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## -*- texinfo -*-
## @deftypefn  {} {@var{y} =} poly (@var{A})
## @deftypefnx {} {@var{y} =} poly (@var{x})
## If @var{A} is a square @math{N}-by-@math{N} matrix, @code{poly (@var{A})}
## is the row vector of the coefficients of @code{det (z * eye (N) - A)},
## the characteristic polynomial of @var{A}.
##
## For example, the following code finds the eigenvalues of @var{A} which are
## the roots of @code{poly (@var{A})}.
##
## @example
## @group
## roots (poly (eye (3)))
##     @result{} 1.00001 + 0.00001i
##        1.00001 - 0.00001i
##        0.99999 + 0.00000i
## @end group
## @end example
##
## In fact, all three eigenvalues are exactly 1 which emphasizes that for
## numerical performance the @code{eig} function should be used to compute
## eigenvalues.
##
## If @var{x} is a vector, @code{poly (@var{x})} is a vector of the
## coefficients of the polynomial whose roots are the elements of @var{x}.
## That is, if @var{c} is a polynomial, then the elements of
## @code{@var{d} = roots (poly (@var{c}))} are contained in @var{c}.  The
## vectors @var{c} and @var{d} are not identical, however, due to sorting and
## numerical errors.
## @seealso{roots, eig}
## @end deftypefn

function y = poly (x)

  if (nargin < 1)
    print_usage ();
  endif

  if (! isnumeric (x))
    error ("poly: input must be numeric, not type %s", class (x));
  elseif (ndims (x) > 2)
    error ("poly: input must be a vector or a square matrix");
  elseif (isempty (x))
    y = 1;
    return;
  elseif (isvector (x))
    n = numel (x);
    v = x;
  elseif (! issquare (x))
    error ("poly: input matrix must be square");
  else
    n = size (x,1);
    v = eig (x);
  endif

  y = zeros (1, n+1);
  y(1) = 1;
  for j = 1:n
    y(2:(j+1)) -= v(j) .* y(1:j);
  endfor

  ## Real, or complex conjugate inputs, should result in real output
  if (isreal (x))
    y = real (y);
  else
    pos_imag = sort (v(imag (v) > 0));
    neg_imag = sort (conj (v(imag (v) < 0)));
    if (size_equal (pos_imag, neg_imag))
      is_equal = (pos_imag == neg_imag);
      if (! isempty (is_equal) && all (is_equal))
        y = real (y);
      endif
    endif
  endif

endfunction


%!assert (poly ([]), 1)
%!assert (poly ([1, 2, 3]), [1, -6, 11, -6])
%!assert (poly ([1, 2; 3, 4]), [1, -5, -2], sqrt (eps))

%!test <*53897>
%! x = [1, sqrt(2)/2+sqrt(2)/2*i, 1i, -sqrt(2)/2+sqrt(2)/2*i, -1, ...
%!      -sqrt(2)/2-sqrt(2)/2*i, -1i, sqrt(2)/2-sqrt(2)/2*i];
%! y = poly (x);
%! assert (isreal (y), true);

%!test <*53897>
%! x = [1 + 1i, 1 + 2i, 3, 4];
%! y = poly (x);
%! assert (y, [1 + 0i, -9 - 3i, 25 + 24i, -17 - 57i, -12 + 36i]);

%!error <Invalid call> poly ()
%!error <input must be numeric> poly ("foo")
%!error <input must be numeric> poly ({1, "foo"; "bar", 1})
%!error <input must be a vector or a square matrix> poly (ones (2, 2, 2))
%!error <matrix must be square> poly ([1, 2, 3; 4, 5, 6])
