Cube Root Algorithm

Date: 04/04/97 at 12:37:41
From: Andrew Walters
Subject: Cube Root Algorithm

Hi, Dr. Math!

Is there an algorithm for working out the cube root of numbers without 
a calculator? My teacher said there was, but told me to research the 
answer.

Thanks for your help,

Andrew

Date: 04/04/97 at 15:48:27
From: Doctor Anthony
Subject: Re: Cube Root Algorithm

Cubic equations can be solved without recourse to approximate methods, 
but they can be very difficult and involve the use of complex numbers.  
Judging by your question, I would guess that you have not yet met 
complex numbers (they make use of sqrt(-1)= i).  For the time being 
you would need to stick to approximate methods.

If you had to solve x^3 = 86, you could proceed by Newton's Method as 
follows:

       f(x) = x^3 - 86 = 0
      f'(x) = 3x^2

Then we can use an iterative formula to improve on the accuracy 
of the root. If x0 is our first guess at the root, then a better 
approximation is given by:

   x1 = x0 - f(x0)/f'(x0)
   x1 = x0 - (x0^3-86)/(3x0^2)  

Now 4^3 = 64 and 5^3 = 125, so we know the root lies between 4 and 5, 
closer to 4.  So let us start with x = 4.3

   x1 = 4.3 - (4.3^3 - 86)/[3(4.3)^2] 
      = 4.417
   x2 = 4.414
   x3 = 4.414005   and  x4 = 4.414004962

If you use a calculator to work out cube root(86) you get the same 
value as x4, so the method gets the right answer quite quickly. 

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/