Mercurial > hg > octave-nkf
annotate scripts/polynomial/roots.m @ 14138:72c96de7a403 stable
maint: update copyright notices for 2012
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Mon, 02 Jan 2012 14:25:41 -0500 |
| parents | 614505385171 |
| children | f3d52523cde1 |
| rev | line source |
|---|---|
|
14138
72c96de7a403
maint: update copyright notices for 2012
John W. Eaton <jwe@octave.org>
parents:
14104
diff
changeset
|
1 ## Copyright (C) 1994-2012 John W. Eaton |
| 2313 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
| 8 ## your option) any later version. | |
| 2313 | 9 ## |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 7016 | 16 ## along with Octave; see the file COPYING. If not, see |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 1025 | 18 |
| 3368 | 19 ## -*- texinfo -*- |
| 20 ## @deftypefn {Function File} {} roots (@var{v}) | |
| 3426 | 21 ## |
| 3499 | 22 ## For a vector @var{v} with @math{N} components, return |
| 3368 | 23 ## the roots of the polynomial |
| 24 ## @tex | |
| 25 ## $$ | |
| 26 ## v_1 z^{N-1} + \cdots + v_{N-1} z + v_N. | |
| 27 ## $$ | |
| 28 ## @end tex | |
| 6850 | 29 ## @ifnottex |
| 3426 | 30 ## |
| 3368 | 31 ## @example |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
32 ## v(1) * z^(N-1) + @dots{} + v(N-1) * z + v(N) |
| 3368 | 33 ## @end example |
|
10821
693e22af08ae
Grammarcheck documentation of m-files
Rik <octave@nomad.inbox5.com>
parents:
9211
diff
changeset
|
34 ## |
| 6850 | 35 ## @end ifnottex |
| 36 ## | |
| 37 ## As an example, the following code finds the roots of the quadratic | |
| 38 ## polynomial | |
| 39 ## @tex | |
| 40 ## $$ p(x) = x^2 - 5. $$ | |
| 41 ## @end tex | |
| 42 ## @ifnottex | |
|
10821
693e22af08ae
Grammarcheck documentation of m-files
Rik <octave@nomad.inbox5.com>
parents:
9211
diff
changeset
|
43 ## |
| 6850 | 44 ## @example |
| 45 ## p(x) = x^2 - 5. | |
| 46 ## @end example | |
|
10821
693e22af08ae
Grammarcheck documentation of m-files
Rik <octave@nomad.inbox5.com>
parents:
9211
diff
changeset
|
47 ## |
| 6850 | 48 ## @end ifnottex |
|
10821
693e22af08ae
Grammarcheck documentation of m-files
Rik <octave@nomad.inbox5.com>
parents:
9211
diff
changeset
|
49 ## |
| 6850 | 50 ## @example |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
51 ## @group |
| 6850 | 52 ## c = [1, 0, -5]; |
|
14104
614505385171
doc: Overhaul docstrings for polynomial functions.
Rik <octave@nomad.inbox5.com>
parents:
11587
diff
changeset
|
53 ## roots (c) |
| 6850 | 54 ## @result{} 2.2361 |
| 55 ## @result{} -2.2361 | |
|
9051
1bf0ce0930be
Grammar check TexInfo in all .m files
Rik <rdrider0-list@yahoo.com>
parents:
8920
diff
changeset
|
56 ## @end group |
| 6850 | 57 ## @end example |
|
10821
693e22af08ae
Grammarcheck documentation of m-files
Rik <octave@nomad.inbox5.com>
parents:
9211
diff
changeset
|
58 ## |
| 6850 | 59 ## Note that the true result is |
| 60 ## @tex | |
| 61 ## $\pm \sqrt{5}$ | |
| 62 ## @end tex | |
| 63 ## @ifnottex | |
| 64 ## @math{+/- sqrt(5)} | |
| 65 ## @end ifnottex | |
| 66 ## which is roughly | |
| 67 ## @tex | |
| 68 ## $\pm 2.2361$. | |
| 69 ## @end tex | |
| 70 ## @ifnottex | |
| 71 ## @math{+/- 2.2361}. | |
| 72 ## @end ifnottex | |
|
14104
614505385171
doc: Overhaul docstrings for polynomial functions.
