三分法求極值

二分法用來解決單調函數的極值問題，而三分法用來解決凸函數的極值問題。

所以在使用三分的時候要注意函數具有凹凸性

模板：

double cal(){//.....具體計算根據題目要求實現return 0.0;}void solve(){double left,right,mid1,mid2,ans,flg1,flg2;left=0.0;right=1000.0;while(right-left>eps){mid1=(2*left+right)/3;mid2=(left+2*right)/3;flg1=cal();//具體實現flg2=cal();if(flg1>flg2)right=mid2;elseleft=mid1;}ans=cal();//left}

例題：

hdu 3400

題意：平面上有兩條線段AB和CD，在AB上行走的速度為P，CD上行走的速度為Q，其他地區行走的速度為R，問從A走到D的最短時間。

//============================================================================// Name        : hdu3400三分.cpp// Author      : xinge008// Version     :// Copyright   : Your copyright notice// Description : Hello World in C++, Ansi-style//============================================================================#include <iostream>#include <cstring>#include <cstdio>#include <cmath>#include <algorithm>#include <string>using namespace std;const double eps=1e-12;double P,Q,R;struct Node{double x,y;};Node A,B,C,D;Node dist(Node n1,Node n2){Node ans;ans.x=(n1.x+n2.x)/2;ans.y=(n1.y+n2.y)/2;return ans;}double cal(Node n1,Node n2){double t1=(n1.x-n2.x)*(n1.x-n2.x);double t2=(n1.y-n2.y)*(n1.y-n2.y);return sqrt(t1+t2);}double run1(Node da){Node left=C,right=D;double max=100,min=0;//double max=cal(left,right)/Q,min=0;while(fabs(max-min)>eps){Node mid1=dist(left,right);Node mid2=dist(mid1,right);min=cal(mid1,da)/R+cal(mid1,D)/Q;max=cal(mid2,da)/R+cal(mid2,D)/Q;if(min<=max)right=mid2;elseleft=mid1;}returnmax;}double run2(){Node left=A,right=B;double max=100,min=0;//double min=0,max=cal(A,B)/P;while(fabs(max-min)>eps){Node mid1=dist(left,right);Node mid2=dist(mid1,right);min=cal(mid1,A)/P+run1(mid1);max=cal(mid2,A)/P+run1(mid2);if(min<=max)right=mid2;elseleft=mid1;}return max;}void gao(){int T;scanf("%d",&T);while(T--){scanf("%lf%lf%lf%lf",&A.x,&A.y,&B.x,&B.y);scanf("%lf%lf%lf%lf",&C.x,&C.y,&D.x,&D.y);scanf("%lf%lf%lf",&P,&Q,&R);printf("%.2lf\n",run2());}}int main() {gao();return 0;}

HDU3714

題意：函數F[x]=max{Si(x)} 
（i=0，1，2，3，...，n）    其中Si(x)為二次函數， 求函數F[X]的最小值。

//============================================================================// Name        : 三分.cpp// Author      : xinge008// Version     :// Copyright   : Your copyright notice// Description : Hello World in C++, Ansi-style//============================================================================#include <iostream>#include <cstring>#include <string>#include <cmath>#include <algorithm>#include <cstdio>using namespace std;const int maxn=10010;const double eps = 1e-12;struct node{double a,b,c;}data[maxn];double _Max(double a,double b){if(a>b)return a;return b;}double cal1(int i,double da){return data[i].a*da*da+data[i].b*da+data[i].c;}double cal2(double x,int n){double ans=-1e10;for(int i=1;i<=n;i++)ans=_Max(ans,cal1(i,x));return ans;}void cal(int n){double left,right,ans,mid1,mid2;left=0;right=1000.0;while(right-left>eps){//在此是高精度的要求mid1=(2*left+right)/3;mid2=(left+2*right)/3;double flg1=cal2(mid1,n);double flg2=cal2(mid2,n);if(flg1>flg2){left=mid1;}else{right=mid2;}}ans=cal2(left,n);printf("%.4lf\n",ans);}int main() {int T;scanf("%d",&T);while(T--){int num;scanf("%d",&num);for(int i=1;i<=num;i++){scanf("%lf%lf%lf",&data[i].a,&data[i].b,&data[i].c);}cal(num);}return 0;}