1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
|
########################################################################
##
## Copyright (C) 2000-2024 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{q} =} polygcd (@var{b}, @var{a})
## @deftypefnx {} {@var{q} =} polygcd (@var{b}, @var{a}, @var{tol})
##
## Find the greatest common divisor of two polynomials.
##
## This is equivalent to the polynomial found by multiplying together all the
## common roots. Together with deconv, you can reduce a ratio of two
## polynomials.
##
## The tolerance @var{tol} defaults to @code{sqrt (eps)}.
##
## @strong{Caution:} This is a numerically unstable algorithm and should not
## be used on large polynomials.
##
## Example code:
##
## @example
## @group
## polygcd (poly (1:8), poly (3:12)) - poly (3:8)
## @result{} [ 0, 0, 0, 0, 0, 0, 0 ]
## deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))) - poly (1:2)
## @result{} [ 0, 0, 0 ]
## @end group
## @end example
## @seealso{poly, roots, conv, deconv, residue}
## @end deftypefn
function x = polygcd (b, a, tol)
if (nargin < 2)
print_usage ();
endif
if (nargin == 2)
if (isa (a, "single") || isa (b, "single"))
tol = sqrt (eps ("single"));
else
tol = sqrt (eps);
endif
endif
## FIXME: No input validation of tol if it was user-supplied
if (length (a) == 1 || length (b) == 1)
if (a == 0)
x = b;
elseif (b == 0)
x = a;
else
x = 1;
endif
else
a /= a(1);
while (1)
[d, r] = deconv (b, a);
nz = find (abs (r) > tol);
if (isempty (nz))
x = a;
break;
else
r = r(nz(1):length (r));
endif
b = a;
a = r / r(1);
endwhile
endif
endfunction
%!test
%! poly1 = [1 6 11 6]; # (x+1)(x+2)(x+3);
%! poly2 = [1 3 2]; # (x+1)(x+2);
%! poly3 = polygcd (poly1, poly2);
%! assert (poly3, poly2, sqrt (eps));
%!assert (polygcd (poly (1:8), poly (3:12)), poly (3:8), sqrt (eps))
%!assert (deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))),
%! poly (1:2), sqrt (eps))
%!test
%! for ii=1:100
%! ## Exhibits numerical problems for multipliers of ~4 and greater.
%! p = (unique (randn (10, 1)) * 3).';
%! p1 = p(3:end);
%! p2 = p(1:end-2);
%! assert (polygcd (poly (-p1), poly (-p2)),
%! poly (- intersect (p1, p2)), sqrt (eps));
%! endfor
|
