The ancient Greeks knew that the iteration x_n+1 = 1/2 (x_n + 2/x_n) would conve

Question

The ancient Greeks knew that the iteration x_n+1 = 1/2 (x_n + 2/x_n) would converge to squareroot2 if x_0 = 1. Write a Python program that takes a starting value as input and determines it the sequence converges or not. The sequence converges if |x_n+1 - x_n| lessthanorequalto 10^-7. If it doesn't converge in 100 iterations, then we'll say the sequence doesn't converge. The definition statement of your program should be def squareroot2(x0) Your output statement should be of the form: The sequence starting at X0X0X0 converges to XXXX in NNNN iterations. Or The sequence starting at X0X0X0 doesn't converge. where X0X0X0 is the initial value x_0, XXXX is the value the sequence converges to and NNNN is the number of iterations it takes to converge.

Explanation / Answer

def sqrt2(x0):
   eps = 0.0000001
   ctr = 0
   while True:
       if ctr > 100:
           print "The sequence starting at ", x0, " doesnt converge"
           break
       else:
           ctr +=1
           x1 = 1/2 * (x0 + (2/x0))
           if abs(x1 - x0) <= eps:
               print "The sequence starting at ", x0, " converges to ", abs(x1 - x0), " in ", ctr, " iterations"
               break
           x1 = x0

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