Show that applying Newton’s Method to f(x) = x2 − A produces the iteration of Example 1.6.
Calculate by using FPI.
An ancient method for determining square roots can be expressed as an FPI. Suppose we want to find the first 10 digits of . Start with the initial guess x0 = 1. This guess is obviously too low; therefore, 2/1 = 2 is too high. In fact, any initial guess 0 <>0 < 2,="" together="" with="">0, form a bracket for . Because of that, it is reasonable to average the two to get a better guess: