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[CreaPhase.git] / octave_packages / m / polynomial / roots.m

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+## Copyright (C) 1994-2012 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} {} roots (@var{v})
+##
+## For a vector @var{v} with @math{N} components, return
+## the roots of the polynomial
+## @tex
+## $$
+## v_1 z^{N-1} + \cdots + v_{N-1} z + v_N.
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## v(1) * z^(N-1) + @dots{} + v(N-1) * z + v(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
+
+## Author: KH <Kurt.Hornik@wu-wien.ac.at>
+## Created: 24 December 1993
+## Adapted-By: jwe
+
+function r = roots (v)
+
+  if (nargin != 1 || min (size (v)) > 1)
+    print_usage ();
+  elseif (any (isnan(v) | isinf(v)))
+    error ("roots: inputs must not contain Inf or NaN");
+  endif
+
+  n = numel (v);
+  v = v(:);
+
+  ## If v = [ 0 ... 0 v(k+1) ... v(k+l) 0 ... 0 ], we can remove the
+  ## leading k zeros and n - k - l roots of the polynomial are zero.
+
+  if (isempty (v))
+    f = v;
+  else
+    f = find (v ./ max (abs (v)));
+  endif
+  m = numel (f);
+
+  if (m > 0 && n > 1)
+    v = v(f(1):f(m));
+    l = max (size (v));
+    if (l > 1)
+      A = diag (ones (1, l-2), -1);
+      A(1,:) = -v(2:l) ./ v(1);
+      r = eig (A);
+      if (f(m) < n)
+        tmp = zeros (n - f(m), 1);
+        r = [r; tmp];
+      endif
+    else
+      r = zeros (n - f(m), 1);
+    endif
+  else
+    r = [];
+  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(all (all (abs (roots ([1, -6, 11, -6]) - [3; 2; 1]) < sqrt (eps))));
+
+%!assert(isempty (roots ([])));
+
+%!error roots ([1, 2; 3, 4]);
+
+%!assert(isempty (roots (1)));
+
+%!error roots ([1, 2; 3, 4]);
+
+%!error roots ([1 Inf 1]);
+
+%!error roots ([1 NaN 1]);
+
+%!assert(roots ([1e-200, -1e200, 1]), 1e-200)
+%!assert(roots ([1e-200, -1e200 * 1i, 1]), -1e-200 * 1i)