【单峰函数，三分搜索算法(Ternary_Search)】UVa 1476 - Error Curves

Josephina is a clever girl and addicted to Machine Learning recently. She pays much attention to a method called Linear Discriminant Analysis, which has many interesting properties.

In order to test the algorithm's efficiency, she collects many datasets. What's more, each data is divided into two parts: training data and test data. She gets the parameters of the model on training data and test the model on test data.

To her surprise, she finds each dataset's test error curve is just a parabolic curve. A parabolic curve corresponds to a quadratic function. In mathematics, a quadratic function is a polynomial function of the form f(x) = ax² + bx + c. The quadratic will degrade to linear function if a = 0.

 

It's very easy to calculate the minimal error if there is only one test error curve. However, there are several datasets, which means Josephina will obtain many parabolic curves. Josephina wants to get the tuned parameters that make the best performance on all datasets. So she should take all error curves into account, i.e., she has to deal with many quadric functions and make a new error definition to represent the total error. Now, she focuses on the following new function's minimal which related to multiple quadric functions.

The new function F(x) is defined as follow:

F(x) = max(S_i(x)), i = 1...n. The domain of x is [0, 1000]. S_i(x) is a quadric function.

Josephina wonders the minimum of F(x). Unfortunately, it's too hard for her to solve this problem. As a super programmer, can you help her?

Input

The input contains multiple test cases. The first line is the number of cases T (T < 100). Each case begins with a number n(n ≤ 10000). Following n lines, each line contains three integers a (0 ≤ a ≤ 100), b (|b| ≤ 5000), c (|c| ≤ 5000), which mean the corresponding coefficients of a quadratic function.

Output

For each test case, output the answer in a line. Round to 4 digits after the decimal point.

Sample Input

2
1
2 0 0
2
2 0 0
2 -4 2

Sample Output

0.0000
0.5000


题意：给出n条二次曲线S(x) = ax2 + bx + c(a >= 0, 0 <= x <= 1000),定义F(x)=max{Si(x)},即F(x)为取x值时n条二次曲线对应值的最大值;则你需要求出x在[0, 1000]范围内F(x)值的最小值；

分析：x在[0,1000]范围内所有F(x)值可以连成一条曲线。由于每条二次曲线S(x)都是下凸单峰函数，则F(x)=max{Si(x)}的曲线也是下凸单峰。对于单峰函数求极值，一般选用三分搜索算法。

所谓三分:把区间分为长度相等的三段进行查找，称为三分查找，三分查找通常用来迅速确定最值。

　众所周知，二分算法的要求是搜索的序列是单调序列，而三分法所面向的搜索序列的要求是：序列为一个凸性函数。

 

 

 

　与二分法类似，三分算法先把区间分为长度相等的三段，那么L与R之间就有两个点，分别是：m1=L+(R-L)/3; m2=R-(R-L)/3;

 

　如果m1比m2更靠近最值，我们就舍弃右区间，否则我们舍弃左区间。

 

　代码如下：

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cmath>
#include<algorithm>
using namespace std;
const double eps = 1e-14;
const int maxn = 10010;
int a[maxn], b[maxn], c[maxn];
int n;

double F(double x)
{
    double ans = a[0]*x*x + b[0]*x + c[0];
    for(int i = 1; i < n; i++)
    {
        ans = max(ans, a[i]*x*x + b[i]*x + c[i]);
    }
    return ans;
}


void Ternary_Search()
{
    double L = 0.0, R = 1000.0;
    for(int i = 0; i < 100; i++)
    {
        double m1 = L+(R-L)/3;
        double m2 = R-(R-L)/3;

        if(F(m1) < F(m2)) R = m2;
        else L = m1;
    }
    printf("%.4lf\n", F(L));
}
int main()
{

    int T; scanf("%d", &T);
    while(T--)
    {
        scanf("%d", &n);
        for(int i = 0; i < n; i++)
            scanf("%d%d%d", &a[i], &b[i], &c[i]);

        Ternary_Search();
    }
    return 0;
}