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########################################################################
##
## Copyright (C) 1994-2024 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{r} =} roots (@var{c})
##
## Compute the roots of the polynomial @var{c}.
##
## For a vector @var{c} with @math{N} components, return the roots of the
## polynomial
## @tex
## $$
## c_1 x^{N-1} + \cdots + c_{N-1} x + c_N.
## $$
## @end tex
## @ifnottex
##
## @example
## c(1) * x^(N-1) + @dots{} + c(N-1) * x + c(N)
## @end example
##
## @end ifnottex
##
## As an example, the following code finds the roots of the quadratic
## polynomial
## @tex
## $$ p(x) = x^2 - 5. $$
## @end tex
## @ifnottex
##
## @example
## p(x) = x^2 - 5.
## @end example
##
## @end ifnottex
##
## @example
## @group
## c = [1, 0, -5];
## roots (c)
## @result{} 2.2361
## @result{} -2.2361
## @end group
## @end example
##
## Note that the true result is
## @tex
## $\pm \sqrt{5}$
## @end tex
## @ifnottex
## @math{+/- sqrt(5)}
## @end ifnottex
## which is roughly
## @tex
## $\pm 2.2361$.
## @end tex
## @ifnottex
## @math{+/- 2.2361}.
## @end ifnottex
## @seealso{poly, compan, fzero}
## @end deftypefn
function r = roots (c)
if (nargin < 1 || (! isvector (c) && ! isempty (c)))
print_usage ();
elseif (any (! isfinite (c)))
error ("roots: inputs must not contain Inf or NaN");
endif
c = c(:);
n = numel (c);
## If c = [ 0 ... 0 c(k+1) ... c(k+l) 0 ... 0 ],
## we can remove the leading k zeros,
## and n - k - l roots of the polynomial are zero.
c_max = max (abs (c));
if (isempty (c) || c_max == 0)
r = [];
return;
endif
f = find (c ./ c_max);
m = numel (f);
c = c(f(1):f(m));
l = numel (c);
if (l > 1)
A = diag (ones (1, l-2), -1);
A(1,:) = -c(2:l) ./ c(1);
r = eig (A);
if (f(m) < n)
r = [r; zeros(n - f(m), 1)];
endif
else
r = zeros (n - f(m), 1);
endif
endfunction
%!test
%! p = [poly([3 3 3 3]), 0 0 0 0];
%! r = sort (roots (p));
%! assert (r, [0; 0; 0; 0; 3; 3; 3; 3], 0.001);
%!assert (isempty (roots ([])))
%!assert (isempty (roots ([0 0])))
%!assert (isempty (roots (1)))
%!assert (roots ([1, -6, 11, -6]), [3; 2; 1], sqrt (eps))
%!assert (roots ([1e-200, -1e200, 1]), 1e-200)
%!assert (roots ([1e-200, -1e200 * 1i, 1]), -1e-200 * 1i)
%!error <Invalid call> roots ()
%!error roots ([1, 2; 3, 4])
%!error <inputs must not contain Inf or NaN> roots ([1 Inf 1])
%!error <inputs must not contain Inf or NaN> roots ([1 NaN 1])
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