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1 ## Copyright (C) 1994, 1995, 1996, 1997, 1999, 2000, 2004, 2005, 2006, |
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2 ## 2007 John W. Eaton |
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3 ## |
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4 ## This file is part of Octave. |
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5 ## |
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6 ## Octave is free software; you can redistribute it and/or modify it |
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7 ## under the terms of the GNU General Public License as published by |
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8 ## the Free Software Foundation; either version 3 of the License, or (at |
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9 ## your option) any later version. |
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10 ## |
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11 ## Octave is distributed in the hope that it will be useful, but |
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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14 ## General Public License for more details. |
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15 ## |
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16 ## You should have received a copy of the GNU General Public License |
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17 ## along with Octave; see the file COPYING. If not, see |
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18 ## <http://www.gnu.org/licenses/>. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {} roots (@var{v}) |
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22 ## |
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23 ## For a vector @var{v} with @math{N} components, return |
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24 ## the roots of the polynomial |
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25 ## @iftex |
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26 ## @tex |
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27 ## $$ |
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28 ## v_1 z^{N-1} + \cdots + v_{N-1} z + v_N. |
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29 ## $$ |
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30 ## @end tex |
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31 ## @end iftex |
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32 ## @ifnottex |
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33 ## |
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34 ## @example |
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35 ## v(1) * z^(N-1) + ... + v(N-1) * z + v(N) |
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36 ## @end example |
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37 ## @end ifnottex |
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38 ## |
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39 ## As an example, the following code finds the roots of the quadratic |
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40 ## polynomial |
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41 ## @iftex |
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42 ## @tex |
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43 ## $$ p(x) = x^2 - 5. $$ |
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44 ## @end tex |
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45 ## @end iftex |
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46 ## @ifnottex |
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47 ## @example |
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48 ## p(x) = x^2 - 5. |
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49 ## @end example |
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50 ## @end ifnottex |
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51 ## @example |
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52 ## c = [1, 0, -5]; |
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53 ## roots(c) |
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54 ## @result{} 2.2361 |
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55 ## @result{} -2.2361 |
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56 ## @end example |
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57 ## Note that the true result is |
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58 ## @iftex |
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59 ## @tex |
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60 ## $\pm \sqrt{5}$ |
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61 ## @end tex |
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62 ## @end iftex |
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63 ## @ifnottex |
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64 ## @math{+/- sqrt(5)} |
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65 ## @end ifnottex |
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66 ## which is roughly |
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67 ## @iftex |
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68 ## @tex |
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69 ## $\pm 2.2361$. |
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70 ## @end tex |
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71 ## @end iftex |
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72 ## @ifnottex |
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73 ## @math{+/- 2.2361}. |
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74 ## @end ifnottex |
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75 ## @seealso{compan} |
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76 ## @end deftypefn |
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77 |
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78 ## Author: KH <Kurt.Hornik@wu-wien.ac.at> |
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79 ## Created: 24 December 1993 |
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80 ## Adapted-By: jwe |
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81 |
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82 function r = roots (v) |
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83 |
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84 if (nargin != 1 || min (size (v)) > 1) |
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85 print_usage (); |
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86 endif |
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87 |
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88 n = length (v); |
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89 v = reshape (v, 1, n); |
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90 |
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91 ## If v = [ 0 ... 0 v(k+1) ... v(k+l) 0 ... 0 ], we can remove the |
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92 ## leading k zeros and n - k - l roots of the polynomial are zero. |
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93 |
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94 f = find (v); |
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95 m = max (size (f)); |
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96 |
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97 if (m > 0 && n > 1) |
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98 v = v(f(1):f(m)); |
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99 l = max (size (v)); |
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100 if (l > 1) |
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101 A = diag (ones (1, l-2), -1); |
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102 A(1,:) = -v(2:l) ./ v(1); |
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103 r = eig (A); |
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104 if (f(m) < n) |
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105 tmp = zeros (n - f(m), 1); |
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106 r = [r; tmp]; |
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107 endif |
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108 else |
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109 r = zeros (n - f(m), 1); |
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110 endif |
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111 else |
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112 r = []; |
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113 endif |
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114 |
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115 endfunction |
