gmp_gcdext()
(PHP 4 >= 4.0.4, PHP 5, PHP 7)
Calculate GCD and multipliers
说明
gmp_gcdext(GMP$a, GMP$b): array
Calculates g, s, and t, such thata*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
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.