1. 简介

牛顿迭代法是求近似根的一种方法。

以求平方根为例。
如$x^2=m$
令$f(x)=x^2-m$
则$f^{\prime }(x)=2x$
函数$f(x)$在$x_0$处的切线方程为
$g(x)=f^{\prime }(x_0)(x-x_0)+f(x_0)$

令该切线与x轴交点为$(x_1,0)$
$x_1=x_0-\frac{f(x_0)}{f^{\prime }(x_0)}$

$x_1=x_0-\frac{x_0^2-m}{2x_0}$

重复上述迭代过程，直到$x_{n+1}-x_n$小于某一精度

2. 实现

迭代即可

• cpp

#include <iostream>
#include <cstring>
#include <cmath>double my_sqrt(double x)
{double x0 = 1;double x1 = x0 - (x0 * x0 - x)/(2*x0);while ( std::abs(x0-x1) > 1e-6) {x0 = x1;x1 -= (x1*x1 - x) / (2*x1);}return x0;
}int main(int argc, char *argv[])
{std::cout << my_sqrt(2) << std::endl;std::cout << my_sqrt(3) << std::endl;std::cout << my_sqrt(5) << std::endl;return 0;
}

• go

package mainimport ("fmt""math"
)func New_ton(x ,z float64) float64{return z - (z*z - x) / (2 * z)
}func Sqrt(x float64) float64 {z := 1.0for z0:= New_ton(x, z); math.Abs(z-z0) > 1e-6; z0 = New_ton(x,z0) {z = z0}return z
}func main() {fmt.Println(Sqrt(2))fmt.Println(Sqrt(3))fmt.Println(Sqrt(5))
}

3. Ref

go_tutorial