牛顿法特殊形式——开方公式与二分逼近开方的

数值分析-开方公式

// 开方公式
#include<bits/stdc++.h>using namespace std;int main()
{double c, x0, jingdu;printf("请输入被开方数c：");scanf("%lf", &c);printf("请输入初值x0：");scanf("%lf", &x0);printf("请输入精度（用小数表示）：");scanf("%lf", &jingdu);double xk, xk1 = x0;int i = 0;do {i++;xk = xk1;xk1 = 0.5 * (xk + c / xk);printf("x%d = %lf\n", i, xk1);} while (abs(xk1 - xk) > jingdu);printf("故 √c ≈ x%d = %lf\n", i, xk1);return 0;
}

二分逼近开方

#include<iostream>using namespace std;
using ll = long long;const int N = 5000 + 10;int n;int main()
{double c, jingdu;printf("------------------二分逼近------------------\n");printf("请输入被开方数c：");scanf("%lf", &c);printf("请输入精度（用小数表示）：");scanf("%lf", &jingdu);double l = 0, r = max(1.0, c), mid = 0;int i = 0;while(abs(mid * mid - c) > jingdu) {i++;mid = (l + r) / 2.0;if (mid * mid < c) l = mid;else r = mid;printf("x%d = %lf\n", i, mid);}printf("故 √c ≈ x%d = %lf\n", i, mid);return 0;
}

对比总结

二分逼近的复杂度是O(logN)*
开方公式的时间复杂度是O(k)=O(loglogN)**
开方公式的收敛速度更快。