C语言实现定积分求解方法

求定积分的方法有很多种，下面是我总结的几种比较常用的方法。

 

#include <stdio.h>

#include <stdlib.h>

#include <math.h>

#include <time.h>



#define N		3



double fun(double x)

{

	double y;

	

	y = sqrt(4-(x)*(x));

	//y = sin(x);

	

	return y;

}



/*随机点法求定积分*/

double Darts(int n)

{

	double x, y;

	time_t t;

	int i = 0;

	int count = 0;

	

	srand((unsigned)time(&t));

	

	for (i=0; i<n; i++)

	{

		x = rand()%100/100.0;

		y = rand()%100/100.0;

		

		if (y <= 1-pow(x,2))

		{

			count++;	

		}	

	}

	

	return (double)count/(double)n;

}



/*左矩形法求定积分*/

double LeftRect(double down, double up, int n)

{

	double h, s;

	int i;

	

	/*计算步长*/

	h = (up-down)/n;

	

	s = fun(down)*h;

	

	for (i=1; i<n; i++)

	{

		s = s + fun(down+i*h)*h;	

	} 

	

	return s;

}



/*梯形公式求定积分*/

double Trape(double down, double up, int n)

{

	double h, s;

	int i = 0;

	

	/*计算步长*/

	h = (up-down)/n;

	

	s = 0.5*(fun(down)+fun(down+h))*h;

	

	for (i=1; i<n; i++)

	{

		s = s + 0.5 * (fun(down+i*h) + fun(down+(i+1)*h))*h;

	}

	

	return s;	

} 



/*复合梯形公式*/

double T(double x, double y, int z)

{

	double h, Tn;

	int i = 0;

	

	h = (y-x)/z;

	Tn = (fun(x)+fun(y))/2;

	

	for (i=0; i<z; i++)

	{

		Tn = Tn+fun(x+i*h);	

	}	

	

	Tn = Tn*h;

	

	return Tn;

} 



/*辛普生公式求定积分，公式为：S[n]=(4*T[2*n]-T[n])/3，其中T[2n],T[n]为梯形公式计算结果*/

double Simposn(double down, double up, int n)

{

	double s;

	

	/*辛普生公式*/

	s = (4*T(down, up, 2*n) - T(down, up, n))/3;

	

	return s;	

}



/*高斯公式求定积分*/

double Gass(double (*func)(double x), double a, double b, int n)

{

	int i = 0;

	

	//高斯点及其求积系数列表

	float x1[1]={0.0};                                                  

	float A1[1]={2};

	float x2[2]={-0.5573503,0.5573503};                                

	float A2[2]={1,1};

	float x3[3]={-0.7745967,0.0,0.7745967};                             

	float A3[3]={0.555556,0.888889,0.555556};

	float x4[4]={0.3399810,-0.3399810,0.8611363,-0.8611363};            

	float A4[4]={0.6521452,0.6521452,0.3478548,0.3478548};

	float x5[5]={0.0,0.5384693,-0.5384693,0.9061799,-0.9061799};       

	float A5[5]={0.5688889,0.4786287,0.4786287,0.2369269,0.2369269};   





	float *p, *t;

	switch (n)

	{

		case 1:          

			p = x1;

			t = A1;

			break;

		case 2:          

			p = x2;

			t = A2;

			break;

		case 3:          

			p = x3;

			t = A3;

			break;

		case 4:          

			p = x4;

			t = A4;

			break;

		case 5:          

			p = x5;

			t = A5;

			break;

		default :   

			printf("intput wrong!");

	}

	

	float g = 0;

	for (i=0; i<n; i++)

	{

		g += (*func)((b-a)*p[i]/2+(a+b)/2)*t[i];

	} 

	g *= (b-a)/2;

	

	return g;

}



int main(int argc, char *argv[])

{

	printf("随机点法积分值%f\n", Darts(10000));

	

	double down, up;

	int n;

	double sum = 0;

	

	printf("积分下限:\n");

	scanf("%lf", &down);

	printf("积分上限:\n");

	scanf("%lf", &up);

	printf("分隔数目:\n");

	scanf("%d", &n);

	

	sum = LeftRect(down, up, n);

	printf("左矩形法积分值为:%f\n", sum);

	

	sum = Trape(down, up, n);

	printf("梯形公式积分值为:%f\n", sum);

	

	sum = Simposn(down, up, n);

	printf("辛普生公式积分值为:%f\n", sum);

	

	sum = Gass(fun, down, up, N);

	printf("高斯公式积分值为:%f\n", sum);

	

	return 0;

}