[BZOJ 1502] [NOI2005] 月下柠檬树 【Simpson积分】

题目链接: BZOJ - 1502

题目分析

这是我做的第一道 Simpson 积分的题目。Simpson 积分是一种用 (fl + 4*fmid + fr) / 6 * (r - l) 来拟合 fl...fr 的方法。自适应 Simpson 的自适应指的是，如果分成左右两端分别 Simpson 的和与对整段 Simpson 的差值不超过一个 Eps，那么就接受这个值，否则递归下去求两段的 Simpson 值。

代码

 #include <iostream>
 #include <cstdlib>
 #include <cstring>
 #include <cstdio>
 #include <cmath>
 #include <algorithm>

 using namespace std;

 typedef double LF;

 const LF Eps = 1e-;

 inline LF gmin(LF a, LF b) {return a < b ? a : b;}
 inline LF gmax(LF a, LF b) {return a > b ? a : b;}
 inline LF Sqr(LF x) {return x * x;}

 const int MaxN =  + ;

 int n;

 LF Alpha, talpha, Ht, Lp, Rp, Ans;
 LF A[MaxN], P[MaxN], Rad[MaxN], Ls[MaxN], Rs[MaxN], Lh[MaxN], Rh[MaxN];

 inline LF f(LF x)
 {
     LF ret = 0.0;
     for (int i = ; i <= n; ++i)
     {
         if (fabs(x - P[i]) < Rad[i]) ret = gmax(ret, sqrt(Sqr(Rad[i]) - Sqr(x - P[i])));
         if (x > Ls[i] && x < Rs[i]) ret = gmax(ret, Lh[i] + (Rh[i] - Lh[i]) * (x - Ls[i]) / (Rs[i] - Ls[i]));
     }
     return ret;
 }

 inline LF Simpson(LF l, LF r, LF fl, LF fmid, LF fr)
 {
     return (fl + fmid * 4.0 + fr) / 6.0 * (r - l);
 }

 LF RSimpson(LF l, LF r, LF fl, LF fmid, LF fr)
 {
     LF mid, p, q, x, y, z;
     mid = (l + r) / 2.0;
     p = f((l + mid) / 2.0);
     q = f((mid + r) / 2.0);
     x = Simpson(l, r, fl, fmid, fr);
     y = Simpson(l, mid, fl, p, fmid);
     z = Simpson(mid, r, fmid, q, fr);
     if (fabs(x - y - z) < Eps) return y + z;
     else return RSimpson(l, mid, fl, p, fmid) + RSimpson(mid, r, fmid, q, fr);
 }

 int main()
 {
     scanf("%d%lf", &n, &Alpha);
     talpha = tan(Alpha);
     Ht = 0.0;
     for (int i = ; i <= n + ; ++i)
     {
         scanf("%lf", &A[i]);
         Ht += A[i];
         P[i] = Ht / talpha;
     }
     Lp = P[]; Rp = P[n + ];
     for (int i = ; i <= n; ++i)
     {
         scanf("%lf", &Rad[i]);
         Lp = gmin(Lp, P[i] - Rad[i]);
         Rp = gmax(Rp, P[i] + Rad[i]);
     }
     Rad[n + ] = 0.0;
     for (int i = ; i <= n; ++i)
     {
         if (P[i + ] - P[i] > fabs(Rad[i + ] - Rad[i]))
         {
             Ls[i] = P[i] + Rad[i] * (Rad[i] - Rad[i + ]) / (P[i + ] - P[i]);
             Rs[i] = P[i + ] + Rad[i + ] * (Rad[i] - Rad[i + ]) / (P[i + ] - P[i]);
             Lh[i] = sqrt(Sqr(Rad[i]) - Sqr(Ls[i] - P[i]));
             Rh[i] = sqrt(Sqr(Rad[i + ]) - Sqr(Rs[i] - P[i + ]));
         }
         else
         {
             Ls[i] = -;
             Rs[i] = -;
         }
     }
     Ans = RSimpson(Lp, Rp, f(Lp), f((Lp + Rp) / 2.0), f(Rp)) * ;
     printf("%.2lf\n", Ans);
     return ;
 }