LA3485 Bridge[(辛普森自适应)微积分]

做此题完全是为了练积分。

【普通求导版】

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#include<cstdio>

#include<cmath>

using namespace std;

typedef double real;

const real eps=1e-5;

int cas,Cas,D,H,B,L;

real F(real a,real x){

	real a2=a*a;

	real x2=x*x;

	return (x*sqrt(a2+x2)+a2*log(fabs(x+sqrt(a2+x2))))/2.0;

}

real calc(real w,real h){

	real a=4.0*h/(w*w);

	real A=1.0/(2.0*a);

	return (F(A,w/2.0)-F(A,0))*4.0*a;

}

int main(){

	for(scanf("%d",&cas);cas--;){

		scanf("%d%d%d%d",&D,&H,&B,&L);

		int n=(B+D-1)/D;

		real D1=(real)B/n;

		real L1=(real)L/n;

		real l=0.0,r=H,mid;

		while(l+eps<r){

			mid=(l+r)/2.0;

			if(calc(D1,mid)<L1) l=mid;else r=mid;

		}

		printf("Case %d:\n%.2lf\n",++Cas,H-l);

		if(cas) putchar('\n');

	}

	return 0;

}

【辛普森自适应版】

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#include<cstdio>

#include<cmath>

using namespace std;

typedef double real;

const real eps=1e-5;

int cas,Cas,D,H,B,L;

real a;

real F(real x){

	real a2=a*a;

	real x2=x*x;

	return sqrt(1.0+4.0*a2*x2);

}

real simpson(real a,real b){

	real c=(a+b)/2.0;

	return (F(a)+4.0*F(c)+F(b))*(b-a)/6.0;

}

real asr(real a,real b,real eps,real A){

	real c=(a+b)/2.0;

	real L=simpson(a,c),R=simpson(c,b);

	if(fabs(L+R-A)<=15*eps) return L+R+(L+R-A)/15;

	return asr(a,c,eps/2,L)+asr(c,b,eps/2,R);

}

real asr(real a,real b,real eps){

	return asr(a,b,eps,simpson(a,b));

}

real calc(real w,real h){

	a=4.0*h/(w*w);

	return asr(0,w/2.0,eps)*2.0;

}

int main(){

	for(scanf("%d",&cas);cas--;){

		scanf("%d%d%d%d",&D,&H,&B,&L);

		int n=(B+D-1)/D;

		real D1=(real)B/n;

		real L1=(real)L/n;

		real l=0.0,r=H,mid;

		while(l+eps<r){

			mid=(l+r)/2.0;

			if(calc(D1,mid)<L1) l=mid;else r=mid;

		}

		printf("Case %d:\n%.2lf\n",++Cas,H-l);

		if(cas) putchar('\n');

	}

	return 0;

}