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Useful Computational Methods

Cube-roots via Newton-Raphson Method

According to this method, the cube root of a number a is obtained by starting with a guess x1 of the cube root and using the formula
x2 = (1/3)(2 x1 + a/x12).
Thus, to find the cube root of 5, we take x1 as 3/2. This guess is based on the reasoning that a value of 2 will be too high since the cube of 2 is 8 wheres as a value of 1 is too low since the cube of 1 is 1.
So x2 = (1/3)(2(3/2) + 5/(9/4)) = 1 + 20/27 = 47/27. We then let x1 = 47/27 and repeat the process, successively finding better and better approximations to the cube root of 5.

How does this method compare with the method of logarithms? The method of logarithms gives the result in a single step. However, that step entails having to find the value of the antilogarithm of a number. But a calculator or a printed list gives antilogarithms only to a few decimal digits. This means that the Newton-Raphson Method is better when we are looking for accuracy to as many decimal digits as we need.

Observe that the expression for x2 here is precisely that in the Babylonian Algorithm extended to cube roots. So The Babylonian Algorithm extended to cube roots is really the Newton-Raphson method for cube roots. From this equivalence of the two methods and from our discussion of convergence of the former algorithm, we find that the Newton-Raphson Method for cube roots is a quadratically convergent algorithm.

Clearly, the closer x0 is to the cube root sought, the faster you will reach values very close to the cube root. But you do not need to worry much about how close you are. Any initial value x0 > 0 will do in the method. This statement does not hold in finding the roots of polynomials with "critical" points. The polynomial x3 - N which is used in the Newton-Raphson method for cube roots is fortunately not one of them and hence the cube root approximation is convergent to the cube root. You will learn about critical points in calculus. The Newton-Raphson Method for general root-finding is also covered in calculus courses.




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Last updated - November 8, 2003