Rik <octave@nomad.inbox5.com>
parents:
11587
diff
changeset
|
73 ## @seealso{poly, compan, fzero} |
| 3368 | 74 ## @end deftypefn |
| 2311 | 75 |
| 5428 | 76 ## Author: KH <Kurt.Hornik@wu-wien.ac.at> |
| 2312 | 77 ## Created: 24 December 1993 |
| 78 ## Adapted-By: jwe | |
| 79 | |
| 787 | 80 function r = roots (v) |
| 904 | 81 |
| 6372 | 82 if (nargin != 1 || min (size (v)) > 1) |
| 6046 | 83 print_usage (); |
|
7558
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
84 elseif (any (isnan(v) | isinf(v))) |
| 8664 | 85 error ("roots: inputs must not contain Inf or NaN"); |
| 1598 | 86 endif |
| 2325 | 87 |
|
7558
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
88 n = numel (v); |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
89 v = v(:); |
| 1025 | 90 |
| 2303 | 91 ## If v = [ 0 ... 0 v(k+1) ... v(k+l) 0 ... 0 ], we can remove the |
| 2325 | 92 ## leading k zeros and n - k - l roots of the polynomial are zero. |
| 904 | 93 |
|
7558
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
94 if (isempty (v)) |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
95 f = v; |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
96 else |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
97 f = find (v ./ max (abs (v))); |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
98 endif |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
99 m = numel (f); |
| 2325 | 100 |
| 1598 | 101 if (m > 0 && n > 1) |
| 102 v = v(f(1):f(m)); | |
| 787 | 103 l = max (size (v)); |
| 104 if (l > 1) | |
| 105 A = diag (ones (1, l-2), -1); | |
| 1598 | 106 A(1,:) = -v(2:l) ./ v(1); |
| 2325 | 107 r = eig (A); |
| 1598 | 108 if (f(m) < n) |
| 109 tmp = zeros (n - f(m), 1); | |
| 3426 | 110 r = [r; tmp]; |
| 787 | 111 endif |
| 112 else | |
| 113 r = zeros (n - f(m), 1); | |
| 114 endif | |
| 115 else | |
| 1598 | 116 r = []; |
| 787 | 117 endif |
| 2325 | 118 |
| 787 | 119 endfunction |
| 7411 | 120 |
|
7558
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
121 %!test |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
122 %! p = [poly([3 3 3 3]), 0 0 0 0]; |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
123 %! r = sort (roots (p)); |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
124 %! assert (r, [0; 0; 0; 0; 3; 3; 3; 3], 0.001) |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
125 |
| 7411 | 126 %!assert(all (all (abs (roots ([1, -6, 11, -6]) - [3; 2; 1]) < sqrt (eps)))); |
| 127 | |
| 128 %!assert(isempty (roots ([]))); | |
| 129 | |
| 130 %!error roots ([1, 2; 3, 4]); | |
|
11587
c792872f8942
all script files: untabify and strip trailing whitespace
John W. Eaton <jwe@octave.org>
parents:
11523
diff
changeset
|
131 |
|
7558
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
132 %!assert(isempty (roots (1))); |
| 7411 | 133 |
|
10821
693e22af08ae
Grammarcheck documentation of m-files
Rik <octave@nomad.inbox5.com>
parents:
9211
diff
changeset
|
134 %!error roots ([1, 2; 3, 4]); |
|
11587
c792872f8942
all script files: untabify and strip trailing whitespace
John W. Eaton <jwe@octave.org>
parents:
11523
diff
changeset
|
135 |
|
7558
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
136 %!error roots ([1 Inf 1]); |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
137 |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
138 %!error roots ([1 NaN 1]); |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
139 |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
140 %!assert(roots ([1e-200, -1e200, 1]), 1e-200) |
|
690c91f741b8
Apply a scaling factor to leading zero removal in roots.m
Sebastien Loisel
parents:
7411
diff
changeset
|
141 %!assert(roots ([1e-200, -1e200 * 1i, 1]), -1e-200 * 1i) |