蒙特卡洛(Monte Carlo)自动求函数积分的C++算法实现与测试

#include       //printf
#include        //用时间作为随机种子
#include      //rand和srand
#include        //数学函数

//宏定义
#define max(a, b) ((a) > (b) ? (a) : (b))
#define min(a, b) ((a) > (b) ? (b) : (a))

inline double random()
{
	return rand() % 32768 / 32767.0;
}

//Monte carro求给定函数的定积分
//极限：[-1, 1.414213562373095]
inline double f1(double x, double xmin, double xmax)
{
	if (x >= xmin && x <= xmax)
		return x * x * sqrt(1 + x * x * x) - 1;	
}

//极限：[0, 0.693147180559945]
inline double f2(double x, double xmin, double xmax)
{
	if (x >= xmin && x <= xmax)
		return x * log(1 + x);	
}

//极限：[0, 1]
inline double f3(double x, double xmin, double xmax)
{
	if (x >= xmin && x <= xmax)
		return sqrt(1 - x * x);
}

//极限：[-0.35969769413186028259906339255702, 0.1]
inline double f4(double x, double xmin, double xmax)
{
	if (x >= xmin && x <= xmax)
		return cos(x) - 0.9;
}

//定义函数指针类型
typedef double (*Func)(double x, double xmin, double xmax);

//定义结构体存储函数的上下限
typedef struct Limit {
	double lVal, hVal;
	//C++结构体构造函数
	Limit(double lv, double rv)
	{
		lVal = lv;
		hVal = rv;
	}
} Limit;

//取得函数的上下极限——其精确度影响很大---
//由于用于积分，故若y>0，底限0；若y<0，则高限0；若y穿越x轴，则不修改默认极限
Limit getLimit(Func func, double xmin, double xmax)
{	
	//边界极限
	double t1 = func(xmin, xmin, xmax), t2 = func(xmax, xmin, xmax);
	Limit limit(min(t1, t2), max(t1, t2));			
	double delta = 1e-3, lastX, lastY, y;
	int i = 0;
	for(double x = xmin; x <= xmax; x += delta, i++)
	{		
		y = func(x, xmin, xmax);			
		if (y < limit.lVal)
			limit.lVal = y;		
		if (y > limit.hVal)
			limit.hVal = y;
		if (i == 0)
		{
			lastX = x;
			lastY = y;
		}
		else
		{
			double tx = (lastX + x) / 2;
			double ty = func(tx, xmin, xmax);			
			if (fabs(lastY) > fabs(y))
			{
				if (fabs(lastY) < fabs(ty))
				{
					lastX = tx;
					lastY = ty;				
					if (ty < 0 && limit.lVal > ty)
						limit.lVal = ty;
					else if (ty >= 0 && limit.hVal <= ty)
						limit.hVal = ty;
				}				
			}			
			if (fabs(lastY) < fabs(y))
			{
				if (fabs(y) < fabs(ty))
				{
					lastX = tx;
					lastY = ty;				
					if (ty < 0 && limit.lVal > ty)
						limit.lVal = ty;
					else if (ty >= 0 && limit.hVal <= ty)
						limit.hVal = ty;
				}				
			}						
		}
	}
	if (limit.lVal > 0)
		limit.lVal = 0;
	if (limit.hVal < 0)
		limit.hVal = 0;
	return limit;
}

//积分函数——func：函数指针
double cumulate(Func func, int N, double xmin, double xmax)
{
	Limit limit = getLimit(func, xmin, xmax);
	double x, y, y0;
	int cntA = 0, cntB = 0;
	for(int i = 0; i < N; i++)
	{
		//随机在平面上投掷筛子，位置(x, y)
		x = xmin + (xmax - xmin) * random();
		y = limit.lVal + (limit.hVal - limit.lVal) * random();
		//调用函数取x点位置的标准y值
		y0 = func(x, xmin, xmax);
		//若y和yr都在x轴同侧
		if ( y0 >= 0 && y >= 0 && y <= y0)      //正积分计数
			cntA++;
		else if (y0 <= 0 && y <= 0 && y >= y0)  //负积分计数
			cntB++;
	}
	printf("Limit(%.15f, %.15f\n", limit.lVal, limit.hVal);
	return (cntA - cntB) * (xmax - xmin) * (limit.hVal - limit.lVal) / N;
}

int main()
{	
	double rst;
	//每个函数随机投筛子N次
	int N = 1e7;
	//函数指针数组
	Func fs[] = {f1, f2, f3, f4};
	//初始化随机种子
	srand (time(NULL));   
	//经Matlab对上述函数积分得到的结果
	double ans[] = {-0.593682861167513, 0.25, 3.141592653589793/4, -0.058529015192103493347497678369701};
	//分别调用4个函数
	for(int i = 0; i < 4; i++)
	{
		rst = cumulate(fs[i], N, 0, 1);
		printf("标准：%.15f, 程序结果：%.15f\n", ans[i], rst);
	}
	return 0;
}