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########################################################################
##
## Copyright (C) 1994-2025 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- 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])
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