Expert: Ahmed Salami - 1/23/2009

Question
I'm dont know how to apply the formula & i'm lack of examples from my teacher & textbooks so i'm stuck when doing this. Can you guide me? The iterative formula Xr+1= 0.5(Xr+N/(X^2 subscript r)) can be used to find an approximate cube root of a number N.Letting Xo=2,find 5^1/3 correct to 3 significant figures. Thanks a lot.

Answer
Hi Ling,
The iterative formula states,
X(r+1) = 0.5[Xr + N/(Xr)^2]
This means that for every of Xr corresponding to the rth iteration, we can use the formula to get the value X(r+1), the (r+1)th iteration, which gives us a better approximation to the cube root of the number N.
Now,
N = 5  (since we're finding the cube root of 5)
And so, the formula becomes
X(r+1) = 0.5[Xr + 5/(Xr)^2]
We have X0 = 2, we use it to find X1. We can then use X1 to find X2 and so on until we're satisfied with the answer i.e when we dont notice any significant change in the value of X after iterations, especially considering 3 significant figures.
Using the formula,
X1 = 0.5[X0 + 5/(X0)^2]
  = 0.5[2 + 5/(2)^2]
  = 0.5(2 + 5/4)
  = 0.5(2 + 1.25)
  = 0.5(3.25)
  = 1.625

X2 = 0.5[X1 + 5/(X1)^2]
  = 0.5[1.625 + 5/(1.625)^2]
  = 0.5(1.625 + 5/2.641)
  = 0.5(1.625 + 1.894)
  = 0.5(3.519)
  = 1.759

X3 = 0.5[X2 + 5/(X2)^2]
  = 0.5[1.759 + 5/(1.759)^2]
  = 0.5(1.759 + 5/3.095)
  = 0.5(1.759 + 1.616)
  = 0.5(3.375)
  = 1.687
and so on.

Regards