$(8$ points$)$ The Babylonian method for approximating $\sqrt[\phantom{2}]{S}$ is based on the idea that if we guess

a value, say $x,$ and $x^2>S,$ then $\frac{S}{x}$ will be an underestimate of $\sqrt[\phantom{2}]{S},$ 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 $\sqrt[\phantom{2}]{S}$ within some required accuracy. The algorithm is presented below:

$x=1$

while $(|x^2-S|>$eps$)$

$q,x=\frac{1}{2}(x+\frac{S}{x})$

end

return $x$

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

real valued argument, say $S,$ and computes $\sqrt[\phantom{2}]{S}$ using the Babylonian method described

above. For eps 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 mysqrit$($S$)$

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