HDU - 3714

题目
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 minimum which related to multiple quadric functions. The new function F(x) is defined as follows: F(x) = max(Si(x)), i = 1…n. The domain of x is [0, 1000]. Si(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.
题目大意
有n个开口向上的二次函数函数Si(x),设F(x)=max{Si(x)}(i=1,2,3…n).求F(x)的最小值。
解题思路
观察得F(x)的图像是一个类似开口向上的二次函数,存在一点x0,使得F(x)在x=x0处取到最小值。当x<x0时,F(x)单调减小,当x>x0时,F(x)单调增加。
此时可以运用三分法(三分法主要解决开口向上的二次函数在区间(l,r)的最小值问题)。
三分法原理:在区间(l,r)中,存在点(x1,F(x1)),点(x2,F(x2)) (x1<x2)
如果F(x1)<F(x2)那么最小值点在区间(l,x2)之间
如果F(x1)>F(x2)那么最小值点在区间(x1,r)之间
如果F(x1)=F(x2)那么最小值点在区间(x1,x2)之间
不断重复这个过程即可。
代码实现

#include <cstdio>
#include <iostream>
using namespace std;
const double esp=1e-10;
const double mx=100*1000*1000+5000*1000+5000;
int n;
double ans,kkk;
struct data
{
	double a,b,c;
}a[10101];
bool check(double m1,double m2)
{
	double mx1=-mx,mx2=-mx;
	for (int i=1;i<=n;i++)
	{
		double x1=a[i].a*m1*m1+a[i].b*m1+a[i].c;
		double x2=a[i].a*m2*m2+a[i].b*m2+a[i].c;
		if (x1>mx1) mx1=x1;
		if (x2>mx2) mx2=x2;	
	}
	if (mx1>=mx2) ans=mx2;
	else ans=mx1;
	if (mx1>=mx2) return true;
	else return false;
}
int main()
{
	int T;
	scanf("%d",&T);
	while (T--)
	{
		scanf("%d",&n);
		for (int i=1;i<=n;i++)
		  cin>>a[i].a>>a[i].b>>a[i].c;
		double l=0.0,r=1000.0;
		while (l+esp<=r)
		{
			double mid=(r-l)/3.0;
		    double m1=l+mid;
		    double m2=m1+mid;
		    if (check(m1,m2)) l=m1;
		    else r=m2;
		}  
		bool xxx=check(l,r); 
		printf("%.4f\n",ans);
	}
} 
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转载自blog.csdn.net/weixin_45723759/article/details/103962152