Title: Estimating
the value of pi (π) using Monte Carlo Simulation.
Basic
Theory:
In this lab, we are
trying to find the value of pi(π) using Monte Carlo
Simulation. The Monte Carlo simulation deploys the concept of a large
collection of random data that satisfy some property to perform simulations in
diverse fields including computer science. In this case, we are applying it for
Mathematical analysis of estimating the value of pi(π).
Procedure:
For estimating the value
of pi(π),
we first specify a square area and plots random points inside it. We have used
13,186 points in this simulation (a lot of points!). It is still very simple
because we plot the points in the region bounded by 1 to 1 in both X and Y
axes. This can be described using a random function in Excel as,
X
= 12*RAND() and same for y
Now, we plot the points
which fall within the region satisfied by the equation of a circle. So, for the
circle the description of the Excel formula will be,
X
= IF((X^2+Y^2)<1,X,0), Y = IF((X^2+Y^2)<1,Y,0)
This formula shows that
the points will be plotted only if they satisfy x^{2}+y^{2}<1
i.e within the radius of the circle. Note that the circle is inside the square
bounded by 1 to 1 in both axes, so radius of the circle is also 1. Now we can
perform our calculation as,
Area of circle / Area of
Square = Points that satisfy circle equation / Total points plotted
π
r^2 / (r^2+ r^2+ r^2+ r^2)
= Random value / 13186
So, π
r^2/4r^2 = COUNTIF(X0:Xn,"<>0"
) / 13186
Here, we found this as π r^2/4
r^2 = 0.785312425 (from our random data)
π = 4*0.785312425 = 3.141249701
Hence, we found the value
of π.
Sample
Data:
Table1: Sample Data for Monte
Carlo Simulation for finding pi (π).
Rectangle

Circle


X

Y

X

Y

0.700356

0.32303

0.700356

0.32303

0.165

0.58146

0.165

0.58146

0.973976

0.76557

0

0

0.92966

0.373582

0

0

0.22543

0.51423

0.22543

0.51423

0.853136

0.11127

0.853136

0.11127

Output:
Below are the graph plots
for the points in the circle and square.
Conclusion:
Here, we could see that
the estimated value of π is very close to real
value. We can further increase the accuracy of the simulation by increasing the
number of random points. However, the excel worksheet will become too slow if
we generate a lot of random numbers because of large number of calculations.
No comments:
Post a Comment
Was this post helpful? Ask any questions you have, I will try to answer them for you.