牛顿迭代法(Newton's Method)

简介

牛顿迭代法（简称牛顿法）由英国著名的数学家牛顿爵士最早提出。但是，这一方法在牛顿生前并未公开发表。

牛顿法举例

下面程序使用牛顿法求解平方根。

const float EPS = 0.00001; 
int sqrt(double x) { 
	if(x == 0) return 0; 
	double result = x; /*Use double to avoid possible overflow*/ 
	double lastValue; 
	do{ 
		lastValue = result; 
		result = result / 2.0f + x / 2.0f / result; 
	}while(abs(result - lastValue) > EPS);
 return (double)result;
 }

更快的方法

float Q_rsqrt( float number ) { 
	long i; float x2, y; const float threehalfs = 1.5F;
	x2 = number * 0.5F; 
	y = number; 
	i = * ( long * ) &y; // evil floating point bit level hacking 
	i = 0x5f3759df - ( i >> 1 ); // what the fuck? 
	y = * ( float * ) &i; 
	y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration 
	// y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, this can be removed
	#ifndef Q3_VM #
	ifdef __linux__ assert( !isnan(y) ); // bk010122 - FPE?
	#endif
	#endif return y; 
}

int sqrt(float x) { 
	if(x == 0) return 0; 
	float result = x; 
	float xhalf = 0.5f*result; 
	int i = *(int*)&result; 
	i = 0x5f375a86- (i>>1); // what the fuck? 
	result = *(float*)&i; 
	result = result*(1.5f-xhalf*result*result); // Newton step, repeating increases accuracy 
	result = result*(1.5f-xhalf*result*result); 
	return 1.0f/result; 
}

参考文献:
1.wikipedia.org
2.http://www.2cto.com/kf/201206/137256.html

牛顿迭代法(Newton's Method)

原文：http://blog.csdn.net/wangxiaojun911/article/details/18203333