The Babylonian method for approximating $$ is based on the idea that if we guess a value, say $,$ and $2>,$ then $$ will be an underestimate of $,$ and vice versa. Hence the

$$

average of these two numbers provide a better estimate. One can repeat this averaging to get a close approximation to $$ within some required accuracy. The algorithm is presented below:

x$=1$

while$(|2|>)$

$=1(+)2$

end return x

$(1)$ Write a MATLAB user$-$defined function $($named mysqrt$)$ that takes a single positive

real valued argument, say S$,$ and computes $$ using the Babylonian method described above. For $$ you can use eps$($single$(1)).$ Use $$help eps$$ to learn more about this function.

$(2)$ Write a MATLAB script file that uses a while loop to repeatedly prompt the user to enter a positive real value number, say S$,$ and

a$)$ if S is negative it outputs $$Invalid input!$$ and asks the user to re$-$enter the number,

b$)$ if S is positive it outputs:

The square root of S is x$.$

where x is mysqrt$($S$)$

c$)$ if S is $0$ it stops and outputs $$Goodbye$!.$