Newton Raphson iterative method

history

Newton's method was first developed by Isaac · Newton is in 《 Stream number method 》（Method of Fluxions,1671 Years to complete , After Newton's death 1736 Published in ）. Joseph · Lafferson also 1690 In Analysis Aequationum This method is proposed in .

Algorithm

For example, to calculate a Greater than 0 The square root of （a>0）, You can define $$f(x)=x^2-5$$
Take a value close to the real solution first, such as 2, Write it down as x0 Then it is always calculated according to the following recursive formula xn:$$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$$
For example, use C Language can write like this ：

double eps=1e-10;
	double x1=2,x0=2;
	do{
    
		x1 = x0-(pow(x0,2)-5)/(2*x0);
		x0=x1;
	}while((x0-sqrt(5))>eps||(x0-sqrt(5))<(-eps));// It can't be used here abs function , because abs Back to plastic surgery 
	
	printf("x=%.10f",x0-sqrt(5);

give the result as follows ：

Limiting conditions

Not all functions can be solved in this way , Not all of them x0 Can be used as the initial value , Only functions converged by Newton's method can be used .