Java Reference
In-Depth Information
4.8.3 Case Study: Monte Carlo Simulation
Monte Carlo simulation uses random numbers and probability to solve problems. This
method has a wide range of applications in computational mathematics, physics, chemistry,
and finance. This section gives an example of using Monte Carlo simulation for estimating
To estimate
p
.
p
using the Monte Carlo method, draw a circle with its bounding square as
shown below.
y
1
x
-1
1
-1
Assume the radius of the circle is
1
. Therefore, the circle area is and the square area is
4
.
Randomly generate a point in the square. The probability for the point to fall in the circle is
circleArea / squareArea =
π
/4
.
Write a program that randomly generates 1,000,000 points in the square and let
numberOfHits
denote the number of points that fall in the circle. Thus,
numberOfHits
is
approximately
1000000 * (
π
/4)
.
π
can be approximated as
4 * numberOfHits /
1000000
. The complete program is shown in Listing 4.11.
p
L
ISTING
4.11
MonteCarloSimulation.java
1
public class
MonteCarloSimulation {
2
public static void
main(String[] args) {
3
final int
NUMBER_OF_TRIALS =
10000000
;
4
int
numberOfHits =
0
;
5
6
for
(
int
i =
0
; i < NUMBER_OF_TRIALS; i++) {
generate random points
7
8
double
x = Math.random() *
2.0
-
1
;
double
y = Math.random() *
2.0
-
1
;
check inside circle
9
if
(x * x + y * y <=
1
)
10
11 }
12
13
14 System.out.println(
"PI is "
+ pi);
15 }
16 }
numberOfHits++;
double
pi =
4.0
* numberOfHits / NUMBER_OF_TRIALS;
estimate pi
PI is 3.14124
The program repeatedly generates a random point (
x
,
y
) in the square in lines 7-8:
double
x = Math.random() *
2.0
-
1
;
double
y = Math.random() *
2.0
-
1
;
x
2
y
2
If the point is inside the circle and
numberOfHits
is incremented by
1
.
is approximately
4 * numberOfHits / NUMBER_OF_TRIALS
(line 13).
+
…
1,
p
