Mercurial > octave
view scripts/polynomial/roots.m @ 33567:9f0f7a898b73 bytecode-interpreter tip
maint: Merge default to bytecode-interpreter
author | Arun Giridhar <arungiridhar@gmail.com> |
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date | Fri, 10 May 2024 17:57:29 -0400 |
parents | 2e484f9f1f18 |
children |
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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])