The Monte Carlo method is a statistical technique that

uses random sampling to obtain numerical results. In this

case, we will use it to estimate $$ by randomly generating

points in a square and determining how many of these

points fall within a quarter circle inscribed in the square.

Consider a square with sides of length $1,$ and a quarter

circle inscribed inside it as shown in the figure. All points

inside the square are made of two numbers $x$ and $y$

between $0$ and $1.$ If a set of $x,y$ coordinates fall within the

circle then,

$x^2+y^{2}1$

Eq $1$

If a large number of points are generated inside the square, the ratio points that fall inside the circle to

the total number of points will approximately equal the ratio of the area of the quarter circle to the area

of the square. Therefore,

${}_{\text{e}\text{s}\text{t}}=\frac{\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}of\text{p}\text{o}\text{i}\text{n}\text{t}\text{s}\text{i}\text{n}\text{s}\text{i}\text{d}\text{e}\text{c}\text{i}\text{r}\text{c}\text{l}\text{e}}{\text{t}\text{o}\text{t}\text{a}\text{l}\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}of\text{p}\text{o}\text{i}\text{n}\text{t}\text{s}}4$

error $=|{}_{\text{e}\text{s}\text{t}}-|$

Write a MATLAB script named HW$5$p$2\_$Task$1\_$UCusername.m that will prompt the user to input a

tolerance $($tol$).$ Your script will use a loop to generate random points $(x,y)$ inside the square, count

how many points fall inside the circle, and use Eq $2$ to compute ${}_{\text{e}\text{s}\text{t}}.$ The code should continue

generating more points until the error drops below the inputted tolerance then output the values of ${}_{\text{e}\text{s}\text{t}}$

to $6-$digits and the total number of points generated.

Sample Test Case:

Enter tol: $1$ e$-8$

pi$\_$est $=3\mathrm{.}141593$

$n=96933$

Notes:

MATLAB's in$-$built function rand $(1)$ will generate a random

number between $0$ and $1.$ Also, MATLAB's in$-$built function pi can

be used as the exact value of pi in Eq$.3$

This algorithm involves random numbers, therefore your output for $n$

will likely not match the sample test case. However, the value for

pi$\_$est will equal the Test case value.