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maint: Merge stable to default.
author John W. Eaton <jwe@octave.org>
date Wed, 22 Feb 2017 12:58:07 -0500
parents ef4d915df748 3ac9f9ecfae5
children 194eb4bd202b
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1 ## Copyright (C) 1994-2017 John W. Eaton
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or
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8 ## (at your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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13 ## GNU General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {} {} poly (@var{A})
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21 ## @deftypefnx {} {} poly (@var{x})
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22 ## If @var{A} is a square @math{N}-by-@math{N} matrix, @code{poly (@var{A})}
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23 ## is the row vector of the coefficients of @code{det (z * eye (N) - A)},
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24 ## the characteristic polynomial of @var{A}.
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25 ##
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26 ## For example, the following code finds the eigenvalues of @var{A} which are
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27 ## the roots of @code{poly (@var{A})}.
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28 ##
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29 ## @example
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30 ## @group
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31 ## roots (poly (eye (3)))
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32 ## @result{} 1.00001 + 0.00001i
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33 ## 1.00001 - 0.00001i
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34 ## 0.99999 + 0.00000i
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35 ## @end group
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36 ## @end example
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37 ##
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38 ## In fact, all three eigenvalues are exactly 1 which emphasizes that for
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39 ## numerical performance the @code{eig} function should be used to compute
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40 ## eigenvalues.
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41 ##
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42 ## If @var{x} is a vector, @code{poly (@var{x})} is a vector of the
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43 ## coefficients of the polynomial whose roots are the elements of @var{x}.
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44 ## That is, if @var{c} is a polynomial, then the elements of
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45 ## @code{@var{d} = roots (poly (@var{c}))} are contained in @var{c}. The
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46 ## vectors @var{c} and @var{d} are not identical, however, due to sorting and
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47 ## numerical errors.
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48 ## @seealso{roots, eig}
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49 ## @end deftypefn
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50
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51 ## Author: KH <Kurt.Hornik@wu-wien.ac.at>
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52 ## Created: 24 December 1993
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53 ## Adapted-By: jwe
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54
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55 function y = poly (x)
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57 if (nargin != 1)
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58 print_usage ();
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59 endif
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60
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61 m = min (size (x));
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62 n = max (size (x));
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63 if (m == 0)
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64 y = 1;
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65 return;
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66 elseif (m == 1)
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67 v = x;
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68 elseif (m == n)
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69 v = eig (x);
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70 else
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71 print_usage ();
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72 endif
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73
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74 y = zeros (1, n+1);
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75 y(1) = 1;
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76 for j = 1:n;
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77 y(2:(j+1)) = y(2:(j+1)) - v(j) .* y(1:j);
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78 endfor
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79
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80 if (all (all (imag (x) == 0)))
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81 y = real (y);
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82 endif
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83
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84 endfunction
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87 %!assert (poly ([]), 1)
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88 %!assert (poly ([1, 2, 3]), [1, -6, 11, -6])
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89 %!assert (poly ([1, 2; 3, 4]), [1, -5, -2], sqrt (eps))
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91 %!error poly ([1, 2, 3; 4, 5, 6])