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gmp_gcdext — Calculate GCD and multipliers
$a
, GMP $b
)Calculates g, s, and t, such that a*s + b*t = g = gcd(a,b), where gcd is the greatest common divisor. Returns an array with respective elements g, s and t.
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: a*x + b*y = c. For more information, go to the » "Diophantine Equation" page at MathWorld
a
可以是一个 GMP 数据 resouce ,或一个可以转换为数值的字符串。
b
可以是一个 GMP 数据 resouce ,或一个可以转换为数值的字符串。
An array of GMP numbers.
Example #1 Solving a linear Diophantine equation
<?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
?>
[#1] FatPhil [2003-06-15 06:47:13]
The extended GCD can be used to calculate mutual modular inverses of two
coprime numbers. Internally gmp_invert uses this extended GCD routine,
but effectively throws away one of the inverses.
If gcd(a,b)=1, then r.a+s.b=1
Therefore r.a == 1 (mod s) and s.b == 1 (mod r)
Note that one of r and s will be negative, and so you'll want to
canonicalise it.