PHP: gmp_gcdext

Description

array gmp_gcdext ( resource a, resource 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

Przykład 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

?>


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.

gmp_gcdext

Description

User Contributed Notes