Old question about Newton-Raphson Method

[Anushka]

I know we kinda talked about this before but I came across the problem
again and am not sure how to solve it.

Newton-Raphson method is used to calculate the square root of 2 using
the iteration:
x( i+1 ) = ( x( i ) + 2 / x( i ) ) / 2
If the initial value x(0) is chosen as 1.5, What are the number of
iterations needed to get an accuracy of 10^-14?
(A) 2
(B) 4
(C) 8
(D) 16
(E) 32

Now I know this method converges quadratically but how do you solve
the above question?

I'd appreciate any help,
Anushka

[Joel Booth]

You need 14 decimals of accuracy, which is ceiling(lg 14) = 4.  In general, you need ceiling(lg d) iterations to get d decimal places of accuracy.  The number of correct decimal places doubles after each iteration (quadratic convergence).  

- Joel Booth

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