Use Newtons method to compute sqrt(R) in double precision by

1. x_n+1 = (x_n+R/x_n) (20 points)
2. x_n+1 = [x_n(x_n²+3R)]/[3x_n²+R] (20 points)

For R = 0.001, 0.1, 10, and 1000 with stopping criteria of 1) |x_n+1-x_n|< 10^-14 2) |x_n+1-x_n|< 10^-6.

1. Record the number of iterations required to reach the stopping criteria for each of the two methods for each value of R.
2. Plot the values of x_n for methods a) and b) in the same plot with separate plots for the different R values.
3. How sensitive are the iteration counts to your selected starting value?
4. Which method converge the fastest?
5. Why?