There are many methods and algorithms to calculate the square root of a number, A. One of the oldest methods is known as Babylonian method. You may learn digit-by-digit calculation method in HS algebra which is slower than Babylonian method but easy to calculate. There is a fast algorithm developed by using iterative methods. One of the methods to approximate √A is shown in the following algorism:

1.   Start with an initial guess x1,  x!  ≈ √A

2.   The new (better) approximation is given by

x! 2

y!  =

!

8

3.   Compare x1 and x2 and check the difference. error = |x2  − x! |

4.   If error is larger than the convergence criteria (truncation error), e, repeat step 1-3 till the criteria meets.

xn 2

yn  =

n

8

|xn+!  − xn | ≤ e

a)  Write a function, fn_sqrt that calculates xn based on eq. 3 & 4. Inputs should be any number A and guessed number xn and output should be xn+1

b)  Write a script to calculate √A till it meets the desired convergence criteria e. Ask user to       input any positive integer, A and truncation error, e. Script should check that input number A meets the criteria. Call the function fn_sqrt to compute √A . xn can start from 1. Your script forces to run the step 1-3 until |xn+1 - xn | becomes smaller than e. When it meets the criteria, then loop should stop. Display the answer up to the desired decimal place (depending on     truncation error, e). Use for-loop and if statement to cover all the requirements.