gmp_gcdext

(PHP 4 >= 4.0.4, PHP 5, PHP 7)

gmp_gcdext — Calculate GCD and multipliers

说明

array gmp_gcdext ( GMP $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

参数

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。

返回值

An array of GMP numbers.

范例

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
 ?>

用户评论:

[#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.