三分搜索--hdu3714 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 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.

比较明显的三分，求x，使得给定二次曲线中，si(x)最大值最小。

画图可知，一定有最小值，虽然曲线不光滑，但是可以用三分求。

#include <iostream>

#include <cstdio>

#include <cstring>

using namespace std;

const int maxn = 1e4 + 5;

const double eps = 1e-10;

const int INF = 1 << 30;

int a[maxn],b[maxn],c[maxn];

int n;

double cal(double x)

{

    double my = -INF;//函数值可能为负，不能初始化为0

    for (int i = 0; i < n; i ++) {

        my = max(my,a[i] * 1.0 * x * x + b[i] * 1.0 * x + c[i] * 1.0);

    }

    return my;

}

void solve()

{

    double l = 0.0,r = 1000.0;

    double lmid = 0,rmid = 0;

    double ans1 = 0,ans2 = 0;

    while (r - l >= eps) {

        lmid = l + (r - l) / 3;

        rmid = r - (r - l) / 3;

        ans1 = cal(lmid);

        ans2 = cal(rmid);

        if(ans1 < ans2) r = rmid;

        else l = lmid;

    }

    printf("%.4lf\n",cal((l + r)/ 2));

}

int main()

{

    int T;

    cin >> T;

    while (T --) {

        memset(a, 0, sizeof(a));

        memset(b, 0, sizeof(b));

        memset(c, 0, sizeof(c));

        cin >> n;

        for (int i = 0; i < n; i ++) {

            scanf("%d%d%d",&a[i],&b[i],&c[i]);

        }

        solve();

    }

    return 0;

}