hdu4498 Function Curve 分段simpson

最新推荐文章于 2018-04-14 00:04:31 发布

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最新推荐文章于 2018-04-14 00:04:31 发布

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分类专栏：自适应辛普森法

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本文介绍了一种计算由多个二次函数的最小值构成的复合曲线在指定区间内的弧长的方法。通过找出函数间的交点和与边界值的交点，利用数值积分技术（如辛普森公式）精确计算曲线长度。

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题目：给出k1,k2,…,kn, a1,a2,…,an 和 b1,b2,…,bn
求F(x)=min{100,min{ki*(x-ai)^2+bi | 0 < i <= n}} 在坐标上画出的曲线的长度。

要求x>=0&&x<=100，n<=50

思路：求出所有函数之间的交点，所有函数与y=100的交点，交点排序，累加相邻交点的曲线长度

代码：

#pragma comment(linker, "/STACK:1024000000,1024000000")
#include<iostream>
#include<algorithm>
#include<ctime>
#include<cstdio>
#include<cmath>
#include<cstring>
#include<string>
#include<vector>
#include<map>
#include<set>
#include<queue>
#include<stack>
#include<list>
#include<numeric>
using namespace std;
#define LL long long
#define ULL unsigned long long
#define INF 0x3f3f3f3f
#define mm(a,b) memset(a,b,sizeof(a))
#define PP puts("*********************");
template<class T> T f_abs(T a){ return a > 0 ? a : -a; }
template<class T> T gcd(T a, T b){ return b ? gcd(b, a%b) : a; }
template<class T> T lcm(T a,T b){return a/gcd(a,b)*b;}
// 0x3f3f3f3f3f3f3f3f
// 0x3f3f3f3f

const double eps=1e-5;//根据实际情况修改
const int maxn=105;
double k[maxn],a[maxn],b[maxn];
vector<double> inter;
int n;
double sqr(double x){
    return x*x;
}
int dblcmp(double x){//判断正负0
    if(f_abs(x)<eps) return 0;
    else return x<0?-1:1;
}
void judge(double x){
    if(dblcmp(x)>=0&&dblcmp(x-100)<=0)
        inter.push_back(x);
}
void getinter(){
    inter.clear();
    inter.push_back(0);
    inter.push_back(100);
    for(int i=0;i<n;i++){
        if(b[i]>100) continue;
        double x1=a[i]+sqrt((100-b[i])/k[i]);
        double x2=a[i]-sqrt((100-b[i])/k[i]);
        judge(x1);
        judge(x2);
    }
    for(int i=0;i<n;i++)
        for(int j=i+1;j<n;j++){
            double A=k[i]-k[j];
            double B=2*k[j]*a[j]-2*k[i]*a[i];
            double C=k[i]*a[i]*a[i]+b[i]-k[j]*a[j]*a[j]-b[j];
            if(dblcmp(A)==0){
                if(dblcmp(B)!=0)
                    judge(-C/B);
                continue;
            }
            else{
                double delta=B*B-4*A*C;
                if(dblcmp(delta)<0) continue;
                if(dblcmp(delta)==0) judge(-B/(2*A));
                else{
                    delta=sqrt(delta);
                    double x1=(-B+delta)/(2*A);
                    double x2=(-B-delta)/(2*A);
                    judge(x1);
                    judge(x2);
                }
            }
        }
}
int pos;
double F(double x){//
	//Simpson公式用到的函数
    return sqrt(1+sqr(2*k[pos]*(x-a[pos])));
}
double simpson(double a, double b){
	double c = a + (b - a) / 2;
	return (F(a) + 4 * F(c) + F(b))*(b - a) / 6;
}
double asr(double a, double b, double eps, double A){
	double c = a + (b - a) / 2;
	double L = simpson(a, c), R = simpson(c, b);
	if (fabs(L + R - A) <= 15 * eps) return L + R + (L + R - A) / 15.0;
	return asr(a, c, eps / 2, L) + asr(c, b, eps / 2, R);
}
double asr(double a, double b, double eps){
	return asr(a, b, eps, simpson(a, b));
}
double Function(double x,int i){
    return k[i]*sqr(x-a[i])+b[i];
}
int main(){

    int T;
    scanf("%d",&T);
    while(T--){
        scanf("%d",&n);
        for(int i=0;i<n;i++)
            scanf("%lf%lf%lf",&k[i],&a[i],&b[i]);
        getinter();
        sort(inter.begin(),inter.end());
        int sz=inter.size();
        double ans=0;
        for(int i=1;i<sz;i++){
            double x1=inter[i-1],x2=inter[i];
            if(dblcmp(x1-x2)==0) continue;
            double x=(x1+x2)/2;
            pos=0;
            for(int j=1;j<n;j++){
                if(Function(x,j)<Function(x,pos))
                    pos=j;
            }
            if(dblcmp(Function(x,pos)-100)<0)
                ans+=asr(x1,x2,eps);
            else
                ans+=x2-x1;
        }
        printf("%.2f\n",ans);
    }
    return 0;
}