Problem $6.(2$x$20$ points$)$ USING MATLAB use Newton's method to compute sqrt$($R$)$ in double precision by

a$.x_{n+1}=\frac{1}{2}(x_n+\frac{R}{x_n})(20$ points$)$

b$.x_{n+1}=\frac{x_i(x_x^2+3R)}{3x_n^2+R}(20$ points$)$

For $R=0\mathrm{.}001,0\mathrm{.}1,10,$ and $1000$ with stopping criteria of

i$)$ Record the number of iterations required to reach the stopping criteria for each of the two

methods for each value of $R.$

ii$)$ Plot the values of $x_n$ for methods a$)$ and b$)$ in the same plot with separate plots for the different

$R$ values.

iii$)$ How sensitive are the iteration counts to your selected starting value?

iv$)$ Which method converges the fastest?

v$)$ Why?