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
|
<?xml version="1.0" encoding="iso-8859-1"?>
<!-- $Revision: 1.5 $ -->
<!-- splitted from ./en/functions/gmp.xml, last change in rev 1.1 -->
<refentry id="function.gmp-gcdext">
<refnamediv>
<refname>gmp_gcdext</refname>
<refpurpose>Calculate GCD and multipliers</refpurpose>
</refnamediv>
<refsect1>
<title>Description</title>
<methodsynopsis>
<type>array</type><methodname>gmp_gcdext</methodname>
<methodparam><type>resource</type><parameter>a</parameter></methodparam>
<methodparam><type>resource</type><parameter>b</parameter></methodparam>
</methodsynopsis>
<para>
Calculates g, s, and t, such that <literal>a*s + b*t = g =
gcd(a,b)</literal>, where gcd is the greatest common divisor. Returns
an array with respective elements g, s and t.
</para>
<para>
This function can be used to solve linear Diophantine equations in two
variables. These are equations that allow only integer solutions and have the form:
<literal>a*x + b*y = c</literal>.
For more information, go to the <ulink url="&url.diophantine-equation;">"Diophantine
Equation" page at MathWorld</ulink>
</para>
<para>
<example>
<title>Solving a linear Diophantine equation</title>
<programlisting role="php">
<![CDATA[
<?php
// Solve the equation a*s + b*t = g
// where a = 12, b = 21, g = gcd(12, 21) = 3
$a = gmp_init(12);
$b = gmp_init(21);
$g = gmp_gcd($a, $b);
$r = gmp_gcdext($a, $b);
$check_gcd = (gmp_strval($g) == gmp_strval($r['g']));
$eq_res = gmp_add(gmp_mul($a, $r['s']), gmp_mul($b, $r['t']));
$check_res = (gmp_strval($g) == gmp_strval($eq_res));
if ($check_gcd && $check_res) {
$fmt = "Solution: %d*%d + %d*%d = %d\n";
printf($fmt, gmp_strval($a), gmp_strval($r['s']), gmp_strval($b),
gmp_strval($r['t']), gmp_strval($r['g']));
} else {
echo "Error while solving the equation\n";
}
// output: Solution: 12*2 + 21*-1 = 3
?>
]]>
</programlisting>
</example>
</para>
</refsect1>
</refentry>
<!-- Keep this comment at the end of the file
Local variables:
mode: sgml
sgml-omittag:t
sgml-shorttag:t
sgml-minimize-attributes:nil
sgml-always-quote-attributes:t
sgml-indent-step:1
sgml-indent-data:t
indent-tabs-mode:nil
sgml-parent-document:nil
sgml-default-dtd-file:"../../../../manual.ced"
sgml-exposed-tags:nil
sgml-local-catalogs:nil
sgml-local-ecat-files:nil
End:
vim600: syn=xml fen fdm=syntax fdl=2 si
vim: et tw=78 syn=sgml
vi: ts=1 sw=1
-->
|
