Simulation and Modeling: Simulation of Monte Carlo for determination of PI Using C:

Title:

Monte Carlo Simulation: Determination of π.

Objective:

To find the value of  π for different value of N using Monte Carlo simulation and find the difference between actual value of  and calculated value of π.

Theory:

To calculate value of pi we use a circle.

We know,

Area of circle= π r²

And, the eqⁿ of circle with centre (0,0) is x²+y²=r²

Now, assuming that we draw a circle with radius r in a square board with side 2r and we hit dart on it. Then from Monte Carlo simulation method:

No. of dart hitting on or inside circle ₌     Area of Circle.

No. of dart hitting n side square              Area of Square 

 n   ₌     π r²                                      

N         4r²

  π  ₌    4n

            N

Now, equation of circle,

            x²+y²=r²

Assuming it as a function,

            f(x,y)=x²+y²-r²

if r=1

            f(x,y)=x²+y²-1

If(x,y) lies inside or on the circle then f(x,y)<=0

If(x,y) lies inside or on the square then f(x,y)>0

Source Code:

#include<stdio.h>

#include<conio.h>

#include<stdlib.h>

#define pie 3.1415167

void main()

{

            clrscr();

            randomize();

            int i,N,n=0;

            double x,y,fxy,pi,err;

            printf("\n Enter the no. N:");

            scanf("%d",&N);

            for(i=0;i<N;i++)

            {

                        x=(double) rand()/RAND_MAX;

                        y=(double) rand()/RAND_MAX;

                        fxy=x*x + y*y-1;

                        if(fxy<=0.0)

                                    n++;

            }

            pi=(4*n)/(float)N;

            err=(pie-pi)/pie*100;

            printf("\nThe value of N: %d",N);

            printf(" \n The value of n: %d",n);

            printf("\n %f",pi);

            printf("\n Error is: %lf",err);

            getch();

}

Output:

In this way, the value of π is determined using Monte Carlo Simulation Method.