
%!error <MU must be numeric floating point with 2 values>
%! polyval (1, 1, [], [1]);