How to implement fast multiplication of large numbers using PHP and GMP

How to use PHP and GMP to implement fast multiplication of large numbers

Introduction:
In computer science, integer arithmetic is one of the most basic and commonly used operations. However, when large integers are involved, traditional arithmetic methods become inefficient. This article will introduce how to use the GMP (GNU Multiple Precision) library in PHP to implement fast multiplication of large numbers and provide corresponding code examples.

1. Introduction to GMP library
   The GMP library is a high-precision calculation library that provides functions such as addition, subtraction, multiplication, division, and exponentiation of large integers. The advantage of the GMP library is the efficiency of its algorithm, which can handle very large integers. The GMP extension that comes with PHP is based on the encapsulation of the GMP library and provides a simple and easy-to-use interface.
2. Fast multiplication algorithm
   The fast multiplication algorithm is an optimized algorithm used to reduce the complexity of multiplication operations from $O(n^2)$ to $O(nlog n)$. It is based on the divide-and-conquer strategy, which converts multiplication of large numbers into multiplication of smaller numbers. The following is the basic idea of ​​the fast multiplication algorithm:

1) Decompose the two large numbers $x$ and $y$ to be multiplied into the form of $acdot10^m b$ and $ccdot10^m d$, Among them, $a$ and $c$ are the high-order parts of $x$ and $y$ respectively, $b$ and $d$ are the low-order parts of $x$ and $y$ respectively, and $m$ is the appropriate number of bits.

2) Multiply two large numbers to get $(acdot10^m b)(ccdot10^m d)$, use the formula $accdot10^{2m} [(a b)(c d)-ac-bd] cdot10^m bd$ calculation result.

3) Recursively calculate the three parts $ac$, $bd$ and $(a b)(c d)$ in the multiplication.

4) Reduce the multiplication problem to simple multiplication by recursing multiple times until a base case is reached.

Through the above steps, you can achieve fast multiplication of large numbers.

3. PHP code example
   The following is a code example that uses the GMP library in PHP to implement fast multiplication of large numbers:

<?php
function multiply($x, $y) {
    $x_gmp = gmp_init($x);
    $y_gmp = gmp_init($y);
    
    // 当待乘数小于等于一个阈值时，直接返回乘法结果
    if (gmp_cmp($x_gmp, "1000000") <= 0 || gmp_cmp($y_gmp, "1000000") <= 0) {
        return gmp_strval(gmp_mul($x_gmp, $y_gmp));
    }
    
    // 将待乘数分解为高位部分$a$和低位部分$b$
    $x_str = gmp_strval($x_gmp);
    $split_point = ceil(strlen($x_str) / 2);
    $a = substr($x_str, 0, -$split_point);
    $b = substr($x_str, -$split_point);
    
    // 将乘数对应分解为高位部分$c$和低位部分$d$
    $y_str = gmp_strval($y_gmp);
    $c = substr($y_str, 0, -$split_point);
    $d = substr($y_str, -$split_point);
    
    // 计算子问题的结果
    $ac = multiply($a, $c);
    $bd = multiply($b, $d);
    $abcd = multiply(gmp_add($a, $b), gmp_add($c, $d));
    $ad_bc = gmp_sub($abcd, gmp_add($ac, $bd));
    
    // 计算最终结果并返回
    $result = gmp_add(gmp_mul(gmp_pow(10, 2 * $split_point), $ac), gmp_add(gmp_mul(gmp_pow(10, $split_point), $ad_bc), $bd));
    return gmp_strval($result);
}

// 示例输入
$x = "12345678901234567890";
$y = "98765432109876543210";

// 调用乘法函数
$result = multiply($x, $y);
echo "Result: " . $result . "
";
?>

Using the above code, we can Implement fast multiplication of large numbers.

Conclusion:
This article introduces how to use the GMP library in PHP to implement fast multiplication of large numbers. By using the fast multiplication algorithm, we can reduce the complexity of the multiplication operation from $O(n^2)$ to $O(nlog n)$, thus improving the efficiency of the algorithm. I hope this article will be helpful in understanding and implementing fast multiplication of large numbers.

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