Newton method implemented binary and square root

dichotomy:

Dichotomous thinking is very intuitive, and have continued to do binary, but here attention to the need to set in place a precision 0, because the square root does not necessarily guarantee that prescribing depletion. Here take limit = 0.00002.

 

Newton's method:

Let r is f (x) = root 0, select x0 as r initial approximation, through the point (x0, F (x0)) to make the curve y = f (x) tangent L, L of the equation y = f (x0 ) + f '(x0) (x-x0), the intersection of the x-axis is obtained L abscissa x1 = x0-f (x0) / f' (x0), x1 is called a first approximation of r.

Through the point (x1, f (x1)) to make the curve y = f (x) is the tangent, the abscissa and determining the x2-axis intersection of the tangent and x is = x1-f (x1) / f '(x1), x2 is called secondary approximation of r. Repeat the above procedure to give an approximation of the sequence r, where x (n + 1) = x (n) -f (x (n)) / f '(x (n)), called r n + 1 times of approximation, the type known as Newton iterative formula.

 

The Newton iteration principle, possible to obtain the following iteration formula: X (n + 1) = [X (n) + p / Xn] / 2

 https://blog.csdn.net/leviopku/article/details/82811478

Code:

 1 #include<iostream>
 2 #include<vector>
 3 #include<deque>
 4 #include<string>
 5 #define limit 0.0002
 6 using namespace std;
 7 
 8 // 二分法
 9 double my_sqrt(double n)
10 {
11     double start = 0;
12     double end = n;
13     double mid = (start+end)/2;
14     while(mid*mid>n+limit || mid*mid<n-limit)
15     {
16         mid = (start+end)/2;
17         if(mid*mid>n)
18             end = mid-1;
19         else
20             start = mid+1;
21     }
22     return mid;
23 }
24 
25 // 牛顿法
26 double new_sqrt(double n)
27 {
28     double k = n;
29     while(1)
30     {
31         if(k*k<n+limit && k*k>n-limit)
32         {
33             break;
34         }
35         k = 0.5*(k+n/k);
36     }
37     return k;
38 }
39 int main(){
40     double a = 81;
41     cout<<my_sqrt(a)<<endl;
42     cout<<new_sqrt(a)<<endl;
43     return 0;
44 }